Abstract
We introduce a near-linear complexity (geometric and meshless/algebraic) multigrid/multiresolution method for PDEs with rough ($L^\infty$) coefficients with rigorous a priori accuracy and performance estimates. The method is discovered through a decision/game theory formulation of the problems of (1) identifying restriction and interpolation operators, (2) recovering a signal from incomplete measurements based on norm constraints on its image under a linear operator, and (3) gambling on the value of the solution of the PDE based on a hierarchy of nested measurements of its solution or source term. The resulting elementary gambles form a hierarchy of (deterministic) basis functions of $H^1_0(\Omega)$ (gamblets) that (1) are orthogonal across subscales/subbands with respect to the scalar product induced by the energy norm of the PDE, (2) enable sparse compression of the solution space in $H^1_0(\Omega)$, and (3) induce an orthogonal multiresolution operator decomposition. The operating diagram of the multig...
Highlights
We introduce a near-linear complexity multigrid/multiresolution method for PDEs with rough (L∞) coefficients with rigorous a priori accuracy and performance estimates
If the energy norm is used to quantify accuracy, the recovery problem can be expressed as finding the function θ of the measurements y minimizing the approximation error infθ sup b ≤1 x − θ(y) A/ b with x = A−1b and y = ΦA−1b
As in Remark 4, the assumption of convexity of the subdomains τi(k) is not necessary to the results presented here and is only used to derive sharper/simpler constants
Summary
We introduce a near-linear complexity (geometric and meshless/algebraic) multigrid/multiresolution method for PDEs with rough (L∞) coefficients with rigorous a priori accuracy and performance estimates.
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