Parallel overlapping domain decomposition methods for coupled inverse elliptic problems

Abstract

We study an overlapping domain decomposition method for solving the coupled nonlinear system of equations arising from the discretization of inverse elliptic problems. Most algorithms for solving inverse problems take advantage of the fact that the optimality system has a natural splitting into three components: the state equation for the constraints, the adjoint equation for the Lagrange multipliers, and the equation for the parameter to be identified. Such algorithms often involve interiterations between the three separate solvers, and the intercomponent iteration is sequential. Several fully coupled or so-called one-shot approaches exist, and the main challenges in these approaches are that the system has stronger nonlinearity, and the corresponding Jacobian system is more ill-conditioned, in addition to being three times larger. Here we investigate a class of overlapping NewtonKrylov-Schwarz algorithms for solving such coupled systems, obtained with a pointwise ordering of the variables, and show numerically that, with a reasonably large overlap, the algorithm is capable of finding the solution even with noise and discontinuous coefficients. More importantly, we show that this approach is fully parallel and scalable with respect to the size of the problems.

Two-Level Space–Time Domain Decomposition Methods for Three-Dimensional Unsteady Inverse Source Problems

“…We observe that the reconstructed profiles deteriorate and become oscillatory as the noise level increases. This is expected since the regularized solutions provided by the minimization of functional J ( f ) in (2) become less accurate, so are their numerical approximate solutions obtained from the discretised KKT system (8)- (9). We have tested four different sets of regularization parameters for the H 1 -H 1 regularization in (3), and present the L 2 -norm errors between the reconstructed source f and the exact source function f * at the aforementioned three moments t 1 , t 2 and t 3 .…”

“…Because of its essential sequential feature, the reduced space SQP method is less ideal for parallel computers with a large number of processor cores, compared with the full space SQP methods. Full space methods were studied for steady state problems in [9,11], but for unsteady problems it needs to eliminate the sequential steps in the outer iteration of the SQP and solve the full space-time system as a coupled system. Because of the much larger size of the system, the full space approach may not be suitable for parallel computer systems with a small number of processor cores, but it has fewer sequential steps and thus offers a much higher degree of parallelism required by large scale supercomputers [12].…”

A parallel space-time domain decomposition method for unsteady source inversion problems

“…Various preconditioners have been developed and applied for various elliptic and parabolic systems, such as the (block) Jacobi method, (incomplete) LU factorization, (multiplicative) additive Schwarz method, multigrid method, multilevel method, etc. [7,8,9]. Among these preconditioners the Schwarz type domain decomposition method is shown to have excellent preconditioning effect and parallel scalability [9,31].…”

“…[7,8,9]. Among these preconditioners the Schwarz type domain decomposition method is shown to have excellent preconditioning effect and parallel scalability [9,31].…”

Overlapping domain decomposition methods for linear inverse problems

“…Newton's method was first used in [3] for solving the optimality system of the stabilized minimization of an elliptic identification problem, then an additive Schwarz type preconditioned algorithm was applied to solve the linear system involved at each Newton's iteration. As Newton's method requires the evaluations of the Hessian of the corresponding objective functional, the approach of [3] is applicable only to a very special formulation of the parameter identification problem. In this work we shall develop some DDMs for directly solving the stabilized minimization systems of some typical linear inverse problems so that their convergences do not deteriorate or deteriorate only mildly as the entire degrees of freedom of the optimization system grow.…”