Numerical homogenization and model order reduction for multiscale inverse problems

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 parametrization of the microscale tensor (e.g. volume fraction). Departing from the full fine scale model that 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 with 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 finding.

MATHICSE Technical Report : Numerical homogenization and model order reduction for multiscale inverse problems

Assyr Abdulle, Andrea Di Blasio

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 parametrization of the microscaletensor (e.g.volume fraction). Departing from the full fine scale model that would require meshre solution 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 with 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 finding.

MATHICSE2016

Numerical Homogenization And Model Order Reduction For Multiscale Inverse Problems

Assyr Abdulle, Andrea Di Blasio

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.

SIAM PUBLICATIONS2019

Fully discrete analysis of the heterogeneous multiscale method for elliptic problems with multiple scales

Assyr Abdulle, Yun Bai

A fully discrete analysis of the finite element heterogeneous multiscale method (FE-HMM) for elliptic problems with N+1 well-separated scales is discussed. The FE-HMM is a numerical homogenization method that relies on a macroscopic scheme (macro FEM) for the approximation of the effective solution corresponding to the multiscale problem. The effective data are recovered from micro scale computation (micro FEM) on sampling domains located at appropriate quadrature points of the macroscopic mesh. At the macroscopic level, the numerical method can be seen as an FEM with numerical quadrature for a modified effective problem, hence variational crimes are made when designing this method. Up to now, the method has been analysed for two scales and the micro FEM was assumed to be conforming. For more than two scales, variational crimes are committed also at the intermediate (meso) scales and the effective data of the macroscopic scheme are obtained from a cascade of FEMs with numerical integration, which require a careful analysis. Numerical experiments for three-scale problems illustrate the theoretical convergence rates.

Oxford Univ Press2015