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A Dynamically Adaptive Sparse Grid Method for Quasi-Optimal Interpolation of Multidimensional Analytic Functions
In this work we develop a dynamically adaptive sparse grids (SG) method for
quasi-optimal interpolation of multidimensional analytic functions defined over
a product of one dimensional bounded domains. The goal of such approach is to
construct an interpolant in space that corresponds to the "best -terms"
based on sharp a priori estimate of polynomial coefficients. In the past, SG
methods have been successful in achieving this, with a traditional construction
that relies on the solution to a Knapsack problem: only the most profitable
hierarchical surpluses are added to the SG. However, this approach requires
additional sharp estimates related to the size of the analytic region and the
norm of the interpolation operator, i.e., the Lebesgue constant. Instead, we
present an iterative SG procedure that adaptively refines an estimate of the
region and accounts for the effects of the Lebesgue constant. Our approach does
not require any a priori knowledge of the analyticity or operator norm, is
easily generalized to both affine and non-affine analytic functions, and can be
applied to sparse grids build from one dimensional rules with arbitrary growth
of the number of nodes. In several numerical examples, we utilize our
dynamically adaptive SG to interpolate quantities of interest related to the
solutions of parametrized elliptic and hyperbolic PDEs, and compare the
performance of our quasi-optimal interpolant to several alternative SG schemes
Current-voltage characteristic of parallel-plane ionization chamber with inhomogeneous ionization
The balances of particles and charges in the volume of parallel-plane
ionization chamber are considered. Differential equations describing the
distribution of current densities in the chamber volume are obtained. As a
result of the differential equations solution an analytical form of the
current-voltage characteristic of parallel-plane ionization chamber with
inhomogeneous ionization in the volume is got.Comment: 8 pages, 4 figure
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