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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 $\mathrm H^s$ spaces; the nonlinear $N$-term approximation is linked with a non-linear interpolation result for Sobolev spaces $\mathrm W^{s,p}$ non-compactly included into $\mathrm L^2$. \cite{CDH} introduced the intermediate notion of restricted approximation. Based on this, we construct an optimal order resolution algorithm extending beyond the linear elliptic case of \cite{CDD98}, 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 $\Delta u+\mathrm e^{cu}=0$, $c{>}0$, used to benchmark three nonadaptive multilevel methods \cite{H}.\\ %At the end, we present an efficient search of relevant coefficients of %the second derivative of the father wavelet in its dual wavelet basis; this %search enlarges the range of the Lebesgue index $p$ allowed for the %application. The adaptiveness of our algorithm is highlighted by the addition in this illustration of a singular forcing term. This term is an element of \(\mathrm H^{-1}\) but does not belong to \(\mathrm H^{-\frac56}\): 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 ($\varepsilon^{-\frac d{s-1}}$ where $\varepsilon$ is the final error in $\mathrm H^1$ norm) in both univariate ($d=1$) and bivariate ($d=2$) general cases is shown to have optimal order, with \(s\) less than three.
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Daniel Goujot. 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. Journal of Scientific Computing, Springer Verlag, 2005, 29 (2), pp.257-297. ⟨10.1007/s10915-005-9003-6⟩. ⟨hal-02677479⟩

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