Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

TL;DR

This paper presents a novel variational homogenization method for elliptic, parabolic, and hyperbolic equations with rough coefficients, using polyharmonic splines and localized computations to achieve accurate, sparse solutions without relying on ergodicity or scale separation.

Contribution

It introduces a new variational approach that generalizes polyharmonic splines for rough coefficients, enabling efficient localized homogenization without ergodic assumptions.

Findings

Method achieves $ ext{O}(H)$ accuracy in energy norm.

Localized computations remain sparse and banded.

Applicable to inverse problems with finite point measurements.

Abstract

We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough ($L^\infty$) 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 $\mathcal{O}(H^{-d})$ quadratic (cell) problems on (super-)localized sub-domains of size $\mathcal{O}(H \ln (1/ H))$. The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic for $d\leq 3$, and polyharmonic for $d\geq 4$, for the operator $-\diiv(a\nabla \cdot)$ and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method ($\mathcal{O}(H)$ 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 higher-order Poincar\'{e} 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.