Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

Abstract

We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough (L ∞ ) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution H) minimizing the L 2 norm of the source terms; its (pre-)computation involves minimizing O(H −d ) quadratic (cell) problems on (super-)localized sub-domains of size O(H ln(1/H)). The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic for d ≤ 3, and polyharmonic for d ≥ 4, for the operator − div(a∇·) and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method (O(H) in energy norm and independent from aspect ratios of the mesh formed by the scattered points) is established via the introduction of a new class of higherorder Poincaré inequalities. The method bypasses (pre-)computations on the full domain and naturally generalizes to time dependent problems, it also provides a natural solution to the inverse problem of recovering the solution of a divergence form elliptic equation from a finite number of point measurements.

Numerical upscaling of discrete network models

“…Such a procedure is sometimes referred to as upscaling or homogenization. In this work a numerical upscaling method for discrete networks is presented.There exist several numerical upscaling methods for partial differential equations (PDE) based on the idea of homogenization, such as the Heterogeneous Multiscale Method (HMM) [20], the Multiscale Finite Element Method (MsFEM) [8], and the more recent works [3,17]. The upscaling approach presented in this paper is based on the Localized Orthogonal Decomposition Method (LOD) [15,4], which in turn is inspired by the Variational Multiscale Method (VMM) [9].…”

Fast Eigenpairs Computation with Operator Adapted Wavelets and Hierarchical Subspace Correction

“…Rigorous exponential decay/localization results such as (2.26) have been pioneered in [41] for the localized orthogonal decomposition (LOD) basis functions. Although gamblets are derived from a different perspective (namely, a game theoretic approach), from the numerical point of view, gamblets can be seen as a multilevel generalization of optimal recovery splines [43] and of numerical homogenization basis functions such as RPS (rough polyharmonic splines) [50] and variational multiscale/LOD basis functions [26,41].…”

Multiscale-Spectral GFEM and optimal oversampling

“…These methods include upscaling based on harmonic coordinates and elliptic inequalities [38], [39], elliptic solvers based on H-matrices [10,26], explicit solution of local computations through Bayesian numerical homogenization [37], dimension reduction methods based on global changes of coordinates and MS-FEM for upscaling porous media flows [22], [23], the heterogeneous multiscale methods [19], [20], [24], and an adaptive coarse scale-fine scale projection method [36]. Additional contemporary methods include numerical homogenization based on the flux norm for L ∞ coefficients [12], rough polyharmonic splines [13], subgrid upscaling methods [1] and global Galerkin projection schemes for problems with L ∞ coefficients and homogeneous Dirichlet boundary data [35]. For a coarse mesh of diameter H local bases that deliver order H convergence with O((log(1/H)) d+1 approximation functions are developed in [32].…”