190 research outputs found

    Proximal Galerkin: A structure-preserving finite element method for pointwise bound constraints

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    The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop scalable, mesh-independent algorithms for optimal design. The paper leads to a derivation of the latent variable proximal point (LVPP) algorithm: an unconditionally stable alternative to the interior point method. LVPP is an infinite-dimensional optimization algorithm that may be viewed as having an adaptive barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of the main benefits of this algorithm is witnessed when analyzing the classical obstacle problem. Therein, we find that the original variational inequality can be replaced by a sequence of semilinear partial differential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout this work, we arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field density-based topology optimization. The complete latent variable proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis

    A modified combined active-set Newton method for solving phase-field fracture into the monolithic limit

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    In this work, we examine a numerical phase-field fracture framework in which the crack irreversibility constraint is treated with a primal-dual active set method and a linearization is used in the degradation function to enhance the numerical stability. The first goal is to carefully derive from a complementarity system our primal-dual active set formulation, which has been used in the literature in numerous studies, but for phase-field fracture without its detailed mathematical derivation yet. Based on the latter, we formulate a modified combined active-set Newton approach that significantly reduces the computational cost in comparison to comparable prior algorithms for quasi-monolithic settings. For many practical problems, Newton converges fast, but active set needs many iterations, for which three different efficiency improvements are suggested in this paper. Afterwards, we design an iteration on the linearization in order to iterate the problem to the monolithic limit. Our new algorithms are implemented in the programming framework pfm-cracks [T. Heister, T. Wick; pfm-cracks: A parallel-adaptive framework for phase-field fracture propagation, Software Impacts, Vol. 6 (2020), 100045]. In the numerical examples, we conduct performance studies and investigate efficiency enhancements. The main emphasis is on the cost complexity by keeping the accuracy of numerical solutions and goal functionals. Our algorithmic suggestions are substantiated with the help of several benchmarks in two and three spatial dimensions. Therein, predictor-corrector adaptivity and parallel performance studies are explored as well.Comment: 49 pages, 45 figures, 9 table

    A numerical framework for solving PDE-constrained optimization problems from multiscale particle dynamics

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    In this thesis, we develop accurate and efficient numerical methods for solving partial differential equation (PDE) constrained optimization problems arising from multiscale particle dynamics, with the aim of producing a desired time-dependent state at the minimal cost. A PDE-constrained optimization problem seeks to move one or more state variables towards a desired state under the influence of one or more control variables, and a set of constraints that are described by PDEs governing the behaviour of the variables. In particular, we consider problems constrained by one-dimensional and two-dimensional advection-diffusion problems with a non-local integral term, such as the associated mean-field limit Fokker-Planck equation of the noisy Hegselmann-Krause opinion dynamics model. We include additional bound constraints on the control variable for the opinion dynamics problem. Lastly, we consider constraints described by a two-dimensional robot swarming model made up of a system of advection-diffusion equations with additional linear and integral terms. We derive continuous Lagrangian first-order optimality conditions for these problems and solve the resulting systems numerically for the optimized state and control variables. Each of these problems, combined with Dirichlet, no-flux, or periodic boundary conditions, present unique challenges that require versatility of the numerical methods devised. Our numerical framework is based on a novel combination of four main components: (i) a discretization scheme, in both space and time, with the choice of pseudospectral or fi nite difference methods; (ii) a forward problem solver that is implemented via a differential-algebraic equation solver; (iii) an optimization problem solver that is a choice between a fi xed-point solver, with or without Armijo-Wolfe line search conditions, a Newton-Krylov algorithm, or a multiple shooting scheme, and; (iv) a primal-dual active set strategy to tackle additional bound constraints on the control variable. Pseudospectral methods efficiently produce highly accurate solutions by exploiting smoothness in the solutions, and are designed to perform very well with dense, small matrix systems. For a number of problems, we take advantage of the exponential convergence of pseudospectral methods by discretising in this way not only in space, but also in time. The alternative fi nite difference method performs comparatively well when non-smooth bound constraints are added to the optimization problem. A differential{algebraic equation solver works out the discretized PDE on the interior of the domain, and applies the boundary conditions as algebraic equations. This ensures generalizability of the numerical method, as one does not need to explicitly adapt the numerical method for different boundary conditions, only to specify different algebraic constraints that correspond to the boundary conditions. A general fixed-point or sweeping method solves the system of equations iteratively, and does not require the analytic computation of the Jacobian. We improve the computational speed of the fi xed-point solver by including an adaptive Armijo-Wolfe type line search algorithm for fixed-point problems. This combination is applicable to problems with additional bound constraints as well as to other systems for which the regularity of the solution is not sufficient to be exploited by the spectral-in-space-and-time nature of the Newton-Krylov approach. The recently devised Newton-Krylov scheme is a higher-order, more efficient optimization solver which efficiently describes the PDEs and the associated Jacobian on the discrete level, as well as solving the resulting Newton system efficiently via a bespoke preconditioner. However, it requires the computation of the Jacobian, and could potentially be more challenging to adapt to more general problems. Multiple shooting solves an initial-value problem on sections of the time interval and imposes matching conditions to form a solution on the whole interval. The primal-dual active set strategy is used for solving our non-linear and non-local optimization problems obtained from opinion dynamics problems, with pointwise non-equality constraints. This thesis provides a numerical framework that is versatile and generalizable for solving complex PDE-constrained optimization problems from multiscale particle dynamic

    Dual Newton Proximal Point Algorithm for Solution Paths of the L1-Regularized Logistic Regression

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    The l1-regularized logistic regression is a widely used statistical model in data classification. This paper proposes a dual Newton method based proximal point algorithm (PPDNA) to solve the l1-regularized logistic regression problem with bias term. The global and local convergence of PPDNA hold under mild conditions. The computational cost of a semismooth Newton (Ssn) algoithm for solving subproblems in the PPDNA can be effectively reduced by fully exploiting the second-order sparsity of the problem. We also design an adaptive sieving (AS) strategy to generate solution paths for the l1-regularized logistic regression problem, where each subproblem in the AS strategy is solved by the PPDNA. This strategy exploits active set constraints to reduce the number of variables in the problem, thereby speeding up the PPDNA for solving a series of problems. Numerical experiments demonstrate the superior performance of the PPDNA in comparison with some state-of-the-art second-order algorithms and the efficiency of the AS strategy combined with the PPDNA for generating solution paths

    Modeling, Discretization, Optimization, and Simulation of Phase-Field Fracture Problems

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    This course is devoted to phase-field fracture methods. Four different sessions are centered around modeling, discretizations, solvers, adaptivity, optimization, simulations and current developments. The key focus is on research work and teaching materials concerned with the accurate, efficient and robust numerical modeling. These include relationships of model, discretization, and material parameters and their influence on discretizations and the nonlinear (Newton-type methods) and linear numerical solution. One application of such high-fidelity forward models is in optimal control, where a cost functional is minimized by controlling Neumann boundary conditions. Therein, as a side-project (which is itself novel), space-time phase-field fracture models have been developed and rigorously mathematically proved. Emphasis in the entire course is on a fruitful mixture of theory, algorithmic concepts and exercises. Besides these lecture notes, further materials are available, such as for instance the open-source libraries pfm-cracks and DOpElib. The prerequisites are lectures in continuum mechanics, introduction to numerical methods, finite elements, and numerical methods for ODEs and PDEs. In addition, functional analysis (FA) and theory of PDEs is helpful, but for most parts not necessarily mandatory. Discussions with many colleagues in our research work and funding from the German Research Foundation within the Priority Program 1962 (DFG SPP 1962) within the subproject Optimizing Fracture Propagation using a Phase-Field Approach with the project number 314067056 (D. Khimin, T. Wick), and support of the French-German University (V. Kosin) through the French-German Doctoral college ``Sophisticated Numerical and Testing Approaches" (CDFA-DFDK 19-04) is gratefully acknowledged

    An efficient algorithm for the ℓp\ell_{p} norm based metric nearness problem

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    Given a dissimilarity matrix, the metric nearness problem is to find the nearest matrix of distances that satisfy the triangle inequalities. This problem has wide applications, such as sensor networks, image processing, and so on. But it is of great challenge even to obtain a moderately accurate solution due to the O(n3)O(n^{3}) metric constraints and the nonsmooth objective function which is usually a weighted ℓp\ell_{p} norm based distance. In this paper, we propose a delayed constraint generation method with each subproblem solved by the semismooth Newton based proximal augmented Lagrangian method (PALM) for the metric nearness problem. Due to the high memory requirement for the storage of the matrix related to the metric constraints, we take advantage of the special structure of the matrix and do not need to store the corresponding constraint matrix. A pleasing aspect of our algorithm is that we can solve these problems involving up to 10810^{8} variables and 101310^{13} constraints. Numerical experiments demonstrate the efficiency of our algorithm. In theory, firstly, under a mild condition, we establish a primal-dual error bound condition which is very essential for the analysis of local convergence rate of PALM. Secondly, we prove the equivalence between the dual nondegeneracy condition and nonsingularity of the generalized Jacobian for the inner subproblem of PALM. Thirdly, when q(⋅)=∄⋅∄1q(\cdot)=\|\cdot\|_{1} or ∄⋅∄∞\|\cdot\|_{\infty}, without the strict complementarity condition, we also prove the equivalence between the the dual nondegeneracy condition and the uniqueness of the primal solution

    Non-Smooth Optimization by Abs-Linearization in Reflexive Function Spaces

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    Nichtglatte Optimierungsprobleme in reflexiven BanachrĂ€umen treten in vielen Anwendungen auf. HĂ€ufig wird angenommen, dass alle vorkommenden Nichtdifferenzierbarkeiten durch Lipschitz-stetige Operatoren wie abs, min und max gegeben sind. Bei solchen Problemen kann es sich zum Beispiel um optimale Steuerungsprobleme mit möglicherweise nicht glatten Zielfunktionen handeln, welche durch partielle Differentialgleichungen (PDG) eingeschrĂ€nkt sind, die ebenfalls nicht glatte Terme enthalten können. Eine effiziente und robuste Lösung erfordert eine Kombination numerischer Simulationen und spezifischer Optimierungsalgorithmen. Lokal Lipschitz-stetige, nichtglatte Nemytzkii-Operatoren, welche direkt in der Problemformulierung auftreten, spielen eine wesentliche Rolle in der Untersuchung der zugrundeliegenden Optimierungsprobleme. In dieser Dissertation werden zwei spezifische Methoden und Algorithmen zur Lösung solcher nichtglatter Optimierungsprobleme in reflexiven BanachrĂ€umen vorgestellt und diskutiert. Als erste Lösungsmethode wird in dieser Dissertation die Minimierung von nichtglatten Operatoren in reflexiven BanachrĂ€umen mittels sukzessiver quadratischer ÜberschĂ€tzung vorgestellt, SALMIN. Ein neuartiger Optimierungsansatz fĂŒr Optimierungsprobleme mit nichtglatten elliptischen PDG-BeschrĂ€nkungen, welcher auf expliziter Strukturausnutzung beruht, stellt die zweite Lösungsmethode dar, SCALi. Das zentrale Merkmal dieser Methoden ist ein geeigneter Umgang mit Nichtglattheiten. Besonderes Augenmerk liegt dabei auf der zugrundeliegenden nichtglatten Struktur des Problems und der effektiven Ausnutzung dieser, um das Optimierungsproblem auf angemessene und effiziente Weise zu lösen.Non-smooth optimization problems in reflexive Banach spaces arise in many applications. Frequently, all non-differentiabilities involved are assumed to be given by Lipschitz-continuous operators such as abs, min and max. For example, such problems can refer to optimal control problems with possibly non-smooth objective functionals constrained by partial differential equations (PDEs) which can also include non-smooth terms. Their efficient as well as robust solution requires numerical simulations combined with specific optimization algorithms. Locally Lipschitz-continuous non-smooth non-linearities described by appropriate Nemytzkii operators which arise directly in the problem formulation play an essential role in the study of the underlying optimization problems. In this dissertation, two specific solution methods and algorithms to solve such non-smooth optimization problems in reflexive Banach spaces are proposed and discussed. The minimization of non-smooth operators in reflexive Banach spaces by means of successive quadratic overestimation is presented as the first solution method, SALMIN. A novel structure exploiting optimization approach for optimization problems with non-smooth elliptic PDE constraints constitutes the second solution method, SCALi. The central feature of these methods is the appropriate handling of non-differentiabilities. Special focus lies on the underlying structure of the problem stemming from the non-smoothness and how it can be effectively exploited to solve the optimization problem in an appropriate and efficient way

    Efficient methods for optimal control problems subject to partial differential equations with uncertain coefficients

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    In this thesis, we develop and analyze methods to efficiently solve optimization problems under uncertainty, constrained by partial differential equations (PDEs). The uncertainties may arise due to noisy measurements, unknown or unobservable parameters, model ambiguity, or intrinsic randomness of systems. The goal is to find a control which is robust with respect to variations in the uncertain parameters. We prove error bounds and convergence rates for the developed methods, confirm the theoretically derived results through numerical experiments, and examine the developed concepts with regard to their efficiency. The focus of this work is the application and analysis of quasi-Monte Carlo methods, as well as the use of surrogate models for computationally intensive systems in conjunction with a penalty strategy. We first analyze a general formulation of the optimal control problem for the existence and uniqueness of solutions, and then focus on three example problems of optimal control under uncertainty. The regularity of the problems with respect to the uncertain parameters plays a crucial role in the development and the error analysis of the methods. The numerical treatment of the considered problems requires different approximation methods. The total approximation error is decomposed into its components and each error contribution is then studied separately in a chapter. The error estimates and convergence results developed in these chapters are not limited to problems of optimal control subject to PDE constraints with uncertain coefficients. In addition, further strategies to increase the efficiency of the methods are investigated, such as multilevel strategies and the simultaneous solving of the optimal control problem and learning of surrogate models for computationally intensive models

    Optimal Control of anisotropic Allen–Cahn equations

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    This thesis is concerned with the solution of an optimal control problem governed by an anisotropic Allen-Cahn equation as a model for, e.g., crystal growth. The first part treats the analytical existence theory and first order optimality conditions of the in time continuous and of the time discretized versions. The state equation is discretized implicitly in time with piecewise constant functions. To this end, we consider a more general quasilinear parabolic equation, where the quasilinear term is strongly monotone and obeys a certain growth condition while the lower order term is potentially non-monotone. The existence of the control-to-state operator and its Lipschitz continuity is shown for the time discretized as well as for the time continuous problem. Then we present for both the existence of global minimizers as well as the convergence of a subsequence of time discrete optimal controls to a global minimizer of the time continuous problem. The results hold in arbitrary space dimensions. Under some further restrictions we are able to show Fréchet differentiability of the in time discretized problem and use this to rigorously set up the first order conditions. For this the anisotropies are required to be smooth enough, which in this thesis is achieved by a suitable regularization. Therefore, the convergence behavior of the optimal controls are studied for a sequence of (smooth) approximations of the former quasilinear term. In addition the simultaneous limit in the approximation and the time step size is considered. For a class covering a large variety of anisotropies we introduce a certain regularization and show the previously formulated requirements. Finally, we will show that the results cannot be straightforwardly transferred to a semi-implicit discretization scheme. In the second part a trust region Newton method is presented, that eventually is used to numerically solve the optimal control problem. Different ways of preconditioning the involved Steihaug-CG solver are discussed and the limits of existing approaches in the present case are worked out. Then, several aspects of the implementation are examined, like the solver for the appearing partial differential equations, parallelization and the utility of adaptive meshes in the context of the control problem. In the final part, various numerical results based on the previously mentioned choice of anisotropies are presented. These include convergence with respect to the regularization parameter, numerical evidence for mesh independent behavior and a thorough discussion of the simulation in several relevant settings. We concentrate on two choices for the anisotropies and in addition include the isotropic case for comparison. Among others, crystal formation and topology changes are addressed and we see that the algorithm is able to handle these. Furthermore, the behavior of various quantities over the course of the algorithm is investigated. Here we observe that the number of Steihaug steps, and therefore the execution time per trust region step, growths considerably towards the end of the algorithm. Finally, we look at the impact of some implementational aspects with respect to execution speed. We observe that the implicit and semi-implicit approaches perform comparably fast if the implementation is suitably optimized. We however conclude that the implicit approach is preferable since it is less sensitive with respect to the regularization and is supported by more theoretical results
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