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Concept# Relaxation (iterative method)

Summary

In numerical mathematics, relaxation methods are iterative methods for solving systems of equations, including nonlinear systems.
Relaxation methods were developed for solving large sparse linear systems, which arose as finite-difference discretizations of differential equations. They are also used for the solution of linear equations for linear least-squares problems and also for systems of linear inequalities, such as those arising in linear programming. They have also been developed for solving nonlinear systems of equations.
Relaxation methods are important especially in the solution of linear systems used to model elliptic partial differential equations, such as Laplace's equation and its generalization, Poisson's equation. These equations describe boundary-value problems, in which the solution-function's values are specified on boundary of a domain; the problem is to compute a solution also o

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Alexandre Caboussat, Dimitrios Gourzoulidis

In this article, we address the numerical solution of the Dirichlet problem for the three-dimensional elliptic Monge-Ampere equation using a least-squares/relaxation approach. The relaxation algorithm allows the decoupling of the differential operators from the nonlinearities. Dedicated numerical solvers are derived for the efficient solution of the local optimization problems with cubicly nonlinear equality constraints. The approximation relies on mixed low order finite element methods with regularization techniques. The results of numerical experiments show the convergence of our relaxation method to a convex classical solution if such a solution exists; otherwise they show convergence to a generalized solution in a least-squares sense. These results show also the robustness of our methodology and its ability at handling curved boundaries and non-convex domains.

Two-level domain decomposition methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level preconditioner (or its corresponding smoothing iterative method), which is based on domain decomposition techniques, and a coarse correction step, which relies on a coarse space. The coarse space must properly represent the error components that the chosen one-level method is not capable to deal with. In the literature most of the works introduced efficient coarse spaces obtained as the span of functions defined on the entire space domain of the considered PDE. Therefore, the corresponding two-level preconditioners and iterative methods are defined in volume. In this paper, we use the excellent smoothing properties of domain decomposition methods to define a new class of substructured two-level methods, for which both domain decomposition smoothers and coarse correction steps are defined on the interfaces. This approach has several advantages. On the one hand, the required computational effort is cheaper than the one required by classical volumetric two-level methods. On the other hand, our approach does not require the explicit construction of coarse spaces, and it permits a multilevel extension, which is desirable when the high dimension of the problem or the scarce quality of the coarse space prevents the efficient numerical solution. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.

2021The question addressed is the determination of a glacier’s subglacial topography, given surface topography and mass-balance data. The input data can be obtained relatively easily for a large number of glaciers. Several methods essentially based on the shallow ice approximation are proposed, some of which are extended to Stokes ice flows. Two gradient-free, iterative methods are first introduced, namely the quasi-stationary inverse method, that relies on the apparent surface mass-balance description of glacier dynamics, and the transient inverse method, consisting in the iterative update of the bedrock topography proportionally to the surface topography misfit at the end of the glacier’s considered evolution. Then, an optimal control algorithm is suggested that calculates the bedrock topography and some model parameters from surface observations through the minimization of a regularized misfit functional by means of a Lagrangian method. Numerical validations, along with sensitivity analyses and applications to real-world data are presented for each method.