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EFFICIENT SPECTRAL SPARSE GRID METHODS AND APPLICATIONS TO HIGHDIMENSIONAL ELLIPTIC EQUATIONS II. UNBOUNDED DOMAINS ⋆
"... Abstract. This is the second part in a series of papers on using spectral sparse grid methods for solving higherdimensional PDEs. We extend the basic idea in the first part [18] for solving PDEs in bounded higherdimensional domains to unbounded higherdimensional domains, and apply the new method ..."
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Abstract. This is the second part in a series of papers on using spectral sparse grid methods for solving higherdimensional PDEs. We extend the basic idea in the first part [18] for solving PDEs in bounded higherdimensional domains to unbounded higherdimensional domains, and apply the new method to solve the electronic Schrödinger equation. By using modified mapped Chebyshev functions as basis functions, we construct mapped Chebyshev sparse grid (MCSG) methods which enjoy the following properties: (i) the mapped Chebyshev approach enables us to build sparse grids with Smolyak’s algorithms based on nested, spectrally accurate quadratures, and allows us to build fast transforms between the values at the sparse grid points and the corresponding expansion coefficients; (ii) the mapped Chebyshev basis functions lead to identity mass matrices and very sparse stiffness matrices for problems with constant coefficients, and allow us to construct a matrixvector product algorithm with quasioptimal computational cost even for problems with variable coefficients; and (iii) the resultant linear systems for elliptic equations with constant or variable coefficients can be solved efficiently by using a suitable iterative scheme. Ample numerical results are presented to demonstrate the efficiency and accuracy of the proposed algorithms. Key words: spectral method, sparse grid, unbounded domains, mapped Chebyshev functions, electronic Schrödinger equation 1.
Fast Discrete Fourier Transform on Generalized Sparse Grids
, 2013
"... In this paper, we present an algorithm for trigonometric interpolation of multivariate functions on generalized sparse grids and study its application for the approximation of functions in periodic Sobolev spaces of dominating mixed smoothness. In particular, we derive estimates for the error and t ..."
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In this paper, we present an algorithm for trigonometric interpolation of multivariate functions on generalized sparse grids and study its application for the approximation of functions in periodic Sobolev spaces of dominating mixed smoothness. In particular, we derive estimates for the error and the cost. We construct interpolants with a computational cost complexity which is substantially lower than for the standard full grid case. The associated generalized sparse grid interpolants have the same approximation order as the standard full grid interpolants, provided that certain additional regularity assumptions on the considered functions are fulfilled. Numerical results validate our theoretical findings.
FAST STRUCTURED DIRECT SPECTRAL METHODS FOR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS, I. THE ONEDIMENSIONAL CASE∗
"... Abstract. We study the rank structures of the matrices in Fourier and Chebyshevspectral methods for differential equations with variable coefficients in one dimension. We show analytically that these matrices have a socalled lowrank property, not only for constant or smooth variable coefficients ..."
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Abstract. We study the rank structures of the matrices in Fourier and Chebyshevspectral methods for differential equations with variable coefficients in one dimension. We show analytically that these matrices have a socalled lowrank property, not only for constant or smooth variable coefficients, but also for coefficients with steep gradients and/or high variations (large ratios in their maximumminimum function values). We develop a matrixfree direct spectral solver, which uses only a small number of matrixvector products to construct a structured approximation to the original discretized matrix A, without the need to explicitly form A. This is followed by fast structured matrix factorizations and solutions. The overall direct spectral solver has O(N log2 N) complexity and O(N) memory requirement. Numerical tests for several important but notoriously difficult problems show the superior efficiency and accuracy of our direct spectral solver, especially when iterative methods have severe difficulties in the convergence.
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"... th ip Polynomial chaos Stochastic collocation Nonintrusive spectral projection mi spr etho ans num mat s un tion in tensor product spaces, we construct a sparse pseudospectral approximation method that accurately compu engine el com aking tation simulation results. Additionally, the polynomial surr ..."
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th ip Polynomial chaos Stochastic collocation Nonintrusive spectral projection mi spr etho ans num mat s un tion in tensor product spaces, we construct a sparse pseudospectral approximation method that accurately compu engine el com aking tation simulation results. Additionally, the polynomial surrogate is typically much cheaper to evaluate as a function of the input parametensor grids are formed from univariate point sets with a nesting property, such as the Chebyshev points, the number of points in the union of tensor grids is greatly reduced – although this nesting feature is not necessary for the construction of the sparse grids. The points in the sparse grid can be used as a numerical integration rule [6,7], where the weights are linear combinations of weights from the member tensor grids. Alternatively, the interpolating tensor product Lagrange polynomials constructed on the member tensor grids can be linearly combined in a similar fashion to yield a polynomial surrogate [8], since a linear combination of polynomials is itself a polynomial.
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"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit:
HERMITE SPECTRAL METHOD WITH HYPERBOLIC CROSS APPROXIMATIONS TO HIGHDIMENSIONAL PARABOLIC PDES∗
"... Dedicated to Professor Peter Caines on the occasion of his 68th birthday Abstract. It is wellknown that sparse grid algorithm has been widely accepted as an efficient tool to overcome the “curse of dimensionality ” in some degree. In this note, we first give the error estimate of hyperbolic cross ( ..."
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Dedicated to Professor Peter Caines on the occasion of his 68th birthday Abstract. It is wellknown that sparse grid algorithm has been widely accepted as an efficient tool to overcome the “curse of dimensionality ” in some degree. In this note, we first give the error estimate of hyperbolic cross (HC) approximations with generalized Hermite functions. The exponential convergence in both regular and optimized hyperbolic cross approximations has been shown. Moreover, the error estimate of Hermite spectral method to highdimensional linear parabolic PDEs with HC approximations has been investigated in the properly weighted Korobov spaces. The numerical result verifies the exponential convergence of this approach.
Sparse Pseudospectral Approximation Method
"... Multivariate global polynomial approximations – such as polynomial chaos or stochastic collocation methods – are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudospectral variety of these methods uses a numerical integration rule to approximate the Fouriertyp ..."
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Multivariate global polynomial approximations – such as polynomial chaos or stochastic collocation methods – are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudospectral variety of these methods uses a numerical integration rule to approximate the Fouriertype coefficients of a truncated expansion in orthogonal polynomials. For problems in more than two or three dimensions, a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation. However, when using a sparse rule to approximately integrate these coefficients, one often finds unacceptable errors in the coefficients associated with higher degree polynomials. By reexamining Smolyak’s algorithm and exploiting the connections between interpolation and projection in tensor product spaces, we construct a sparse pseudospectral approximation method that accurately reproduces the coefficients for basis functions that naturally correspond to the sparse grid integration rule. The compelling numerical results show that this is the proper way to use sparse grid integration rules for pseudospectral approximation.