Metric based up-scaling

We consider divergence form elliptic operators in dimension $n\geq 2$ with $L^\infty$ coefficients. Although solutions of these operators are only H\"{o}lder continuous, we show that they are differentiable ($C^{1,\alpha}$) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales by transferring a new metric in addition to traditional averaged (homogenized) quantities from subgrid scales into computational scales and error bounds can be given. This numerical homogenization method can also be used as a compression tool for differential operators.Comment: Final version. Accepted for publication in Communications on Pure and Applied Mathematics. Presented at CIMMS (March 2005), Socams 2005 (April), Oberwolfach, MPI Leipzig (May 2005), CIRM (July 2005). Higher resolution figures are available at http://www.acm.caltech.edu/~owhadi

Developments in the Extended Finite Element Method and Algebraic Multigrid for Solid Mechanics Problems Involving Discontinuities

In this dissertation, some contributions related to computational modeling and solution of solid mechanics problems involving discontinuities are discussed. The main tool employed for discrete modeling of discontinuities is the extended finite element method and the primary solution method discussed is the algebraic multigrid. The extended finite element method has been shown to be effective for both weak and strong discontinuities. With respect to weak discontinuities, a new approach that couples the extended finite element method with Monte Carlo simulations with the goal of quantifying uncertainty in homogenization of material properties of random microstructures is presented. For accelerated solution of linear systems arising from problems involving cracks, several new methods involving the algebraic multigrid are presented. In the first approach, the Schur complement of the linear system arising from XFEM is used to develop a Hybrid-AMG method such that crack-conforming aggregates are formed. Another alternative approach involves transforming the original linear system into a modified system that is amenable for a direct application of algebraic multigrid. It is shown that if only Heaviside-enrichments are present, a simple transformation based on the phantom-node approach is available, which decouples the linear system along the discontinuities such that crack conforming aggregates are automatically generated via smoother aggregation algebraic multigrid. Various numerical examples are presented to verify the accuracy of the resulting solutions and the convergence properties of the proposed algorithms. The parallel scalability performance of the implementation are also discussed

Bayesian Numerical Homogenization

Numerical homogenization, i.e. the finite-dimensional approximation of solution spaces of PDEs with arbitrary rough coefficients, requires the identification of accurate basis elements. These basis elements are oftentimes found after a laborious process of scientific investigation and plain guesswork. Can this identification problem be facilitated? Is there a general recipe/decision framework for guiding the design of basis elements? We suggest that the answer to the above questions could be positive based on the reformulation of numerical homogenization as a Bayesian Inference problem in which a given PDE with rough coefficients (or multi-scale operator) is excited with noise (random right hand side/source term) and one tries to estimate the value of the solution at a given point based on a finite number of observations. We apply this reformulation to the identification of bases for the numerical homogenization of arbitrary integro-differential equations and show that these bases have optimal recovery properties. In particular we show how Rough Polyharmonic Splines can be re-discovered as the optimal solution of a Gaussian filtering problem.Comment: 22 pages. To appear in SIAM Multiscale Modeling and Simulatio