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Register a new userOverlapping Domain Decomposition Preconditioner for Integral Equations
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Added by
Chao Chen
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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The discretization of certain integral equations, e.g., the first-kind Fredholm equation of Laplaces equation, leads to symmetric positive-definite linear systems, where the coefficient matrix is dense and often ill-conditioned. We introduce a new preconditioner based on a novel overlapping domain decomposition that can be combined efficiently with fast direct solvers. Empirically, we observe that the condition number of the preconditioned system is $O(1)$, independent of the problem size. Our domain decomposition is designed so that we can construct approximate factorizations of the subproblems efficiently. In particular, we apply the recursive skeletonization algorithm to subproblems associated with every subdomain. We present numerical results on problem sizes up to $16,384^2$ in 2D and $256^3$ in 3D, which were solved in less than 16 hours and three hours, respectively, on an Intel Xeon Platinum 8280M.
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In ptychography experiments, redundant scanning is usually required to guarantee the stable recovery, such that a huge amount of frames are generated, and thus it poses a great demand of parallel computing in order to solve this large-scale inverse problem. In this paper, we propose the overlapping Domain Decomposition Methods(DDMs) to solve the nonconvex optimization problem in ptychographic imaging. They decouple the problem defined on the whole domain into subproblems only defined on the subdomains with synchronizing information in the overlapping regions of these subdomains,thus leading to highly parallel algorithms with good load balance. More specifically, for the nonblind recovery (with known probe in advance), by enforcing the continuity of the overlapping regions for the image (sample), the nonlinear optimization model is established based on a novel smooth-truncated amplitude-Gaussian metric (ST-AGM). Such metric allows for fast calculation of the proximal mapping with closed form, and meanwhile provides the possibility for the convergence guarantee of the first-order nonconvex optimization algorithm due to its Lipschitz smoothness. Then the Alternating Direction Method of Multipliers (ADMM) is utilized to generate an efficient Overlapping Domain Decomposition based Ptychography algorithm(OD2P) for the two-subdomain domain decomposition (DD), where all subproblems can be computed with close-form solutions.Due to the Lipschitz continuity for the gradient of the objective function with ST-AGM, the convergence of the proposed OD2P is derived under mild conditions. Moreover, it is extended to more general case including multiple-subdomain DD and blind recovery. Numerical experiments are further conducted to show the performance of proposed algorithms, demonstrating good convergence speed and robustness to the noise.
In this paper, we propose a novel overlapping domain decomposition method that can be applied to various problems in variational imaging such as total variation minimization. Most of recent domain decomposition methods for total variation minimization adopt the Fenchel--Rockafellar duality, whereas the proposed method is based on the primal formulation. Thus, the proposed method can be applied not only to total variation minimization but also to those with complex dual problems such as higher order models. In the proposed method, an equivalent formulation of the model problem with parallel structure is constructed using a custom overlapping domain decomposition scheme with the notion of essential domains. As a solver for the constructed formulation, we propose a decoupled augmented Lagrangian method for untying the coupling of adjacent subdomains. Convergence analysis of the decoupled augmented Lagrangian method is provided. We present implementation details and numerical examples for various model problems including total variation minimizations and higher order models.
The magnetohydrodynamics (MHD) equations are generally known to be difficult to solve numerically, due to their highly nonlinear structure and the strong coupling between the electromagnetic and hydrodynamic variables, especially for high Reynolds and coupling numbers. In this work, we present a scalable augmented Lagrangian preconditioner for a finite element discretization of the $mathbf{B}$-$mathbf{E}$ formulation of the incompressible viscoresistive MHD equations. For stationary problems, our solver achieves robust performance with respect to the Reynolds and coupling numbers in two dimensions and good results in three dimensions. We extend our method to fully implicit methods for time-dependent problems which we solve robustly in both two and three dimensions. Our approach relies on specialized parameter-robust multigrid methods for the hydrodynamic and electromagnetic blocks. The scheme ensures exactly divergence-free approximations of both the velocity and the magnetic field up to solver tolerances. We confirm the robustness of our solver by numerical experiments in which we consider fluid and magnetic Reynolds numbers and coupling numbers up to 10,000 for stationary problems and up to 100,000 for transient problems in two and three dimensions.
This paper proposes a deep-learning-based domain decomposition method (DeepDDM), which leverages deep neural networks (DNN) to discretize the subproblems divided by domain decomposition methods (DDM) for solving partial differential equations (PDE). Using DNN to solve PDE is a physics-informed learning problem with the objective involving two terms, domain term and boundary term, which respectively make the desired solution satisfy the PDE and corresponding boundary conditions. DeepDDM will exchange the subproblem information across the interface in DDM by adjusting the boundary term for solving each subproblem by DNN. Benefiting from the simple implementation and mesh-free strategy of using DNN for PDE, DeepDDM will simplify the implementation of DDM and make DDM more flexible for complex PDE, e.g., those with complex interfaces in the computational domain. This paper will firstly investigate the performance of using DeepDDM for elliptic problems, including a model problem and an interface problem. The numerical examples demonstrate that DeepDDM exhibits behaviors consistent with conventional DDM: the number of iterations by DeepDDM is independent of network architecture and decreases with increasing overlapping size. The performance of DeepDDM on elliptic problems will encourage us to further investigate its performance for other kinds of PDE and may provide new insights for improving the PDE solver by deep learning.
In this paper, a two-level additive Schwarz preconditioner is proposed for solving the algebraic systems resulting from the finite element approximations of space fractional partial differential equations (SFPDEs). It is shown that the condition number of the preconditioned system is bounded by C(1+H/delta), where H is the maximum diameter of subdomains and delta is the overlap size among the subdomains. Numerical results are given to support our theoretical findings.
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