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Concept# Numerical method

Summary

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.
Let be a well-posed problem, i.e. is a real or complex functional relationship, defined on the cross-product of an input data set and an output data set , such that exists a locally lipschitz function called resolvent, which has the property that for every root of , . We define numerical method for the approximation of , the sequence of problems
with , and for every . The problems of which the method consists need not be well-posed. If they are, the method is said to be stable or well-posed.
Necessary conditions for a numerical method to effectively approximate are that and that behaves like when . So, a numerical method is called consistent if and only if the sequence of functions pointwise converges to on the set of its solutions:
When on the method is said to be strictly consistent.
Denote by a sequence of admissible perturbations of for some numerical method (i.e. ) and with the value such that . A condition which the method has to satisfy to be a meaningful tool for solving the problem is convergence:
One can easily prove that the point-wise convergence of to implies the convergence of the associated method is function.

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Numerical method

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Let be a well-posed problem, i.e. is a real or complex functional relationship, defined on the cross-product of an input data set and an output data set , such that exists a locally lipschitz function called resolvent, which has the property that for every root of , .

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, , , , , , , , ,

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A new numerical method based on numerical homogenization and model order reduction is introduced for the solution of multiscale inverse problems. We consider a class of elliptic problems with highly oscillatory tensors that varies on a microscopic scale. We assume that the micro structure is known and seek to recover a macroscopic scalar parameterization of the microscale tensor (e.g., volume fraction). Departing from the full fine-scale model, which would require mesh resolution for the forward problem down to the finest scale, we solve the inverse problem for a coarse model obtained by numerical homogenization. The input data, i.e., measurement from the Dirichlet-to-Neumann map, are solely based on the original fine-scale model. Furthermore, reduced basis techniques are used to avoid computing effective coefficients for the forward solver at each integration point of the macroscopic mesh. Uniqueness and stability of the effective inverse problem is established based on standard assumptions for the fine-scale model, and a link to this latter model is established by means of G-convergence. A priori error estimates are established for our method. Numerical experiments illustrate the efficiency of the proposed scheme and confirm our theoretical findings.