Multigrid Algorithms for Inverse Problems with Linear Parabolic PDE Constraints

Adavani, Santi; Biros, George

doi:10.1137/070687426

Cited by 24 publications

(32 citation statements)

References 26 publications

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“…In our implementation we follow a discretize-then-optimize approach, that is, we first discretize the problem and then derive the optimality system to obtain a con- systems are similar to optimality systems from linear-quadratic optimal control problems, they can also be solved using advanced iterative algorithms such as multigrid methods [1,3].…”

Section: Numerical Resultsmentioning

confidence: 99%

Directional Sparsity in Optimal Control of Partial Differential Equations

Herzog¹,

Stadler²,

Wachsmuth³

2012

SIAM J. Control Optim.

102

110

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Abstract. We study optimal control problems in which controls with certain sparsity patterns are preferred. For time-dependent problems the approach can be used to find locations for control devices that allow controlling the system in an optimal way over the entire time interval. The approach uses a nondifferentiable cost functional to implement the sparsity requirements; additionally, bound constraints for the optimal controls can be included. We study the resulting problem in appropriate function spaces and present two solution methods of Newton type, based on different formulations of the optimality system. Using elliptic and parabolic test problems we research the sparsity properties of the optimal controls and analyze the behavior of the proposed solution algorithms.

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Section: Numerical Resultsmentioning

confidence: 99%

Directional Sparsity in Optimal Control of Partial Differential Equations

Herzog¹,

Stadler²,

Wachsmuth³

2012

SIAM J. Control Optim.

102

110

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“…We can now proceed using the semismooth Newton approach, solving linear systems of the form ⎡ (1) . .…”

Section: Problem Formulation and Discretizationmentioning

confidence: 99%

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php B995 It is also possible to employ multigrid approaches to such saddle point problems. This class of methods has previously been shown to demonstrate good performance when applied to solve a number of PDE-constrained optimization problems, subject to both steady and transient PDEs [1,2,8,9,23,24,28,29,57].…”

Section: Problem Formulation and Discretizationmentioning

confidence: 99%

“…Note that these complicated looking matrices are actually straightforward to handle as we assume that the mass matrices are lumped for our work. 1 The block M c , which also forms part of the (1, 1)-block of our matrix systems, may be approximated using Chebyshev semi-iteration [16,17,59] for consistent mass matrices or by simple inversion for lumped mass matrices.…”

Section: Problem Formulation and Discretizationmentioning

confidence: 99%

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Fast Iterative Solution of Reaction-Diffusion Control Problems Arising from Chemical Processes

Pearson

Stoll

2013

SIAM J. Sci. Comput.

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PDE-constrained optimization problems, and the development of preconditioned iterative methods for the efficient solution of the arising matrix systems, is a field of numerical analysis that has recently been attracting much attention. In this paper, we analyze and develop preconditioners for matrix systems that arise from the optimal control of reaction-diffusion equations, which themselves result from chemical processes. Important aspects of our solvers are saddle point theory, mass matrix representation, and effective Schur complement approximation, as well as the incorporation of control constraints and application of the outer (Newton) iteration to take into account the nonlinearity of the underlying PDEs. Introduction.A class of problems which has numerous applications within mathematical and physical problems is that of PDE-constrained optimization problems. One field in which these problems can be posed is that of chemical processes [4,19,20,21,22]. In this case the underlying PDEs are reaction-diffusion equations, and therefore the PDE constraints in our formulation are nonlinear PDEs.When solving such reaction-diffusion control problems using a finite element method, and employing a Lagrange-Newton iteration to take account of the nonlinearity involved in the PDEs, the resulting matrix system for each Newton iteration will be large, sparse, and of saddle point structure. It is therefore desirable to devise preconditioned iterative methods to solve these systems efficiently and in such a way that the structure of the matrix is exploited. Work in constructing preconditioners for PDE-constrained optimization problems has been considered for simpler problems previously, for instance, Poisson control [46,47,53], convection-diffusion control [45], Stokes control [36,50,56], and heat equation control [44,54].In this paper, we will consider an optimal control formulation of a reactiondiffusion problem, which generates a symmetric matrix system upon each Newton iteration. (Such an iteration is required to take into account the nonlinear terms within the underlying PDEs.) We will generally search for block triangular preconditioners for the matrix systems we examine, to be used in conjunction with a suitable iterative solver. In order to do this, we will need to approximate the (1, 1)-block by accurately representing the inverse of mass matrices amongst other things, as well as

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“…The numerical implementation of an efficient solver is, in many cases, tailored towards the nature of the control problem, e.g., by accounting for the type and structure of the PDE constraints; see for instance [1, 15] (elliptic), [2, 37, 60, 69] (parabolic), or [13, 19, 55] (hyperbolic). We refer to [14, 21, 39, 49, 51] for an overview on theoretical and algorithmic developments in optimal control and PDE constrained optimization.…”

Section: Introductionmentioning

confidence: 99%

A Semi-Lagrangian Two-Level Preconditioned Newton--Krylov Solver for Constrained Diffeomorphic Image Registration

Mang

Biros

2017

SIAM J. Sci. Comput.

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We propose an efficient numerical algorithm for the solution of diffeomorphic image registration problems. We use a variational formulation constrained by a partial differential equation (PDE), where the constraints are a scalar transport equation. We use a pseudospectral discretization in space and second-order accurate semi-Lagrangian time stepping scheme for the transport equations. We solve for a stationary velocity field using a preconditioned, globalized, matrix-free Newton-Krylov scheme. We propose and test a two-level Hessian preconditioner. We consider two strategies for inverting the preconditioner on the coarse grid: a nested preconditioned conjugate gradient method (exact solve) and a nested Chebyshev iterative method (inexact solve) with a fixed number of iterations. We test the performance of our solver in different synthetic and real-world two-dimensional application scenarios. We study grid convergence and computational efficiency of our new scheme. We compare the performance of our solver against our initial implementation that uses the same spatial discretization but a standard, explicit, second-order Runge-Kutta scheme for the numerical time integration of the transport equations and a single-level preconditioner. Our improved scheme delivers significant speedups over our original implementation. As a highlight, we observe a 20× speedup for a two dimensional, real world multi-subject medical image registration problem.

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Multigrid Algorithms for Inverse Problems with Linear Parabolic PDE Constraints

Cited by 24 publications

References 26 publications

Directional Sparsity in Optimal Control of Partial Differential Equations

Directional Sparsity in Optimal Control of Partial Differential Equations

Fast Iterative Solution of Reaction-Diffusion Control Problems Arising from Chemical Processes

A Semi-Lagrangian Two-Level Preconditioned Newton--Krylov Solver for Constrained Diffeomorphic Image Registration

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