Blind Wavelet Compression of the Solution of a Nonlinear PDE with Singular Forcing Term Within Optimal Order Cost: Stability of Restricted Approximation to Small Errors

Abstract

Level lifting of the wavelet expansion is related to an interpolation result for Sobolev H ^s spaces; the nonlinear N-term approximation is linked with a nonlinear interpolation result for Sobolev spaces W ^s,p noncompactly included into L ². Cohen et al. (2000, Constr. Approx. 16 (1), 85–113) introduced the intermediate notion of restricted approximation. Based on this, we construct an optimal order resolution algorithm extending beyond the linear elliptic case of Cohen et al. (2001, Math. Comput. 70 (233), 27–75), as we illustrate numerically. We undeline that optimal order adaptivity implies the blind compression of the unknown of the PDE. We illustrate on a univariate version of the bivariate PDE $\Delta u + e^{cu} = 0$, c > 0, used to benchmark three nonadaptive multilevel methods Hackbusch (1992, Z. Angew. Math. Mech. 72 (2), 148–151). The adaptiveness of our algorithm is highlighted by the addition in this illustration of a singular forcing term. This term is an element of H ⁻¹ but does not belong to H ^−5/6: more precisely, it is the second derivative of $t\mapsto |3t -1|^{2/3}$. This illustration passed the numerical implementation test $(C\varepsilon^{-0.505}$ flops). The algorithm’s convergence and cost $(\varepsilon^{-d}/(s-1)$ where ɛ is the final error in H ¹ norm) in both univariate (d=1) and bivariate (d=2) general cases is shown to have optimal order, with s less than three. © John Wiley and Sons, Inc.