Michael Ulbrich

Semismooth Newton methods are a modern class of remarkably powerful and versatile algorithms for solving constrained optimization problems with partial differential equations (PDEs), variational inequalities, and related problems. This book provides a comprehensive presentation of these methods in function spaces, striking a balance between thoroughly developed theory and numerical applications. Although largely self-contained, the book also covers recent developments in the field, such as state-constrained problems and offers new material on topics such as improved mesh independence results. The theory and methods are applied to a range of practically important problems, including optimal control of semilinear elliptic differential equations, obstacle problems, and flow control of instationary Navier-Stokes fluids. In addition, the author covers adjoint-based derivative computation and the efficient solution of Newton systems by multigrid and preconditioned iterative methods. Audience: This book is appropriate for researchers and practitioners in PDE-constrained optimization, nonlinear optimization, and numerical analysis, as well as engineers interested in the current theory and methods for solving variational inequalities. It is also suitable as a text for an advanced graduate-level course in the aforementioned topics or applied functional analysis. Contents: Notation; Preface; Chapter One: Introduction; Chapter Two: Elements of Finite-Dimensional Nonsmooth Analysis; Chapter Three: Newton Methods for Semismooth Operator Equations; Chapter Four: Smoothing Steps and Regularity Conditions; Chapter Five: Variational Inequalities and Mixed Problems; Chapter Six: Mesh Independence; Chapter Seven: Trust-Region Globalization; Chapter Eight: State-Constrained and Related Problems; Chapter Nine: Several Applications; Chapter Ten: Optimal Control of Incompressible Navier-Stokes Flow; Chapter Eleven: Optimal Control of Compressible Navier-Stokes Flow; Appendix; Bibliography; Index.

We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes nonlinear complementarity problem (NCP)-function-based reformulations of infinite-dimensional nonlinear complementarity problems and thus covers a very comprehensive class of applications. Our results generalize semismoothness and $\alpha$-order semismoothness from finite-dimensional spaces to a Banach space setting. For this purpose, a new infinite-dimensional generalized differential is used that is motivated by Qi's finite-dimensional C-subdifferential [Research Report AMR96/5, School of Mathematics, University of New South Wales, Australia, 1996]. We apply these semismoothness results to develop a Newton-like method for nonsmooth operator equations and prove its local q-superlinear convergence to regular solutions. If the underlying operator is $\alpha$-order semismooth, convergence of q-order $1+\alpha$ is proved. We also establish the semismoothness of composite operators and develop corresponding chain rules. The developed theory is accompanied by illustrative examples and by applications to NCPs and a constrained optimal control problem.

In this paper, the filter technique of Fletcher and Leyffer (1997) is used to globalize the primal-dual interior-point algorithm for nonlinear programming, avoiding the use of merit functions and the updating of penalty parameters. The new algorithm decomposes the primal-dual step obtained from the perturbed first-order necessary conditions into a normal and a tangential step, whose sizes are controlled by a trust-region type parameter. Each entry in the filter is a pair of coordinates: one resulting from feasibility and centrality, and associated with the normal step; the other resulting from optimality (complementarity and duality), and related with the tangential step. Global convergence to first-order critical points is proved for the new primal-dual interior-point filter algorithm.

Catheter ablation is an established therapy in patients with symptomatic atrial fibrillation (AF) with increasing popularity. Pericardial effusion requiring intervention (PE) is one of the most threatening adverse outcomes. The aim of this study was to examine rates of PE after catheter ablation in a large ‘real-world’ data set in a German-wide hospital network. Using ICD and OPS codes, administrative data of 85 Helios hospitals from 2010 to 2017 was used to identify AF catheter ablation cases [Helios atrial fibrillation ablation registry (SAFER)]. PE occurred in 0.9% of 21 141 catheter ablation procedures. Patients with PE were significantly older, to a higher percentage female, had more frequently hypertension, mild liver disease, diabetes with chronic complications, and renal disease. Low hospital volume (<50 procedures per year) and radiofrequency ablation (vs. cryoablation) were significantly associated with PE. Using two logistic regression models, age, female gender, hypertension, mild liver disease, diabetes with chronic complications, renal disease, low hospital volume, and radiofrequency ablation remained independent predictors for PE. Overall PE rate was 0.9%. Predictors for PE occurrence involved factors ascribed to the patient (age, gender, comorbidities), the type of catheter ablation (radiofrequency), and the institution (low-volume centres).

The high incidence of sudden cardiac death in heart failure (HF) reflects electrophysiologic changes in response to myocardial failure. We previously showed that short-term variability of QT intervals (STV(QT)) identifies latent repolarization disorders in patients with drug-induced or congenital long QT syndrome. This study sought to determine (1) if STV(QT) is increased in patients with dilated cardiomyopathy (DC) and moderate congestive HF and (2) if increased STV(QT) is associated with ventricular arrhythmia in patients with HF. Sixty patients (53 +/- 12 years of age, 14 women) with DC and moderate HF (New York Heart Association classes II to III) were compared to matched controls. Twenty patients had implantable cardiac defibrillators secondary to a history of ventricular tachycardia (VT). Two cardiologists blinded to diagnosis manually measured QT intervals. Beat-to-beat variability of repolarization was determined from Poincaré plots of 30 consecutive QT intervals as was STV(QT). QTc intervals were comparable in patients and controls (419 +/- 36 vs 415 +/- 32 ms, respectively, p >0.05), whereas STV(QT) was significantly higher in patients with HF (7.8 +/- 3 vs 4.1 +/- 2 ms, respectively, p <0.05). STV(QT) was more increased in patients with a history of VT compared to those without VT (10.1 +/- 2 vs 6.6 +/- 2 ms, respectively, p <0.05). Increased STV(QT) and decreased ejection fraction were associated with a history of VT; however, STV(QT) was the strongest indicator. In conclusion, the present study demonstrates for the first time that STV(QT) is increased in patients with DC with HF. Patients with DC and HF and implantable cardiac defibrillators for secondary prevention had the highest STV(QT). Thus, increased STV(QT) in the context of moderate HF may reflect a latent repolarization disorder and increased susceptibility to sudden death in patients with DC, which is not identified by a prolonged QT interval.

We consider optimal control-based boundary feedback stabilization of flow problems for incompressible fluids. We follow an analytical approach laid out during the last years in a series of papers by Barbu, Lasiecka, Triggiani, Raymond, and others. They have shown that it is possible to stabilize perturbed flows described by Navier-Stokes equations by designing a stabilizing controller based on a corresponding linear-quadratic optimal control problem. For this purpose, algorithmic advances in solving the associated algebraic Riccati equations are needed and investigated here. The computational complexity of the new algorithms is essentially proportional to the simulation of the forward problem.

Due to their property of enhancing the sparsity of solutions, $l_1$-regularized optimization problems have developed into a highly dynamic research area with a wide range of applications. We present a class of methods for $l_1$-regularized optimization problems that are based on a combination of semismooth Newton steps, a filter globalization, and shrinkage/thresholding steps. A multidimensional filter framework is used to control the acceptance and to evaluate the quality of the semismooth Newton steps. If the current Newton iterate is rejected a shrinkage/thresholding-based step with quasi-Armijo stepsize rule is used instead. Global convergence and transition to local q-superlinear convergence for both convex and nonconvex objective functions are established. We present numerical results and comparisons with several state-of-the-art methods that show the efficiency and competitiveness of the proposed method.

We develop and analyze a class of trust-region methods for bound-constrained semismooth systems of equations. The algorithm is based on a simply constrained differentiable minimization reformulation. Our global convergence results are developed in a very general setting that allows for nonmonotonicity of the function values at subsequent iterates. We propose a way of computing trial steps by a semismooth Newton-like method that is augmented by a projection onto the feasible set. Under a Dennis--Moré-type condition we prove that close to a regular solution the trust-region algorithm turns into this projected Newton method, which is shown to converge locally q-superlinearly or quadratically, respectively, depending on the quality of the approximate subdifferentials used. As an important application we discuss how the developed algorithm can be used to solve nonlinear mixed complementarity problems (MCPs). Hereby, the MCP is converted into a bound-constrained semismooth equation by means of an NCP-function. The efficiency of our algorithm is documented by numerical results for a subset of the MCPLIB problem collection.

We present a new relaxation scheme for mathematical programs with equilibrium constraints (MPEC), where the complementarity constraints are replaced by a reformulation that is exact for the complementarity conditions corresponding to sufficiently nondegenerate complementarity components and relaxes only the remaining complementarity conditions. A positive parameter determines to what extent the complementarity conditions are relaxed. The relaxation scheme is such that a strongly stationary solution of the MPEC is also a solution of the relaxed problem if the relaxation parameter is chosen sufficiently small. We discuss the properties of the resulting parameterized nonlinear programs and compare stationary points and solutions. We further prove that a limit point of a sequence of stationary points of a sequence of relaxed problems is Clarke-stationary if it satisfies a so-called MPEC-constant rank constraint qualification, and it is Mordukhovich-stationary if it satisfies the MPEC-linear independence constraint qualification and the stationary points satisfy a second order sufficient condition. From this relaxation scheme, a numerical approach is derived that is applied to a comprehensive test set. The numerical results show that the approach combines good efficiency with high robustness.

We present an end-to-end framework which equips robots with the capability to perform reaching motions in a natural human-like fashion. A markerless, high-accuracy, model- based human motion tracker is used to observe how humans perform everyday activities in real-world scenarios. The obtained trajectories are clustered to represent different types of manipulation and reaching motions occurring in a kitchen environment. Using bilevel optimization methods a combination of physically inspired optimization principles is determined that describes the human motions best. For humanoid robots like the iCub these principles are used to compute reaching motion trajectories which are similar to human behavior and respect the individual requirements of the robotic hardware.

In this paper, we study a few theoretical issues in the discretized Kohn--Sham (KS) density functional theory. The equivalence between either a local or global minimizer of the KS total energy minimization problem and the solution to the KS equation is established under certain assumptions. The nonzero charge densities of a strong local minimizer are shown to be bounded from below by a positive constant uniformly. We analyze the self-consistent field (SCF) iteration by formulating the KS equation as a fixed point map with respect to the potential. The Jacobian of these fixed point maps is derived explicitly. Both global and local convergence of the simple mixing scheme can be established if the gap between the occupied states and unoccupied states is sufficiently large. This assumption can be relaxed in certain cases. Numerical experiments based on the MATLAB toolbox KSSOLV show that it holds on a few simple examples. Although our assumption on the gap is very stringent, our analysis is still valuable for a better understanding of the KS minimization problem, the KS equation, and the SCF iteration.

The prevalence of uncertainty in models of engineering and the natural sciences necessitates the inclusion of random parameters in the underlying partial differential equations (PDEs). The resulting decision problems governed by the solution of such random PDEs are infinite dimensional stochastic optimization problems. In order to obtain risk-averse optimal decisions in the face of such uncertainty, it is common to employ risk measures in the objective function. This leads to risk-averse PDE-constrained optimization problems. We propose a method for solving such problems in which the risk measures are convex combinations of the mean and conditional value-at-risk (CVaR). Since these risk measures can be evaluated by solving a related inequality-constrained optimization problem, we suggest a log-barrier technique to approximate the risk measure. This leads to a new continuously differentiable convex risk measure: the log-barrier risk measure. We show that the log-barrier risk measure fits into the setting of optimized certainty equivalents of Ben-Tal and Teboulle and the expectation quadrangle of Rockafellar and Uryasev. Using the differentiability of the log-barrier risk measure, we derive first-order optimality conditions reminiscent of classical primal and primal-dual interior-point approaches in nonlinear programming. We derive the associated Newton system, propose a reduced symmetric system to calculate the steps, and provide a sufficient condition for local superlinear convergence in the continuous setting. Furthermore, we provide a $\Gamma$-convergence result for the log-barrier risk measures to prove convergence of the minimizers to the original nonsmooth problem. The results are illustrated by a numerical study.

Using a bilevel optimization approach, we investigate the question how humansplan and execute their arm motions.It is known that human motions are (approximately) optimal for suitable andunknown cost functions subject to the dynamics. We investigate the followinginverse problem: Which cost function out of a parameterized family (e.g.,convex combinations of functions suggested in the literature) reproducesrecorded human arm movements best? The lower level problem is an optimalcontrol problem governed by a nonlinear model of the human arm dynamics.The approach is analyzed for a dynamical 3D model of the human arm.Furthermore, results for a two-dimensional experiment with human probandsare presented.

Seismic tomography is a technique to determine the material properties of the Earth's subsurface based on the observation of seismograms. This can be stated as a PDE-constrained optimization problem governed by the elastic wave equation. We present a semismooth Newton-PCG method with a trust-region globalization for full-waveform seismic inversion that uses a Moreau--Yosida regularization to handle additional constraints on the material parameters. We establish results on the differentiability of the parameter-to-state operator and analyze the proposed optimization method in a function space setting. The elastic wave equation is discretized by a high-order continuous Galerkin method in space and an explicit Newmark time-stepping scheme. The matrix-free implementation relies on the adjoint-based computation of the gradient and Hessian-vector products and on an MPI-based parallelization. Numerical results are shown for an application in geophysical exploration at reservoir scale.

Low-rank tensor methods provide efficient representations and computations for high-dimensional problems and are able to break the curse of dimensionality when dealing with systems involving multiple parameters. Motivated by optimal control problems with PDEs under uncertainty, we present algorithms for constrained nonlinear optimization problems that use low-rank tensors. They are applied to optimal control of PDEs with uncertain parameters and to parametrized variational inequalities of obstacle type. These methods are tailored to the usage of low-rank tensor arithmetics and allow us to solve huge scale optimization problems. In particular, we consider a semismooth Newton method for an optimal control problem with pointwise control constraints and an interior point algorithm for an obstacle problem, both with uncertainties in the coefficients.

In this work, we present a globalized stochastic semismooth Newton method for solving stochastic optimization problems involving smooth nonconvex and nonsmooth convex terms in the objective function. We assume that only noisy gradient and Hessian information of the smooth part of the objective function is available via calling stochastic first and second order oracles. The proposed method can be seen as a hybrid approach combining stochastic semismooth Newton steps and stochastic proximal gradient steps. Two inexact growth conditions are incorporated to monitor the convergence and the acceptance of the semismooth Newton steps and it is shown that the algorithm converges globally to stationary points in expectation and almost surely. We present numerical results and comparisons on l1-regularized logistic regression and nonconvex binary classification that demonstrate the efficiency of the algorithm.

A class of interior-point trust-region algorithms for infinite-dimensional nonlinear optimization subject to pointwise bounds in L p-Banach spaces, $2\le p\le\infty$, is formulated and analyzed. The problem formulation is motivated by optimal control problems with L p-controls and pointwise control constraints. The interior-point trust-region algorithms are generalizations of those recently introduced by Coleman and Li [SIAM J. Optim., 6 (1996), pp. 418--445] for finite-dimensional problems. Many of the generalizations derived in this paper are also important in the finite-dimensional context. All first- and second-order global convergence results known for trust-region methods in the finite-dimensional setting are extended to the infinite-dimensional framework of this paper.

In this paper we present an approach to shape optimization which is based on continuous adjoint computations. If the exact discrete adjoint equation is used, the resulting formula yields the exact discrete reduced gradient. We first introduce the adjoint-based shape derivative computation in a Banach space setting. This method is then applied to the instationary Navier-Stokes equations. Finally, we give some numerical results.

In this article we consider shape optimization problems as optimal control problems via the method of mappings. Instead of optimizing over a set of admissible shapes, a reference domain is introduced, and it is optimized over a set of admissible transformations. The focus is on the choice of the set of transformations, which we motivate from a continuous perspective. In order to guarantee local injectivity of the admissible transformations, we enrich the optimization problem by a nonlinear constraint. Numerical results for drag minimization of Stokes flow are presented.

We develop and analyze a superlinearly convergent affine-scaling interior-point Newton method for infinite-dimensional problems with pointwise bounds in Lp-space. The problem formulation is motivated by optimal control problems with Lp-controls and pointwise control constraints. The finite-dimensional convergence theory by Coleman and Li [SIAM J. Optim., 6 (1996), pp. 418--445] makes essential use of the equivalence of norms and the exact identifiability of the active constraints close to an optimizer with strict complementarity. Since these features are not available in our infinite-dimensional framework, algorithmic changes are necessary to ensure fast local convergence. The main building block is a Newton-like iteration for an affine-scaling formulation of the KKT-condition. We demonstrate in an example that a stepsize rule to obtain an interior iterate may require very small stepsizes even arbitrarily close to a nondegenerate solution. Using a pointwise projection instead we prove superlinear convergence under a weak strict complementarity condition and convergence with \mbox{Q-rate $>$1} under a slightly stronger condition if a smoothing step is available. We discuss how the algorithm can be embedded in the class of globally convergent trust-region interior-point methods recently developed by M. Heinkenschloss and the authors. Numerical results for the control of a heating process confirm our theoretical findings.

We propose and analyze a semismooth Newton-type method for the solution of a pointwise constrained optimal control problem governed by the time-dependent incompressible Navier–Stokes equations. The method is based on a reformulation of the optimality system as an equivalent nonsmooth operator equation. We analyze the flow control problem and prove q-superlinear convergence of the method. In the numerical implementation, adjoint techniques are combined with a truncated conjugate gradient method. Numerical results are presented that support our theoretical results and confirm the viability of the approach.

Abstract The control of complex, unsteady flows is a pacing technology for advances in fluid mechanics. Recently, optimal control theory has become popular as a means of predicting best case controls that can guide the design of practical flow control systems. However, most of the prior work in this area has focused on incompressible flow which precludes many of the important physical flow phenomena that must be controlled in practice including the coupling of fluid dynamics, acoustics, and heat transfer. This paper presents the formulation and numerical solution of a class of optimal boundary control problems governed by the unsteady two‐dimensional compressible Navier–Stokes equations. Fundamental issues including the choice of the control space and the associated regularization term in the objective function, as well as issues in the gradient computation via the adjoint equation method are discussed. Numerical results are presented for a model problem consisting of two counter‐rotating viscous vortices above an infinite wall which, due to the self‐induced velocity field, propagate downward and interact with the wall. The wall boundary control is the temporal and spatial distribution of wall‐normal velocity. Optimal controls for objective functions that target kinetic energy, heat transfer, and wall shear stress are presented along with the influence of control regularization for each case. Copyright © 2002 John Wiley &amp; Sons, Ltd.

The present study aimed to distinguish responders to cardiac resynchronization therapy (CRT) from nonresponders, using electrocardiogram-gated 18F-FDG PET/CT.Seven consecutive CRT nonresponders were included in the study, along with 7 age- and sex-matched CRT responders, serving as reference material. Therapy response was defined as clinical improvement (≥1 New York Heart Association class) and evidence of reverse remodeling. Besides PET/CT, we measured brain natriuretic peptide levels and assessed dyssynchrony using transthoracic echocardiography.Compared with nonresponders, CRT responders showed significant differences in the declines of left-ventricular end-systolic volume and brain natriuretic peptide and in left-ventricular dyssynchrony (global left-ventricular entropy), extent of the myocardial scar burden, and biventricular pacemaker leads positioned within viable myocardial regions. Among the nonresponders, further therapy management was guided by the PET/CT results in 4 of 7 patients.Cardiac hybrid imaging using gated 18F-FDG PET/CT enabled the identification of potential reasons for nonresponse to CRT therapy, which can guide subsequent therapy.

The ensemble density functional theory (E-DFT) is valuable for simulations of metallic systems due to the absence of a gap in the spectrum of the Hamiltonian matrices. Although the widely used self-consistent field (SCF) iteration method can be extended to solve the minimization of the total energy functional with respect to orthogonality constraints, there is no theoretical guarantee on the convergence of these algorithms. In this paper, we consider an equivalent model with a single variable and a single spherical constraint by eliminating the dependence on the fractional occupancies. A proximal gradient method is developed by keeping the entropy term but linearizing all other terms in the total energy functional. Convergence to the stationary point is established. Numerical results using the MATLAB toolbox KSSOLV can outperform SCF consistently on many metallic systems.

The self-consistent field (SCF) iteration has been used ubiquitously for solving the Kohn--Sham (KS) equation or the minimization of the KS total energy functional with respect to orthogonality constraints in electronic structure calculations. Although SCF with heuristics such as charge mixing often works remarkably well on many problems, it is well known that its convergence can be unpredictable and there is no general theoretical analysis on its performance. We regularize the SCF iteration and establish rigorous global convergence to the first-order optimality conditions. The Hessian of the total energy functional is further exploited. By adding the part of the Hessian which is not considered in SCF, our methods can always achieve a highly accurate solution on problems for which SCF fails and exhibit a better convergence rate than SCF in the KSSOLV toolbox under the MATLAB environment.

Abstract Background The prospective WEARIT-II-EUROPE registry aimed to assess the value of the wearable cardioverter-defibrillator (WCD) prior to potential ICD implantation in patients with heart failure and reduced ejection fraction considered at risk of sudden arrhythmic death. Methods and results 781 patients (77% men; mean age 59.3 ± 13.4 years) with heart failure and reduced left ventricular ejection fraction (LVEF) were consecutively enrolled. All patients received a WCD. Follow-up time for all patients was 12 months. Mean baseline LVEF was 26.9%. Mean WCD wearing time was 75 ± 47.7 days, mean daily WCD use 20.3 ± 4.6 h. WCD shocks terminated 13 VT/VF events in ten patients (1.3%). Two patients died during WCD prescription of non-arrhythmic cause. Mean LVEF increased from 26.9 to 36.3% at the end of WCD prescription ( p &lt; 0.01). After WCD use, ICDs were implanted in only 289 patients (37%). Forty patients (5.1%) died during follow-up. Five patients (1.7%) died with ICDs implanted, 33 patients (7%) had no ICD (no information on ICD in two patients). The majority of patients (75%) with the follow-up of 12 months after WCD prescription died from heart failure (15 patients) and non-cardiac death (15 patients). Only three patients (7%) died suddenly. In seven patients, the cause of death remained unknown. Conclusions Mortality after WCD prescription was mainly driven by heart failure and non-cardiovascular death. In patients with HFrEF and a potential risk of sudden arrhythmic death, WCD protected observation of LVEF progression and appraisal of competing risks of potential non-arrhythmic death may enable improved selection for beneficial ICD implantation. Graphic abstract

205 cases of clinical poisoning with the mushroom Amanita phalloides (death cap) in the period 1971 to 1980 have been studied retrospectively. The lethality was 22.4%. Age and latency between the ingestion of the mushrooms and the first clinical symptoms were of prognostic significance. The death rate was 51.3% in children below 10 years of age but only 16.5% in patients older than 10 years. The average latency period was 10.3 hours for the fatal cases and 12.6 hours for the surviving patients. Country, year, sex and time of hospitalization did not influence lethality. Prognostic relevance could also be attributed to the thromboplastin time (Quick). 84% of the patients with values below 10% died, while all patients with minimal values of more than 40% survived. The correlation with the outcome was weaker for the serum transaminases and nil for creatinine. The patients underwent on the average 8 therapeutic measures, but up to 20 therapies could be administered to the same patient. Eight of the 30 recorded treatments involved general support, 7 toxin elimination and the remaining 14 could be classified as pharmacotherapy. With the aid of multiple regression analysis taking into account age, latency period and the effects of all the other measures, penicillin and hyperbaric oxygenation were found to contribute independently to a higher survival rate. As compared to penicillin, the combination of penicillin with silybin was associated with still further increased survival. On the other hand, several measures, including exchange transfusion, thiocytic acid, sulfamethoxazole, plasma expanders, haemodialysis, treatment of the hemorrhagic diathesis and THAM/sodium bicarbonate were administered more often to patients who did not survive. For the remaining 20 therapeutic measures our analysis revealed neither a positive nor a negative correlation with the clinical outcome.

Abstract Shape optimization via the method of mappings is investigated for unsteady fluid-structure interaction (FSI) problems that couple the Navier–Stokes equations and the Lamé system. Building on recent existence and regularity theory we prove Fréchet differentiability results for the state with respect to domain variations. These results form an analytical foundation for optimization und inverse problems governed by FSI systems. Our analysis develops a general framework for deriving local-in-time continuity and differentiability results for parameter dependent nonlinear systems of partial differential equations. The main part of the paper is devoted to conducting this analysis for the FSI problem, transformed to a shape reference domain. The underlying shape transformation—actually we work with the corresponding shape displacement instead—represents the shape and the main result proves the Fréchet differentiability of the solution of the FSI system with respect to the shape transformation.

.The sample average approximation (SAA) approach is applied to risk-neutral optimization problems governed by semilinear elliptic partial differential equations with random inputs. After constructing a compact set that contains the SAA critical points, we derive nonasymptotic sample size estimates for SAA critical points using the covering number approach. Thereby, we derive upper bounds on the number of samples needed to obtain accurate critical points of the risk-neutral PDE-constrained optimization problem through SAA critical points. We quantify accuracy using expectation and exponential tail bounds. Numerical illustrations are presented.Keywordsstochastic optimizationPDE-constrained optimization under uncertaintysample average approximationMonte Carlo samplingsample complexityuncertainty quantificationMSC codes90C1590C3090C6049J2049J5549K4549K2035J61

We present an analysis of generalized Nash equilibrium problems in infinite-dimensional spaces with possibly non-convex objective functions of the players. Such settings arise, for instance, in games that involve nonlinear partial differential equation constraints. Due to non-convexity, we work with equilibrium concepts that build on first order optimality conditions, especially Quasi-Nash Equilibria (QNE), i.e. first-order optimality conditions for (Generalized) Nash Equilibria, and Variational Equilibria (VE), i.e. first-order optimality conditions for Normalized Nash Equilibria. We prove existence of these types of equilibria and study characterizations of them via regularized (and localized) Nikaido-Isoda merit functions. We also develop continuity and (continuous) differentiability results for these merit functions under quite weak assumptions, using a generalization of Danskin's theorem. They provide a theoretical foundation for, e.g. using globalized descent methods for computing QNE or VE.

We develop an implementable stochastic proximal point (SPP) method for a class of weakly convex, composite optimization problems.The proposed stochastic proximal point algorithm incorporates a variance reduction mechanism and the resulting SPP updates are solved using an inexact semismooth Newton framework.We establish detailed convergence results that take the inexactness of the SPP steps into account and that are in accordance with existing convergence guarantees of (proximal) stochastic variance-reduced gradient methods.Numerical experiments show that the proposed algorithm competes favorably with other state-of-the-art methods and achieves higher robustness with respect to the step size selection.

We consider the method of mappings for performing shape optimization for unsteady fluid-structure interaction (FSI) problems. In this work, we focus on the numerical implementation. We model the optimization problem such that it takes several theoretical results into account, such as regularity requirements on the transformations and a differential geometrical point of view on the manifold of shapes. Moreover, we discretize the problem such that we can compute exact discrete gradients. This allows for the use of general purpose optimization solvers. We focus on an FSI benchmark problem to validate our numerical implementation. The method is used to optimize parts of the outer boundary and the interface. The numerical simulations build on FEniCS, dolfin-adjoint and IPOPT. Moreover, as an additional theoretical result, we show that for a linear special case the adjoint attains the same structure as the forward problem but reverses the temporal flow of information.

Introduction: The purpose of this study was to examine the reentrant circuit of ventricular tachycardias (VTs) involving the left anterior fascicle (LAF) in nonischemic cardiomyopathy. Methods and Results: Six patients with nonischemic cardiomyopathy presented with VTs involving the LAF. Potentials in the diastolic or presystolic phase of the VT were identified close to the LAF in 3 patients and in the mid or inferior left ventricular (LV) septum in 3 patients. Superimposed on a CARTO or NavX 3‐dimensional voltage map, the diastolic and presystolic potentials were recorded within or at the border of a low‐voltage zone in the LV septum in all cases. In 2 patients, both left bundle fascicles participated in the reentrant circuit including a possible interfascicular VT in one case. Ablation targeting the diastolic or presystolic potentials near the LAF or in the midinferior LV septum eliminated the VTs in all patients with the occurrence of a left posterior fascicular block and the delayed occurrence of a complete atrioventricular block in each one patient. During the follow‐up of 23 ± 20 months after ablation, 4 patients were free of ventricular tachyarrhythmias. Due to detoriation of heart failure, one patient died after 12 months and one patient underwent heart transplantation after 40 months. Conclusions: Slow conduction in diseased myocardium close to the LAF or in the middle and inferior aspects of the LV septum may represent the diastolic pathway of VT involving the LAF.

The walking paths of humans in everyday life exhibit specific characteristics. Our investigation takes the well-established perspective that human locomotion approximately fulfills suitable optimality principles when walking from a starting position to a designated end position. Here, we address the special task of collision avoidance with a crossing interferer. While our model of the dynamics is quite simple, we focus on the task of determining the cost function out of a parametric family, that results in a best fit between the corresponding optimal control-based navigation and given recorded data of human navigation. The resulting bilevel optimization problem combines an optimal control problem on the lower level with a data fitting problem on the upper level. Our solution strategy uses the first-order optimality conditions of the (discretized) optimal control problem to transform the bilevel problem to a standard (one-level) optimization problem. Modeling aspects concerning the walking process and especially the interfering person are discussed. Since human walking motions with a crossing interferer do not seem to be globally optimal, an MPC-like approach distinguishing between obstacle avoidance and free motion is introduced and optimization results using recorded human data are presented.

Distributed control of large-scale dynamical systems poses a new challenge to the field of control driven by the technological advances of modern communication networks. A particular challenge is the distributed design of such control systems. Here, a distributed iterative controller synthesis method for continuous time linear systems using a gradient descent method is presented. One of the main contributions is the determination of the step size according to a distributed Barzilai-Borwein (BB) method. As the control objective, we treat the finite horizon linear quadratic cost functional. The gradient approach uses communication only with direct neighbors and is based on the forward simulation of the system states and the backwards simulation of adjoint states. The effectiveness of the approach is shown by means of numerical simulations.

Background: We describe immediate reinitiation of macroreentry ventricular tachycardia (VT) involving the His‐Purkinje system by ventricular pacing from the electrode of an implantable cardioverter defibrillator (ICD) as a mechanism of VT storm refractory to ICD therapy. Methods and Results: Repetitive reinitiation of bundle branch reentry tachycardia (BBRT), interfascicular tachycardia, or both VTs by ventricular pacing was identified in four ICD patients presenting with VT storm or incessant VT. All patients had a pre‐existing prolonged HV interval (75 ± 9 ms) and left bundle branch block (LBBB) or bifascicular block during sinus rhythm. The VTs included BBRT with LBBB in three patients and interfascicular tachycardia with right bundle branch block (RBBB) and left anterior or left posterior fascicular block in two patients. The paced beats from the ICD electrode exhibited a LBBB pattern of depolarization in two patients and a RBBB contour in V1 and V2 with left axis deviation in two patients. The QRS complex during pacing from the ICD electrode closely resembled that of the recurrent VT in all four patients suggesting that the pacing site of the ICD electrode was in proximity to the myocardial exit site of the bundle fascicle used for antegrade conduction during the reinitiated VT. Ventricular pacing from the ICD electrode after termination of the VT apparently encountered the retrograde refractoriness of this bundle fascicle and allowed immediate re‐propagation of the wavefront orthodromically along the VT circuit. BBRT was eliminated by ablation of the right bundle branch. Successful ablation of the interfascicular tachycardias was achieved by targeting (1) an abnormal potential of the distal left posterior Purkinje network or (2) a diastolic potential during VT in the midinferior left ventricular (LV) septum. Conclusions: Repetitive reinitiation of BBRT and interfascicular tachycardia by ventricular pacing from the ICD electrode should be considered as a mechanism of VT storm refractory to ICD therapy in patients with a pre‐existing conduction delay within the His‐Purkinje system.

We present an approach to shape optimization which is based on transformation to a reference domain with continuous adjoint computations. This method is applied to the instationary Navier-Stokes equations for which we discuss the appropriate setting and discuss Fréchet differentiability of the velocity field with respect to domain transformations. Goal-oriented error estimation is used for an adaptive refinement strategy. Finally, we give some numerical results.

AimsThe purpose of the study was to evaluate the role of coronary venous mapping to identify epicardial ventricular tachycardia (VT) in patients with structural heart disease.

Privacy concerns spark the desire to analyze large-scale interconnected systems in a distributed fashion, that is, without a central entity having global model knowledge. Two different approaches are presented to analyze the stability of interconnected linear time-invariant systems with limited model knowledge. The two algorithms implement sufficient stability conditions and require information exchange only with direct neighbors thus reducing the need to share model data widely and ensuring privacy. The first algorithm is based on an M-matrix condition, and the second one is based on Lyapunov inequalities. Both algorithms rely on distributed optimization using a dual decomposition approach. Numerical investigations are used to validate both approaches.

In this paper an inverse optimal control problem in the form of a mathematical program with complementarity constraints (MPCC) is considered and numerical experiences are discussed. The inverse optimal control problem arises in the context of human navigation where the body is modelled as a dynamical system and it is assumed that the motions are optimally controlled with respect to an unknown cost function. The goal of the inversion is now to find a cost function within a given parametrized family of candidate cost functions such that the corresponding optimal motion minimizes the deviation from given data. MPCCs are known to be a challenging class of optimization problems typically violating all standard constraint qualifications (CQs). We show that under certain assumptions the resulting MPCC fulfills CQs for MPCCs being the basis for theory on MPCC optimality conditions and consequently for numerical solution techniques. Finally, numerical results are presented for the discretized inverse optimal control problem of locomotion using different solution techniques based on relaxation and lifting.

Abstract Aims Catheter ablation (CA) of ventricular arrhythmias is one of the most challenging electrophysiological interventions with an increasing use over the last years. Several benefits must be weighed against the risk of potentially life-threatening complications which necessitates a steady reevaluation of safety endpoints. Therefore, the aims of this study were (i) to investigate overall in-hospital mortality in patients undergoing such procedures and (ii) to identify variables associated with in-hospital mortality in a German-wide hospital network. Methods and results Between January 2010 and September 2018, administrative data provided by 85 Helios hospitals were screened for patients with main or secondary discharge diagnosis of ventricular tachycardia (VT) or premature ventricular contractions (PVCs) in combination with an arrhythmia-related CA using ICD- and OPS codes. In 5052 cases (mean age 60.9 ± 14.3 years, 30.1% female) of 30 different hospitals, in-hospital mortality was 1.27% with a higher mortality in patients ablated for VT (1.99%, n = 2, 955) compared to PVC (0.24%, n = 2, 097, P &lt; 0.01). Mortality rates were 2.06% in patients with ischaemic heart disease (IHD, n = 2, 137), 1.47% in patients with non-ischaemic structural heart disease (NIHD, n = 1, 224), and 0.12% in patients without structural heart disease (NSHD, n = 1, 691). Considering different types of hospital admission, mortality rates were 0.35% after elective (n = 2, 825), 1.60% after emergency admission/hospital transfer &lt;24 h (n = 1, 314) and 3.72% following delayed hospital transfer &gt;24 h after initial admission (n = 861, P &lt; 0.01 vs. elective admission and emergency admission/hospital transfer &lt;24 h). In multivariable analysis, a delayed hospital transfer &gt;24 h [odds ratio (OR) 2.28, 95% confidence interval (CI) 1.59–3.28, P &lt; 0.01], the occurrence of procedure-related major adverse events (OR 6.81, 95% CI 2.90–16.0, P &lt; 0.01), Charlson Comorbidity Index (CCI, OR 2.39, 95% CI 1.56–3.66, P &lt; 0.01) and its components congestive heart failure (OR 8.04, 95% CI 1.71–37.8, P &lt; 0.01), and diabetes mellitus (OR 1.59, 95% CI 1.13–2.22, P &lt; 0.01) were significantly associated with in-hospital death. Conclusions We reported in-hospital mortality rates after CA of ventricular arrhythmias in the largest multicentre, administrative dataset in Germany which can be implemented in quality management programs. Aside from comorbidities, a delayed hospital transfer to a CA performing centre is associated with an increased in-hospital mortality. This deserves further studies to determine the optimal management strategy.

Most results on distributed control design of large-scale interconnected systems assume a central designer with global model knowledge. The wish for privacy of subsystem model data raises the desire to find control design methods to determine an optimal control law without centralized model knowledge, i.e. in a distributed fashion. In this paper we present a distributed control design method with guaranteed stability to minimize an infinite horizon LQ cost functional. The introduction of adjoint states allows to iteratively optimize the feedback matrix using a gradient descent method in a distributed way, based on a finite horizon formulation. Inspired by ideas on stabilizing model predictive control, a terminal cost term is used, which gives a bound on the infinite horizon cost functional and ensures stability. A method is presented to determine that term in a distributed fashion. The results are validated using numerical experiments.

In this paper a bundle method for nonconvex nonsmooth optimization in infinite-dimensional Hilbert spaces is developed and analyzed. The algorithm requires only inexact function value and subgradient information. Global convergence to approximately stationary points is proved, where the final accuracy depends on the error level in the function and subgradient data. The method is then applied to an optimal control problem governed by the obstacle problem. For adaptively controlling the inexactness, implementable conditions are developed, first on a general level and then for the concrete case of a FEM discretization for optimal control of an obstacle problem. Numerical results are presented.

We consider shape optimization problems governed by the unsteady Navier--Stokes equations by applying the method of mappings, where the problem is transformed to a reference domain ${\Omega_{{ref}}}$ and the physical domain is given by $\Omega=\tau({\Omega_{{ref}}})$ with a domain transformation $\tau\in W^{1,\infty}({\Omega_{{ref}}})$. We show the Fréchet differentiability of $\tau\mapsto (v,p)(\tau)$ in a neighborhood of $\tau={id}$ under as low regularity requirements on ${\Omega_{{ref}}}$ and $\tau$ as possible. We propose a general analytical framework beyond the implicit function theorem to show the Fréchet differentiability of the transformation-to-state mapping conveniently. It can be applied to other shape optimization or optimal control problems and takes care of the usual norm discrepancy needed for nonlinear problems to show differentiability of the state equation and invertibility of the linearized operator. By applying the framework to the unsteady Navier--Stokes equations, we show that for Lipschitz domains ${\Omega_{{ref}}}$ and arbitrary $r>1$, $s>0$ the mapping $\tau\in ({W}^{1,\infty}\cap {W}^{1+s,r})({\Omega_{{ref}}})\mapsto (v,p)(\tau)\in (W(0,T;V)+W(0,T;{H}_0^1))\times (L^2(0,T;L_0^2)+W^{1,1}(0,T;{cl}_{(H^1)^*}(L^2_0))^*)$ is Fréchet differentiable at $\tau={id}$ and the mapping $\tau\in ({W}^{1,\infty}\cap {W}^{1+s,r})({\Omega_{{ref}}})\mapsto (v,p)(\tau)\in (L^2(0,T;{H}_0^1)\cap C([0,T];{L}^2)\times (L^2(0,T;L_0^2)+W^{1,1}(0,T;{cl}_{(H^1)^*}(L^2_0))^*)$ is Fréchet differentiable on a neighborhood of ${id}$, where $V\subset {H}_0^1({\Omega_{{ref}}})$ is the subspace of solenoidal functions and $W(0,T;V)$ is the usual space of weak solutions. A crucial role in the analysis plays the handling of the incompressibility condition and the low time regularity of the pressure for weak solutions.

We propose an infeasible interior point method for pointwise state constrained optimal control problems with linear elliptic PDEs. A smoothed constraint violation functional is used to develop a self-concordant barrier approach in an infinite-dimensional setting. We provide a detailed convergence analysis in function space for this approach. The quality of the smoothing is described by a parameter. By fixing this parameter we obtain a perturbed version of the original problem. We establish complexity estimates and convergence rates for the methods that we propose to solve a given perturbed problem. We also estimate the distance between the optimal solution of the perturbed problem and the optimal solution of the original problem. Moreover, our approach yields a rigorous measure for the proximity of the actual iterate to the minimizer of the perturbed and the original problems. We report on numerical experiments to illustrate efficiency and mesh independence.

Which criteria determine the formation of rest-to-rest arm movements when interacting with virtual mass-damper dynamics? A novel bilevel optimization approach is used to find the optimal linear combination of common optimization criteria for human trajectory formation such that the resulting trajectory comes closest to the human-performed one. The goal is to utilize this optimal combination to predict human motions in robot control. Experimental results show that subject-dependent criteria combinations can be found for different dynamics.

Theoretical and practical issues arising in optimal boundary control of the unsteady two-dimensional compressible Navier-Stokes equations are discussed. Assuming a sufficiently smooth state, formal adjoint and gradient equations are derived. For a vortex rebound model problem wall normal suction and blowing is used to minimize cost functionals of interest, here the kinetic energy at the final time.

Partial differential equation (PDE) constrained optimization is designed to solve control, design, and inverse problems with underlying physics. A distinguishing challenge of this technique is the handling of large numbers of optimization variables in combination with the complexities of discretized PDEs. Over the last several decades, advances in algorithms, numerical simulation, software design, and computer architectures have allowed for the maturation of PDE constrained optimization (PDECO) technologies with subsequent solutions to complicated control, design, and inverse problems. This special journal edition, entitled “PDE-Constrained Optimization”, features eight papers that demonstrate new formulations, solution strategies, and innovative algorithms for a range of applications. In particular, these contributions demonstrate the impactfulness on our engineering and science communities. This paper offers brief remarks to provide some perspective and background for PDECO, in addition to summaries of the eight papers.

Abstract Introduction Concealed structural abnormalities were detected by delayed enhancement ‐ magnetic resonance imaging (DE‐MRI) in patients with apparently idiopathic tachycardia of left ventricular (LV) origin. Basal septal fibrosis was evaluated as a potential arrhythmia substrate in patients with left ventricular outflow tract (LVOT) arrhythmias. Methods and Results A total of 22 patients with LVOT arrhythmias, including frequent monomorphic premature ventricular complexes (PVCs) in 15 patients and ventricular tachycardia (VT) in 7 patients, underwent catheter ablation and DE‐MRI. A total of 19 patients with frequent PVCs and 17 patients with idiopathic VT of other origin served as a control group. Basal septal intramural fibrosis as thin strip‐shaped intramyocardial DE or as marked intramyocardial DE involving &gt;25% of wall thickness was detected more frequently in patients with LVOT arrhythmias (41% and 32%) than in patients with non LVOT arrhythmias (14% and 3%). After successful ablation, 4/16 patients with basal septal intramural fibrosis and LVOT PVCs (n = 3) or LVOT VT (n = 1) compared with no patient without basal septal fibrosis experienced episodes of sustained VT with similar or different QRS morphology resulting in ICD therapy in three patients. Follow‐up DE‐MRI after PVC ablation (17 ± 7 months) revealed an increase in LV ejection fraction from 49 ± 5% to 56 ± 5% (n = 9) but the amount of septal DE remained unchanged. Conclusions Basal septal intramural fibrosis may serve as the arrhythmia substrate in a substantial part of patients with premature ventricular complexes (PVCs) and VT originating from the LVOT and identifies patients with continued risk for VT recurrence after initially successful ablation of LVOT arrhythmias.

We consider distributionally robust optimization problems (DROPs) with nonlinear and nonconcave dependence on uncertain parameters. The DROP can be written as a nonsmooth, nonlinear program with a bilevel structure; the objective function and each of the constraint functions are suprema of expected values of parametric functions taken over an ambiguity set of probability distributions. We define ambiguity sets through moment constraints, and to make the computation of first order stationary points tractable, we approximate nonlinear functions using quadratic expansions w.r.t. parameters, resulting in lower-level problems defined by trust-region problems and semidefinite programs. Subsequently, we construct smoothing functions for the approximate lower level functions which are computationally tractable, employing strong duality for trust-region problems, and show that gradient consistency holds. We formulate smoothed DROPs and apply a homotopy method that dynamically decreases smoothing parameters and establish its convergence to stationary points of the approximate DROP under mild assumptions. Through our scheme, we provide a new approach to robust nonlinear optimization as well. We perform numerical experiments and comparisons to other methods on a well-known test set, assuming design variables are subject to implementation errors, which provides a representative set of numerical examples.

To analyse the long-term safety of implantable cardioverter defibrillators (ICDs) in patients discharged within 24 h or after 2- 5-day hospitalization, respectively, after complication-free implantation, in circumstances of actual care.Patients in the multicentre, nationwide German DEVICE registry were contacted 12-15 months after their first ICD implantation or device replacement. Data were collected on complications, potential arrhythmic events, syncope, resuscitation, ablation procedures, cardiac events, hospitalizations, heart failure status, change of medication, and quality of life. Of 2356 patients from 43 centres, 527 patients were discharged within 24 h and 1829 were hospitalized routinely for >24 h after complication-free implantations. The disease profiles and rates of co-morbidities were similar at baseline for both cohorts. During between 384 and 543 days of follow-up, there were no significant differences between the groups in terms of complications, hospitalizations, or quality of life. One-year rates of death were 4.5% in patients discharged early compared with 7.2% in hospitalized patients (hazard ratio 0.65; 95% confidence interval 0.42-1.02; P = 0.052). Rates of major adverse cardiovascular events or defibrillator events were not higher in patients discharged after 24 h. In both groups, a high rate of patients declared that they would opt for the procedure again in the same situation.Data from a large-scale registry reflecting current day-to-day practice in Germany suggest that most patients can be discharged safely within 24 h of successful ICD implantation if there are no procedure-related events. Follow-up data up to 1.5 years after implantation did not raise long-term safety concerns.

Background: The entrainment mapping algorithm is used for ablation of ventricular tachycardia (VT) in right ventricular (RV) cardiomyopathy, but ablation at endocardial isthmus sites has only a moderate success rate. This study was performed to identify additional local electrogram characteristics associated with successful ablation. Patients and Methods: Using entrainment mapping, 45 reentry circuit isthmus sites were detected in 11 patients with RV cardiomyopathy presenting with 13 monomorphic VTs. Local bipolar electrograms were retrospectively analyzed at reentry circuit isthmus sites during VT, sinus rhythm, and programmed stimulation from the right ventricular apex (RVA), and compared between successful and unsuccessful ablation sites. Results: Ablation was successful at 10 reentry circuit isthmus sites and unsuccessful at 35 isthmus sites. During VT, a longer endocardial activation time relative to QRS onset, an increased electrogram‐QRS interval as a percentage of VT cycle length, and a longer electrogram duration were found at successful in comparison to unsuccessful ablation sites. The presence of isolated diastolic potentials during sinus rhythm at reentry circuit isthmus sites, consistent with slow conduction or unidirectional conduction block, was associated with successful catheter ablation. Prolongation of the duration of the local multipotential electrogram by &gt;100 ms during programmed RVA pacing at reentry circuit exit sites, indicating functional conduction disorder was also a marker of successful ablation. Conclusions: The demonstration of multipotential electrogram characteristics indicating fixed or functional conduction block may increase the likelihood of successful VT ablation at exit and central isthmus sites of reentry circuits in RV cardiomyopathy.

Newton differentiability is an important concept for analyzing generalized Newton methods for nonsmooth equations. In this work, for a convex function defined on an infinite-dimensional space, we discuss the relation between Newton and Bouligand differentiability and upper semicontinuity of its subdifferential. We also construct a Newton derivative of an operator of the form $(Fx)(p) = f(x,p)$ for general nonlinear operators $f$ that possess a Newton derivative with respect to $x$ and also for the case where $f$ is convex in $x$.

Many applications in mathematical modeling and optimal control lead to problems that are posed in function spaces and contain pointwise complementarity conditions. In this paper, a projected Newton method for nonlinear complementarity problems in the infinite dimensional function space L p is proposed and analyzed. Hereby, an NCP-function is used to reformulate the problem as a nonsmooth operator equation. The method stays feasible with respect to prescribed bound-constraints. The convergence analysis is based on semismoothness results for superposition operators in function spaces. The proposed algorithm is shown to converge locally q-superlinearly to a regular solution. As an important tool for applications, we establish a sufficient condition for regularity. The application of the algorithm to the distributed bound-constrained control of an elliptic partial differential equation is discussed in detail. Numerical results confirm the efficiency of the method.

In the current research on distributed control of interconnected large-scale dynamical systems an often neglected issue is the desire to ensure privacy of subsystems. This gives motivation for the presented distributed controller design method which requires communication and the exchange of model data only with direct neighbors. Thus, no global system knowledge is required. An important property of many large-scale systems is the presence of algebraic conservation constraints, for example in terms of energy or mass flow. Therefore, the presented controller design takes these constraints explicitly into account while preserving the sparsity structure of the distributed system necessary for a distributed design. The computation is based on the simulation of the system states and of adjoint states. The control objective is represented by the finite horizon linear quadratic cost functional.

Motivated by applications in subsurface CO2 sequestration, we investigate constrained optimal control problems with partially miscible two-phase flow in porous media. The objective is, e.g., to maximize the amount of trapped CO2 in an underground reservoir after a fixed period of CO2 injection, where the time-dependent injection rates in multiple wells are used as control parameters. We describe the governing two-phase two-component Darcy flow PDE system and formulate the optimal control problem. For the discretization we use a variant of the BOX method, a locally conservative control-volume FE method. The timestep-wise Lagrangian of the control problem is implemented as a functional in the PDE toolbox Sundance, which is part of the HPC software Trilinos. The resulting MPI parallelized Sundance state and adjoint solvers are linked to the interior point optimization package IPOPT. Finally, we present some numerical results in a heterogeneous model reservoir.

We present a Newton-CG method for full-waveform seismic inversion. Our method comprises the adjoint-based computation of the gradient and Hessian-vector products of the reduced problem and a preconditioned conjugate gradient method to solve the Newton system in matrix-free fashion. A trust-region globalization strategy and a multi-frequency inversion approach are applied. The governing equations are given by a coupled system of the acoustic and the elastic wave equation for the numerical simulation of wave propagation in solid and fluid media. We show numerical results for the application of our method to marine geophysical exploration.

For implantable cardioverter defibrillator a 10 J safety margin between the defibrillation threshold (DFT) and the maximum output of the device is intended. In complex cases, the additional placement of a subcutaneous array lead is a common strategy for lowering the DFT. We report the successful use of transvenous coil electrodes as single element subcutaneous array leads in order to lower the DFT.

FCT POCI/MAT/59442/2004, PTDC/MAT/64838/2006; ESA contract AS-2007-09-003; Sonderforschungsbereich 666 funded by Deutsche Forschungsgemeinschaft

The aim of this work is to develop an inexact bundle method for nonsmooth nonconvex minimization in Hilbert spaces and to investigate its application to optimal control problems with deterministic or stochastic obstacle problems as constraints. A central requirement is that (approximate) subgradients can be obtained at given points. The second part of the paper thus studies in detail how subgradients can be obtained for optimal control problems governed by (stochastic) obstacle problems.

Recently, the stochastic Polyak step size (SPS) has emerged as a competitive adaptive step size scheme for stochastic gradient descent. Here we develop ProxSPS, a proximal variant of SPS that can handle regularization terms. Developing a proximal variant of SPS is particularly important, since SPS requires a lower bound of the objective function to work well. When the objective function is the sum of a loss and a regularizer, available estimates of a lower bound of the sum can be loose. In contrast, ProxSPS only requires a lower bound for the loss which is often readily available. As a consequence, we show that ProxSPS is easier to tune and more stable in the presence of regularization. Furthermore for image classification tasks, ProxSPS performs as well as AdamW with little to no tuning, and results in a network with smaller weight parameters. We also provide an extensive convergence analysis for ProxSPS that includes the non-smooth, smooth, weakly convex and strongly convex setting.

Introduction: Delayed enhancement - magnetic resonance imaging (DE-MRI) has demonstrated that nonischemic cardiomyopathy is mainly characterized by intramural or epicardial fibrosis whereas global endomyocardial fibrosis suggests cardiac involvement in autoimmune rheumatic diseases or amyloidosis. Conduction disorders and sudden cardiac death are important manifestations of autoimmune rheumatic diseases with cardiac involvement but the substrates of ventricular arrhythmias in autoimmune rheumatic diseases have not been fully elucidated. Methods and Results: 20 patients with autoimmune rheumatic diseases presenting with ventricular tachycardia (VT) (n=11) or frequent ventricular extrasystoles (n=9) underwent DE-MRI and / or endocardial electroanatomical mapping of the left ventricle (LV). 10 patients with autoimmune rheumatic diseases underwent VT ablation. Global endomyocardial fibrosis without myocardial thickening and unrelated to coronary territories was detected by DE-MRI or electroanatomical voltage mapping in 9 of 20 patients with autoimmune rheumatic diseases. In the other patients with autoimmune rheumatic diseases, limited regions of predominantly epicardial (n = 4) and intramyocardial (n = 5) fibrosis or only minimal fibrosis (n = 2) were found using DE-MRI. Endocardial low-amplitude diastolic potentials and pre-systolic Purkinje or fascicular potentials, mostly within fibrotic areas, were identified as the targets of successful VT ablation in 7 of 10 patients with autoimmune rheumatic diseases. Conclusion: Global endomyocardial fibrosis can be a tool to diagnose severe cardiac involvement in autoimmune rheumatic diseases and may serve as the substrate of ventricular arrhythmias in a substantial part of patients.

A new method for performing density-based topology optimization for Stokes flow is presented. It differs from previous approaches in the way the underlying mixed integer problem is relaxed and in the choice of the space in which the density that describes the topology lives. Initial numerical experiments, using for the density and a discretization by continuous piecewise linear finite elements, showed unsatisfactory convergence properties of state-of-the-art optimization solvers. This motivated the work in this paper, which proposes solutions to these difficulties. We present a theoretically founded new problem formulation based on a space for the density that allows for jumps along hypersurfaces, such as BV or fractional order Sobolev spaces. We extend the existing theory for the generalized Stokes equations and investigate the arising optimization problems concerning existence of solutions, differentiability, and convergence of relaxed solutions toward solutions of the original problem. We motivate a localized fractional order Sobolev norm as an approximation of the -norm for -valued functions and discuss its discretization by piecewise constant finite elements. Building on these theoretical findings, we present some numerical realizations and show computational results.

Delayed enhancement-magnetic resonance imaging (DE-MRI) has demonstrated that nonischemic cardiomyopathy is mainly characterized by intramural or epicardial fibrosis whereas global endomyocardial fibrosis suggests cardiac involvement in autoimmune rheumatic diseases or amyloidosis. Conduction disorders and sudden cardiac death are important manifestations of autoimmune rheumatic diseases with cardiac involvement but the substrates of ventricular arrhythmias in autoimmune rheumatic diseases have not been fully elucidated.20 patients with autoimmune rheumatic diseases presenting with ventricular tachycardia (VT) (n = 11) or frequent ventricular extrasystoles (n = 9) underwent DE-MRI and/or endocardial electroanatomical mapping of the left ventricle (LV). Ten patients with autoimmune rheumatic diseases underwent VT ablation. Global endomyocardial fibrosis without myocardial thickening and unrelated to coronary territories was detected by DE-MRI or electroanatomical voltage mapping in 9 of 20 patients with autoimmune rheumatic diseases. In the other patients with autoimmune rheumatic diseases, limited regions of predominantly epicardial (n = 4) and intramyocardial (n = 5) fibrosis or only minimal fibrosis (n = 2) were found using DE-MRI. Endocardial low-amplitude diastolic potentials and pre-systolic Purkinje or fascicular potentials, mostly within fibrotic areas, were identified as the targets of successful VT ablation in 7 of 10 patients with autoimmune rheumatic diseases.Global endomyocardial fibrosis can be a tool to diagnose severe cardiac involvement in autoimmune rheumatic diseases and may serve as the substrate of ventricular arrhythmias in a substantial part of patients.

A structure-exploiting forward mode is discussed that achieves almost optimal complexity for functions given by Kantorovič trees. It is based on approriate representations of the gradient and the Hessian. After a brief exposition of the forward and reverse mode of automatic differentiation for derivatives up to second order and compact proofs of their complexities, the new forward mode is presented and analyzed. It is shown that in the case of functions f: IR n → IR with a tree as Kantorovič graph the algorithm is only O(ln(n)) times as expensive as the reverse mode. Except for the fact that the new method is a very efficient implementation of the forward mode, it can be used to significantly reduce the length of characterizing sequences before applying the memory expensive reverse mode. For the Hessian all discussed algorithms are shown to be efficiently parallelizable. Some numerical examples confirm the advantages of the new forward mode.

One of the main challenges in a classical mesh-based FEM-approach is the representation of complex geometries. This challenge is often tackled by a computationally costly mesh generation process, where the resulting mesh’s facets represent the boundary. An alternative approach, that we employ here, is the immersed boundary (IB) approach. This uses instead a computationally cheaper structured adaptive Cartesian mesh and an explicit boundary representation, where the challenge mainly lies in the boundary condition (BC) imposition on the mesh cells intersected by the geometry’s boundary. One IB method is Nitsche’s method that we employ here for fluid-structure interaction (FSI) and shape optimization problems. The simulation of such complex physical systems modeled by PDEs requires a combination of sophisticated numerical methods. Implementing a FEM-based simulation software that computes a particular PDE’s solution often requires the reusage of existing methods. In order to make our approach public and also to prove the modularity of it, we integrated our IB methods in an existing FEM-based PDE toolbox of the Trilinos project, called Sundance.

Abstract We analyze optimal control problems for two-phase Navier-Stokes equations with surface tension. Based on $$L_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> -maximal regularity of the underlying linear problem and recent well-posedness results of the problem for sufficiently small data we show the differentiability of the solution with respect to initial and distributed controls for appropriate spaces resulting from the $$L_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> -maximal regularity setting. We consider first a formulation where the interface is transformed to a hyperplane. Then we deduce differentiability results for the solution in the physical coordinates. Finally, we state an equivalent Volume-of-Fluid type formulation and use the obtained differentiability results to derive rigorosly the corresponding sensitivity equations of the Volume-of-Fluid type formulation. For objective functionals involving the velocity field or the discontinuous pressure or phase indciator field we derive differentiability results with respect to controls and state formulas for the derivative. The results of the paper form an analytical foundation for stating optimality conditions, justifying the application of derivative based optimization methods and for studying the convergence of discrete sensitivity schemes based on Volume-of-Fluid discretizations for optimal control of two-phase Navier-Stokes equations.

We develop a sampling-free approximation scheme for distributionally robust PDE-constrained optimization problems, which are min-max control problems. We define the ambiguity set through moment and entropic constraints. We use second-order Taylor's expansions of the reduced objective function w.r.t. uncertain parameters, allowing us to compute the expected value of the quadratic function explicitly. The objective function of the approximated min-max problem separates into a trust-region problem and a semidefinite program. We construct smoothing functions for the optimal value functions defined by these problems. We prove the existence of optimal solutions for the distributionally robust control problem, and the approximated and smoothed problems, and show that a worst-case distribution exists. For the numerical solution of the approximated problem, we develop a homotopy method that computes a sequence of stationary points of smoothed problems while decreasing smoothing parameters to zero. The adjoint approach is used to compute derivatives of the smoothing functions. Numerical results for two nonlinear optimization problems are presented.

Abstract We derive a generic system that constructs an optimization model for an emergency stop scenario on the highway, based on map data from high definition maps that are used in Advanced Driver Assistance Systems (ADAS) and in Highly Automated Driving (HAD). New additional situative and scenario-based information is computed by applying a global maximization approach to the model. For this purpose, we develop two new rigorous and deterministic branch-and-bound algorithms that both determine the certified global optimal value up to a predefined tolerance. The underlying interval optimization algorithm, which uses first-order techniques, is enhanced by one of two second-order methods that are applied for specifically selected intervals. We investigate two approaches that either compute a concave overestimator for the objective function or approximate the function with a quadratic polynomial using Taylor expansion. We show the limits of interval arithmetic in our problem, especially for the interval versions of the derivatives, and present a local linearization of the curve data that improves the results significantly. The presented novel method for deriving secondary information is compared to state of the art methods on two exemplary and for the automotive context representative scenarios to show the advantages of our approach.

Wir betrachten das allgemeine nichtlineare Optimierungsproblem (NLP) 15.1 % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBqj3BWbIqubWexLMBb50ujbqegm0B % 1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqr % Ffpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0F % irpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaa % GcbaWaaCbeaeaaciGGTbGaaiyAaiaac6gaaSqaaiaadIhacqGHiiIZ % tuuDJXwAK1uy0HMmaeXbfv3ySLgzG0uy0HgiuD3BaGqbaiab-1risn % aaCaaameqabaGaamOBaaaaaSqabaGccaWGMbGaaiikaiaadIhacaGG % PaacbaGaa4hiaiaa+bcacaGFGaGaa4xDaiaa+5cacaGFKbGaa4Nlai % aa+5eacaGFUaGaa4hiaiaa+bcacaGFGaGaam4zaiaacIcacaWG4bGa % aiykaiabgsMiJkaaicdacaGGSaGaa4hiaiaa+bcacaGFGaGaamiAai % aacIcacaWG4bGaaiykaiabg2da9iaaicdaaaa!63E1! $$ \mathop {\min }\limits_{x \in \mathbb{R}^n } f(x) u.d.N. g(x) \leqslant 0, h(x) = 0 $$ mit stetig differenzierbaren Funktionen f: ℝ n → ℝ, g: ℝ n → ℝ m , h: ℝ n → ℝ p .

In this work, we present a globalized stochastic semismooth Newton method for solving stochastic optimization problems involving smooth nonconvex and nonsmooth convex terms in the objective function. We assume that only noisy gradient and Hessian information of the smooth part of the objective function is available via calling stochastic first and second order oracles. The proposed method can be seen as a hybrid approach combining stochastic semismooth Newton steps and stochastic proximal gradient steps. Two inexact growth conditions are incorporated to monitor the convergence and the acceptance of the semismooth Newton steps and it is shown that the algorithm converges globally to stationary points in expectation. Moreover, under standard assumptions and utilizing random matrix concentration inequalities, we prove that the proposed approach locally turns into a pure stochastic semismooth Newton method and converges r-superlinearly with high probability. We present numerical results and comparisons on $\ell_1$-regularized logistic regression and nonconvex binary classification that demonstrate the efficiency of our algorithm.

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Battery electric vehicles (BEVs) are playing an increasingly important role in personal mobility due to the wish to counteract climate change and political regulations concerning carbon dioxide emissions. Nevertheless, there are obstacles that need to be overcome. Especially long-distance journeys are problematic due to long charging stops and range anxiety It is a driver’s wish to fulfill a given driving task in a time-optimal way. But in the BEV case, driving faster does not necessarily lead to a decreased total travel time. The vehicle routing and charging problem is formulated as a mixed-integer nonlinear program (MINLP) and solved using mathematical optimization methods. First, time-minimizing vehicle routing with charging stations providing different power levels is discussed. The program returning the exact result is significantly faster than previous ones. Afterwards, the model is extended: driving speed becomes adjustable. A combined time minimal optimization of which route to take, how fast to drive, where and how much to recharge is the result. The combination of these four parameters has never been studied before. It is shown that up to 14.48% of driving time can be saved in our examples by incorporating the choice of a driving speed.

We analyze optimal control problems for two-phase Navier-Stokes equations with surface tension. Based on $L_p$-maximal regularity of the underlying linear problem and recent well-posedness results of the problem for sufficiently small data we show the differentiability of the solution with respect to initial and distributed controls for appropriate spaces resulting form the $L_p$-maximal regularity setting. We consider first a formulation where the interface is transformed to a hyperplane. Then we deduce differentiability results for the solution in the physical coordinates. Finally, we state an equivalent Volume-of-Fluid type formulation and use the obtained differentiability results to derive rigorosly the corresponding sensitivity equations of the Volume-of-Fluid type formulation. For objective functionals involving the velocity field or the discontinuous pressure or phase indciator field we derive differentiability results with respect to controls and state formulas for the derivative. The results of the paper form an analytical foundation for stating optimality conditions, justifying the application of derivative based optimization methods and for studying the convergence of discrete sensitivity schemes based on Volume-of-Fluid discretizations for optimal control of two-phase Navier-Stokes equations.

In this paper, we study a few theoretical issues in the discretized Kohn-Sham (KS) density functional theory (DFT). The equivalence between either a local or global minimizer of the KS total energy minimization problem and the solution to the KS equation is established under certain assumptions. The nonzero charge densities of a strong local minimizer are shown to be bounded below by a positive constant uniformly. We analyze the self-consistent field (SCF) iteration by formulating the KS equation as a fixed point map with respect to the potential. The Jacobian of these fixed point maps is derived explicitly. Both global and local convergence of the simple mixing scheme can be established if the gap between the occupied states and unoccupied states is sufficiently large. This assumption can be relaxed if the charge density is computed using the Fermi-Dirac distribution and it is not required if there is no exchange correlation functional in the total energy functional. Although our assumption on the gap is very stringent and is almost never satisfied in reality, our analysis is still valuable for a better understanding of the KS minimization problem, the KS equation and the SCF iteration.

Dieses Lehrbuch beschäftigt sich mit der Analyse und der numerischen Behandlung endlichdimensionaler stetiger Optimierungsprobleme. Hierunter verstehen wir die Aufgabenstellung, eine stetige Zielfunktion f : X → ℝ auf dem nichtleeren zulässigen Bereich X ⊂ℝ n zu minimieren (wir werden später gewisse strukturelle Anforderungen an X stellen). Wir schreiben dies kurz in folgender Form: (1.1) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyBaiaacM % gacaGGUbGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiaabcca % caqG1bGaaeOlaiaabsgacaqGUaGaaeOtaiaab6cacaqGGaGaamiEai % abgIGiolaadIfacaGGUaaaaa!4650! $$\min f\left( x \right){\text{ u}}{\text{.d}}{\text{.N}}{\text{. }}x \in X.$$

In this chapter we collect several results of finite-dimensional nonsmooth analysis that are required for our investigations. In particular, finite-dimensional semismoothness and semismooth Newton methods are considered. The concepts introduced in this section will serve as a motivation and guideline for the developments in subsequent sections.

6.1 IntroductionAn important motivation for investigating optimization methods in infinite dimensions is developing algorithms that are mesh independent. Here, mesh independence means the following: Suppose that for the infinite-dimensional problem (P), a local convergence theory for an abstract solution algorithm A is available. This algorithm A could, e.g., be the Newton method if (P) is an operator equation. For the numerical implementation, the problem (P) needs to be discretized, which results in a discrete problem ( Ph ) ( h>0 denoting the mesh size or, more generally, the accuracy of the discretization).

3.1 IntroductionIt was shown in Chapter 1 that semismooth NCP- and MCP-functions can be used to refor-mulate the VIP (1.14) as (one or more) nonsmooth operator equation(s) of the formΦ(u)=0,whereΦ(u)(ω)=ϕ(G(u)(ω))onΩ,(3.1)with G mapping u∈Lp(Ω) to a vector of Lebesgue functions. In particular, for NCPs we have G(u)=(u, F(u)) with F : Lp(Ω)→Lp′ (Ω) , p, p′ ∈ (1, ∞]. In finite dimensions this reformulation technique is well investigated and yields a semismooth system of equations, which can be solved by semismooth Newton methods. Naturally, the question arises if it is possible to develop a similar semismoothness theory for operators of the form (3.1).