PDE-constrained Optimization Research Articles

Superlinear Convergence of the GMRES for PDE-Constrained Optimization Problems

ABSTRACTOptimal control problems for PDEs arise in many important applications. A main step in the solution process is the solution of the arising linear system, where the crucial point is usually finding a proper preconditioner. We propose both proper block diagonal and more involved preconditioners, and derive mesh independent superlinear convergence of the preconditioned GMRES iterations based on a compact perturbation property of the underlying operators.

Some preconditioners for elliptic PDE-constrained optimization problems

For the structured systems of linear equations arising from the Galerkin finite element discretizations of elliptic PDE-constrained optimization problems, some preconditioners are proposed to accelerate the convergence rate of Krylov subspace methods such as GMRES for both cases of the Tikhonov parameter β not very small (equal or greater than 1e−6) and sufficiently small (less than 1e−6), respectively. We derive the explicit expressions for the eigenvalues and eigenvectors of the corresponding preconditioned matrices. Numerical results show that the corresponding preconditioned GMRES methods perform and match well with the theoretical results.

Higher-Order Moving Mesh Methods for PDE-Constrained Shape Optimization

We present a new approach to discretizing shape optimization problems that generalizes standard moving mesh methods to higher-order mesh deformations and that is naturally compatible with higher-order finite element discretizations of PDE-constraints. This shape optimization method is based on discretized deformation diffeomorphisms and allows for arbitrarily high resolution of shapes with arbitrary smoothness. Numerical experiments show that this method allows the solution of PDE-constrained shape optimization problems to high accuracy.

Existence and Optimality Conditions for Risk-Averse PDE-Constrained Optimization

Uncertainty is ubiquitous in virtually all engineering applications, and, for such problems, it is inadequate to simulate the underlying physics without quantifying the uncertainty in unknown or random inputs, boundary and initial conditions, and modeling assumptions. In this work, we introduce a general framework for analyzing risk-averse optimization problems constrained by partial differential equations (PDEs). In particular, we postulate conditions on the random variable objective function as well as the PDE solution that guarantee existence of minimizers. Furthermore, we derive optimality conditions and apply our results to the control of an environmental contaminant. Finally, we introduce a new risk measure, called the conditional entropic risk, that fuses desirable properties from both the conditional value-at-risk and the entropic risk measures.

A Randomized Maximum A Posteriori Method for Posterior Sampling of High Dimensional Nonlinear Bayesian Inverse Problems

We present a randomized maximum a posteriori (rMAP) method for generating approximate samples of posteriors in high dimensional Bayesian inverse problems governed by large-scale forward problems. We derive the rMAP approach by (1) casting the problem of computing the MAP point as a stochastic optimization problem; (2) interchanging optimization and expectation; and (3) approximating the expectation with a Monte Carlo method. For a specific randomized data and prior mean, rMAP reduces to the randomized maximum likelihood (RML) approach. It can also be viewed as an iterative stochastic Newton method. An analysis of the convergence of the rMAP samples is carried out for both linear and nonlinear inverse problems. Each rMAP sample requires solution of a PDE-constrained optimization problem; to solve these problems, we employ a state-of-the-art trust region inexact Newton conjugate gradient method with sensitivity-based warm starts. An approximate Metropolization approach is presented to reduce the bias in rMA...

A New Lagrange-Newton-Krylov Solver for PDE-constrained Nonlinear Model Predictive Control

Abstract Real-time optimization of systems governed by partial differential equations (PDEs) presents significant computational challenges to nonlinear model predictive control (NMPC). The large-scale nature of PDEs often limits the use of standard nested black-box optimizers that require repeated forward simulations and expensive gradient computations. Hence, to ensure online solutions at relevant time-scales, large-scale NMPC algorithms typically require powerful, customized PDE-constrained optimization solvers. To this end, this paper proposes a new Lagrange-Newton-Krylov (LNK) method that targets the class of time-dependent nonlinear diffusion-reaction systems arising from chemical processes. The LNK solver combines a high-order spectral Petrov-Galerkin (SPG) method with a new, parallel preconditioner tailored for the large-scale saddle-point systems that form subproblems of Sequential Quadratic Programming (SQP) methods. To establish proof-of-concept, a case study uses a simple parallel MATLAB implementation of the preconditioner with 10 cores. As a step towards real-time control, the results demonstrate that large-scale diffusion-reaction optimization problems with more than 106 unknowns can be solved efficiently in less than a minute.

A Posteriori Error Estimation for a PDE-Constrained Optimization Problem Involving the Generalized Oseen Equations

We derive globally reliable a posteriori error estimators for a linear-quadratic optimal control problem involving the generalized Oseen equations as state equations; control constraints are also considered. The corresponding local error indicators are locally efficient. The assumptions under which we perform the analysis are such that they can be satisfied for a wide variety of stabilized finite element methods as well as for standard finite element methods. When stabilized methods are considered, no a priori relation between the stabilization terms for the state and adjoint equations is required. If a lower bound for the inf-sup constant is available, a posteriori error estimators that are fully computable and provide guaranteed upper bounds on the norm of the error can be obtained. We illustrate the theory with numerical examples.

An efficient duality-based approach for PDE-constrained sparse optimization

In this paper, elliptic optimal control problems involving the $L^1$-control cost ($L^1$-EOCP) is considered. To numerically discretize $L^1$-EOCP, the standard piecewise linear finite element is employed. However, different from the finite dimensional $l^1$-regularization optimization, the resulting discrete $L^1$-norm does not have a decoupled form. A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the $L^1$-norm. It is clear that this technique will incur an additional error. To avoid the additional error, solving $L^1$-EOCP via its dual, which can be reformulated as a multi-block unconstrained convex composite minimization problem, is considered. Motivated by the success of the accelerated block coordinate descent (ABCD) method for solving large scale convex minimization problems in finite dimensional space, we consider extending this method to $L^1$-EOCP. Hence, an efficient inexact ABCD method is introduced for solving $L^1$-EOCP. The design of this method combines an inexact 2-block majorized ABCD and the recent advances in the inexact symmetric Gauss-Seidel (sGS) technique for solving a multi-block convex composite quadratic programming whose objective contains a nonsmooth term involving only the first block. The proposed algorithm (called sGS-imABCD) is illustrated at two numerical examples. Numerical results not only confirm the finite element error estimates, but also show that our proposed algorithm is more efficient than (a) the ihADMM (inexact heterogeneous alternating direction method of multipliers), (b) the APG (accelerated proximal gradient) method.

Properties of Chance Constraints in Infinite Dimensions with an Application to PDE Constrained Optimization

Chance constraints represent a popular tool for finding decisions that enforce the satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in finite dimensions. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties as well as a stability result to an infinite dimensional setting. The abstract results are applied to a simple PDE constrained control problem subject to (uniform) state chance constraints.

A rational deferred correction approach to parabolic optimal control problems

The accurate and efficient solution of time-dependent PDE-constrained optimization problems is a challenging task, in large part due to the very high dimension of the matrix systems that need to be solved. We devise a new deferred correction method for coupled systems of time-dependent PDEs, allowing one to iteratively improve the accuracy of low-order time stepping schemes. We consider two variants of our method, a splitting and a coupling version, and analyze their convergence properties. We then test our approach on a number of PDE-constrained optimization problems. We obtain solution accuracies far superior to that achieved when solving a single discretized problem, in particular in cases where the accuracy is limited by the time discretization. Our approach allows for the direct reuse of existing solvers for the resulting matrix systems, as well as state-of-the-art preconditioning strategies.

Efficient time domain decomposition algorithms for parabolic PDE-constrained optimization problems

Optimization with time-dependent partial differential equations (PDEs) as constraints appears in many science and engineering applications. The associated first-order necessary optimality system consists of one forward and one backward time-dependent PDE coupled with optimality conditions. An optimization process by using the one-shot method determines the optimal control, state and adjoint state at once, with the cost of solving a large scale, fully discrete optimality system. Hence, such a one-shot method could easily become computationally prohibitive when the time span is long or time step is small. To overcome this difficulty, we propose several time domain decomposition algorithms for improving the computational efficiency of the one-shot method. In these algorithms, the optimality system is split into many small subsystems over a much smaller time interval, which are coupled by appropriate continuity matching conditions. Both one-level and two-level multiplicative and additive Schwarz algorithms are developed for iteratively solving the decomposed subsystems in parallel. In particular, the convergence of the one-level, non-overlapping algorithms is proved. The effectiveness of our proposed algorithms is demonstrated by both 1D and 2D numerical experiments, where the developed two-level algorithms show convergence rates that are scalable with respect to the number of subdomains.

Analysis on inexact block diagonal preconditioners for elliptic PDE-constrained optimization problems

By using the Galerkin finite element method, the distributed control problems can be discretized into a saddle point problem with a coefficient matrix of block three-by-three, which can be reduced to a linear system with lower order. We first introduce a class of inexact block diagonal preconditioners and estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices, respectively. Based on the Cholesky decomposition of the known matrices, we also analyze a lower triangular preconditioner to accelerate the minimal residual method for the reduced linear system and discuss its real and complex eigenvalues respectively. Moreover, these bounds do not rely on the regularization parameter and the eigenvalues of the matrices in the discrete system. Numerical experiments are also presented to demonstrate the effectiveness and robustness of the two new preconditioners.

Preconditioning PDE-constrained optimization with [formula omitted]-sparsity and control constraints

PDE-constrained optimization aims at finding optimal setups for partial differential equations so that relevant quantities are minimized. Including nonsmooth L1 sparsity promoting terms in the formulation of such problems results in more practically relevant computed controls but adds more challenges to the numerical solution of these problems. The needed L1-terms as well as additional inclusion of box control constraints require the use of semismooth Newton methods. We propose robust preconditioners for different formulations of the Newton equation. With the inclusion of a line-search strategy and an inexact approach for the solution of the linear systems, the resulting semismooth Newton’s method is reliable for practical problems. Our results are underpinned by a theoretical analysis of the preconditioned matrix. Numerical experiments illustrate the robustness of the proposed scheme.

Some applications of weighted norm inequalities to the error analysis of PDE-constrained optimization problems

The purpose of this work is to illustrate how the theory of Muckenhoupt weights, Muckenhoupt weighted Sobolev spaces and the corresponding weighted norm inequalities can be used in the analysis and discretization of PDE constrained optimization problems. We consider: a linear quadratic constrained optimization problem where the state solves a nonuniformly elliptic equation; a problem where the cost involves pointwise observations of the state and one where the state has singular sources, e.g. point masses. For all three examples we propose and analyze numerical schemes and provide error estimates in two and three dimensions. While some of these problems might have been considered before in the literature, our approach allows for a simpler, Hilbert space-based, analysis and discretization and further generalizations.

Finite horizon model predictive control of electrowetting on dielectric with pinning

A time-discrete spatially-continuous electrowetting on dielectric (EWOD) model with contact line pinning is considered as the state system in an optimal control framework. The pinning model is based on a complementarity condition. In addition to the physical variables describing velocity, pressure, and voltage, the solid-liquid-air interface, i.e., the contact line, arises as a geometric variable that evolves in time. Due to the complementarity condition, the resulting optimal control of a free boundary problem is thus a mathematical program with equilibrium constraints (MPEC) in function space. In order to cope with the geometric variable, a finite horizon model predictive control approach is proposed. Dual stationarity conditions are derived by applying a regularization procedure, exploiting techniques from PDE-constrained optimization, and then passing to the limit in the regularization parameters. Moreover, a function-space-based numerical procedure is developed by following the theoretical limit argument used in the derivation of the dual stationarity conditions. The performance of the algorithm is demonstrated by several examples; including barycenter matching and trajectory tracking.

Fast interior point solution of quadratic programming problems arising from PDE-constrained optimization

Interior point methods provide an attractive class of approaches for solving linear, quadratic and nonlinear programming problems, due to their excellent efficiency and wide applicability. In this paper, we consider PDE-constrained optimization problems with bound constraints on the state and control variables, and their representation on the discrete level as quadratic programming problems. To tackle complex problems and achieve high accuracy in the solution, one is required to solve matrix systems of huge scale resulting from Newton iteration, and hence fast and robust methods for these systems are required. We present preconditioned iterative techniques for solving a number of these problems using Krylov subspace methods, considering in what circumstances one may predict rapid convergence of the solvers in theory, as well as the solutions observed from practical computations.

Reproducing kernel Hilbert spaces and variable metric algorithms in PDE-constrained shape optimization

In this paper we investigate and compare different gradient algorithms designed for the domain expression of the shape derivative. Our main focus is to examine the usefulness of kernel reproducing Hilbert spaces for PDE-constrained shape optimization problems. We show that radial kernels provide convenient formulas for the shape gradient that can be efficiently used in numerical simulations. The shape gradients associated with radial kernels depend on a so-called smoothing parameter that allows a smoothness adjustment of the shape during the optimization process. Besides, this smoothing parameter can be used to modify the movement of the shape. The theoretical findings are verified in a number of numerical experiments.

Adaptive multilevel trust-region methods for time-dependent PDE-constrained optimization

We present a class of adaptive multilevel trust-region methods for the efficient solution of optimization problems governed by time-dependent nonlinear partial differential equations with control constraints. The algorithm is based on the ideas of the adaptive multilevel inexact SQP-method from \[26], \[27]. It is in particular well suited for problems with time-dependent PDE constraints. Instead of the quasi-normal step in a classical SQP method which results in solving the linearized PDE sufficiently well, in this algorithm a (nonlinear) solver is applied to the current discretization of the PDE. Moreover, different discretizations and solvers for the PDE and the adjoint PDE may be applied. The resulting inexactness of the reduced gradient in the current discretization is controlled within the algorithm. Thus, highly efficient PDE solvers can be coupled with the proposed optimization framework. The algorithm starts with a coarse discretization of the underlying optimization problem and provides during the optimization process implementable criteria for an adaptive refinement strategy of the current discretization based on error estimators. We prove global convergence to a stationary point of the infinite-dimensional problem. Moreover, we illustrate how the adaptive refinement strategy of the algorithm can be implemented by using a posteriori error estimators for the state and the adjoint equation. Numerical results for a semilinear parabolic PDE-constrained problem with pointwise control constraints are presented.

A comparative study of full-wave inversion and local time-of-flight approaches in elastography

Most of the current techniques for elastography rely on a pointwise measurement of the induced shear waves. A common assumption in these methods is plane wave propagation in an unbounded domain and local homogeneity of the medium. Because of these simplifying assumptions, complex wave patterns, boundary conditions, and interfaces can present significant challenges to these methods. On the other hand, general nonlinear optimization approaches with PDE constraints relax the underlying assumptions of planar waves and unbounded domains and, hence, can handle very general conditions. However, this generality usually leads to a higher computationally expense as compared to time-of-flight methods. Therefore, the added computational expense and higher complexity of these optimization approaches needs to be justified in the context of elastography. In this work, we present a recent study that compares shear modulus reconstructions obtained with a PDE-constrained optimization approach with a conventional shear wave...

Massively parallel structural acoustics for forward and inverse problems

Three-dimensional structural acoustic simulations on highly complex structural models that are immersed in infinite and semi-infinite acoustic domains typically lead to large numbers of degrees of freedom that cannot be solved with commercial software packages. Typical applications include underwater acoustic simulation of submerged structures, and reverberation testing of aerospace structures. Many of these applications of interest involve large acoustic domains and complex 3D structures, thus making a finite element solution an attractive option. In addition, unknown parameters in the models can be mitigated with the solution of inverse problems. In this talk, we will discuss recent research efforts in Sierra-SD in the area of structural acoustics and will also present a partial differential equation (PDE) constrained optimization approach for solving inverse problems in structural acoustics that uses Sierra-SD for solving the forward and adjoint problems. Inverse problems are commonly encountered in structural acoustics, where accelerometer and/or microphone pressures are measured experimentally, and it is desired to characterize the acoustic sources, material parameters, and/or boundary conditions that produced the measurements. With a PDE-constrained optimization approach, the scalability of Sierra-SD can be leveraged for solving inverse problems. We will present results on the application of Sierra-SD on several large-scale structural acoustic applications examples of interest.