An Efficient Linear Solver for Nonlinear Parameter Identification Problems

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.

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“…We consider an inverse elliptic problem [14]: Find the coefficient function ρ(x) in the system −∇ · (ρ∇u) = f, x ∈ , u(x) = 0, x ∈ ∂ .…”

Section: Msc2000: 65n21 65n55 65y05mentioning

confidence: 99%

“…In this paper, our algorithms are not based on the structure of (13), but on the structure of (15), which is based on the ordering scheme (11) + (14). For the purpose of parallel processing, the mesh points are ordered subdomain by subdomain.…”

Section: Scalable Solversmentioning

confidence: 99%

Parallel overlapping domain decomposition methods for coupled inverse elliptic problems

Cai

Liu

Zou

2009

CAMCoS

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“…This is the case when the saddle-point problems arise from the domain decomposition method with Lagrange multiplier [27], [34], or from the Lagrange multiplier formulations for optimization problems [25] and the parameter identification [16], [33]. But the algorithm still does not seem satisfactory, as its convergence can be guaranteed only under some restriction; see (4.2) in [29].…”

Section: Then Compute the Relaxation Parametermentioning

confidence: 99%

Nonlinear Inexact Uzawa Algorithms for Linear and Nonlinear Saddle-point Problems

Zou

2006

SIAM J. Optim.

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Abstract. This paper proposes some nonlinear Uzawa methods for solving linear and nonlinear saddle-point problems. A nonlinear inexact Uzawa algorithm is first introduced for linear saddlepoint problems. Two different PCG techniques are allowed in the inner and outer iterations of the algorithm. This algorithm is then extended for a class of nonlinear saddle-point problems arising from some convex optimization problems with linear constraints. For this extension, some PCG method used in the inner iteration needs to be carefully constructed so that it converges in a certain energy norm instead of the usual l 2 -norm. It is shown that the new algorithm converges under some practical conditions and there is no need for any a priori estimates on the minimal and maximal eigenvalues of the two local preconditioned systems involved. The two new methods perform more efficiently than the existing methods in the cases where no good preconditioners are available for the Schur complements.

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“…So far, a lot of algorithms for solving the continuous parameter identification problem have been worked out, among which, three kinds of methods can be used. One of them is termed as the traditional mathematic and physical methods [1][2][3][4][5]. The second one is known as the evolutionary algorithms [6][7][8][9][10], and the numerical experiments indicate that these algorithms are good to solve inverse problem.…”

Section: Introductionmentioning

confidence: 99%

Evolutionary Algorithm for Identifying Discontinuous Parameters of Inverse Problems

Jiang

Kang

2007

Computational Science – ICCS 2007

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Abstract. In this paper, we make an investigation into the discontinuous parameter identification in the case of elliptic problem. The discontinuous parameter is identified by evolutionary algorithm for the first time. For this kind of problem, we present a two-level evolutionary algorithm. The first level is the evolution for discontinuous point and the second level is the evolution for parameter. The numerical experiments suggest that the algorithm carries such features as good stability and adaptability and is not very sensitive to the noise in observation data as well.

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An Efficient Linear Solver for Nonlinear Parameter Identification Problems

Cited by 44 publications

References 15 publications

Parallel overlapping domain decomposition methods for coupled inverse elliptic problems

Parallel overlapping domain decomposition methods for coupled inverse elliptic problems

Nonlinear Inexact Uzawa Algorithms for Linear and Nonlinear Saddle-point Problems

Evolutionary Algorithm for Identifying Discontinuous Parameters of Inverse Problems

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