The subject of this talk is targeted grid optimization for elliptic and time-domain problems arising in remote sensing (geophysics, computed topography, etc.), where the solution is needed only at few receiver points. The simplest and most common discretization of the second derivatives for the solution of one-dimensional elliptic equations is the three-point stencil. Such an approximation applied to two- or three-dimensional Laplace equations yields five-point or seven-point schemes, respectively, with second order convergence. Finite-difference schemes with larger stencils are traditionally used to increase the convergence order. The drawback is that the computational cost of such schemes grows proportional to their stencil. Our approach can be viewed as an extension of the conception of the Gaussian quadratures rules to the second order finite-difference schemes. A standard Gaussian k-point quadrature for numerical integration is chosen to be exact for 2k polynomials, and an optimal grid with k nodes is chosen to match some 2k functionals of the solution. To solve this problem we employ methods of rational approximation, linear algebra and inverse spectral problem. The grid optimization yields EXPONENTIAL superconvergence of the standard second order schemes at the prescribed points COST FREE . This approach has been successfully applied to the approximation of elliptic and hyperbolic problems including ones in unbounded domains. We present theoretical analysis together with nontrivial numerical examples and discuss some open questions.

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