The Experts below are selected from a list of 15843 Experts worldwide ranked by ideXlab platform
Sheehan Olver - One of the best experts on this subject based on the ideXlab platform.
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the automatic solution of Partial Differential equations using a global spectral method
Journal of Computational Physics, 2015Co-Authors: Alex Townsend, Sheehan OlverAbstract:A spectral method for solving linear Partial Differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of Partial Differential Operators and the one-dimensional ultraspherical spectral method. If a Partial Differential Operator is of splitting rank 2, such as the Operator associated with Poisson or Helmholtz, the corresponding PDE is solved via a generalized Sylvester matrix equation, and a bivariate polynomial approximation of the solution of degree ( n x , n y ) is computed in O ( ( n x n y ) 3 / 2 ) operations. Partial Differential Operators of splitting rank ?3 are solved via a linear system involving a block-banded matrix in O ( min ? ( n x 3 n y , n x n y 3 ) ) operations. Numerical examples demonstrate the applicability of our 2D spectral method to a broad class of PDEs, which includes elliptic and dispersive time-evolution equations. The resulting PDE solver is written in Matlab and is publicly available as part of Chebfun. It can resolve solutions requiring over a million degrees of freedom in under 60 seconds. An experimental implementation in the Julia language can currently perform the same solve in 10 seconds.
R.j.p. De Figueiredo - One of the best experts on this subject based on the ideXlab platform.
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A unified approach to optimal image interpolation problems based on linear Partial Differential equation models
IEEE Transactions on Image Processing, 1993Co-Authors: G. Chen, R.j.p. De FigueiredoAbstract:The unified approach to optimal image interpolation problems presented provides a constructive procedure for finding explicit and closed-form optimal solutions to image interpolation problems when the type of interpolation can be either spatial or temporal-spatial. The unknown image is reconstructed from a finite set of sampled data in such a way that a mean-square error is minimized by first expressing the solution in terms of the reproducing kernel of a related Hilbert space, and then constructing this kernel using the fundamental solution of an induced linear Partial Differential equation, or the Green's function of the corresponding self-adjoint Operator. It is proved that in most cases, closed-form fundamental solutions (or Green's functions) for the corresponding linear Partial Differential Operators can be found in the general image reconstruction problem described by a first- or second-order linear Partial Differential Operator. An efficient method for obtaining the corresponding closed-form fundamental solutions (or Green's functions) of the Operators is presented. A computer simulation demonstrates the reconstruction procedure.
Alex Townsend - One of the best experts on this subject based on the ideXlab platform.
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the automatic solution of Partial Differential equations using a global spectral method
Journal of Computational Physics, 2015Co-Authors: Alex Townsend, Sheehan OlverAbstract:A spectral method for solving linear Partial Differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of Partial Differential Operators and the one-dimensional ultraspherical spectral method. If a Partial Differential Operator is of splitting rank 2, such as the Operator associated with Poisson or Helmholtz, the corresponding PDE is solved via a generalized Sylvester matrix equation, and a bivariate polynomial approximation of the solution of degree ( n x , n y ) is computed in O ( ( n x n y ) 3 / 2 ) operations. Partial Differential Operators of splitting rank ?3 are solved via a linear system involving a block-banded matrix in O ( min ? ( n x 3 n y , n x n y 3 ) ) operations. Numerical examples demonstrate the applicability of our 2D spectral method to a broad class of PDEs, which includes elliptic and dispersive time-evolution equations. The resulting PDE solver is written in Matlab and is publicly available as part of Chebfun. It can resolve solutions requiring over a million degrees of freedom in under 60 seconds. An experimental implementation in the Julia language can currently perform the same solve in 10 seconds.
Ian H Sloan - One of the best experts on this subject based on the ideXlab platform.
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analysis of quasi monte carlo methods for elliptic eigenvalue problems with stochastic coefficients
Numerische Mathematik, 2019Co-Authors: Alexander D Gilbert, Frances Y. Kuo, Ivan G Graham, Robert Scheichl, Ian H SloanAbstract:We consider the forward problem of uncertainty quantification for the generalised Dirichlet eigenvalue problem for a coercive second order Partial Differential Operator with random coefficients, motivated by problems in structural mechanics, photonic crystals and neutron diffusion. The PDE coefficients are assumed to be uniformly bounded random fields, represented as infinite series parametrised by uniformly distributed i.i.d. random variables. The expectation of the fundamental eigenvalue of this problem is computed by (a) truncating the infinite series which define the coefficients; (b) approximating the resulting truncated problem using lowest order conforming finite elements and a sparse matrix eigenvalue solver; and (c) approximating the resulting finite (but high dimensional) integral by a randomly shifted quasi-Monte Carlo lattice rule, with specially chosen generating vector. We prove error estimates for the combined error, which depend on the truncation dimension s, the finite element mesh diameter h, and the number of quasi-Monte Carlo samples N. Under suitable regularity assumptions, our bounds are of the particular form $${\mathcal {O}}(h^2 + N^{-1 + \delta })$$ , where $$\delta > 0$$ is arbitrary and the hidden constant is independent of the truncation dimension, which needs to grow as $$h\rightarrow 0$$ and $$N \rightarrow \infty $$ . As for the analogous PDE source problem, the conditions under which our error bounds hold depend on a parameter $$p \in (0, 1)$$ representing the summability of the terms in the series expansions of the coefficients. Although the eigenvalue problem is nonlinear, which means it is generally considered harder than the source problem, in almost all cases ( $$p \ne 1$$ ) we obtain error bounds that converge at the same rate as the corresponding rate for the source problem. The proof involves a detailed study of the regularity of the fundamental eigenvalue as a function of the random parameters. As a key intermediate result in the analysis, we prove that the spectral gap (between the fundamental and the second eigenvalues) is uniformly positive over all realisations of the random problem.
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analysis of quasi monte carlo methods for elliptic eigenvalue problems with stochastic coefficients
arXiv: Numerical Analysis, 2018Co-Authors: Alexander D Gilbert, Frances Y. Kuo, Ivan G Graham, Robert Scheichl, Ian H SloanAbstract:We consider the forward problem of uncertainty quantification for the generalised Dirichlet eigenvalue problem for a coercive second order Partial Differential Operator with random coefficients, motivated by problems in structural mechanics, photonic crystals and neutron diffusion. The PDE coefficients are assumed to be uniformly bounded random fields, represented as infinite series parametrised by uniformly distributed i.i.d. random variables. The expectation of the fundamental eigenvalue of this problem is computed by (a) truncating the infinite series which define the coefficients; (b) approximating the resulting truncated problem using lowest order conforming finite elements and a sparse matrix eigenvalue solver; and (c) approximating the resulting finite (but high dimensional) integral by a randomly shifted quasi-Monte Carlo lattice rule, with specially chosen generating vector. We prove error estimates for the combined error, which depend on the truncation dimension $s$, the finite element mesh diameter $h$, and the number of quasi-Monte Carlo samples $N$. Under suitable regularity assumptions, our bounds are of the particular form $\mathcal{O}(h^2+N^{-1+\delta})$, where $\delta>0$ is arbitrary and the hidden constant is independent of the truncation dimension, which needs to grow as $h\to 0$ and $N\to\infty$. Although the eigenvalue problem is nonlinear, which means it is generally considered harder than the analogous source problem, in almost all cases we obtain error bounds that converge at the same rate as the corresponding rate for the source problem. The proof involves a detailed study of the regularity of the fundamental eigenvalue as a function of the random parameters. As a key intermediate result in the analysis, we prove that the spectral gap (between the fundamental and the second eigenvalues) is uniformly positive over all realisations of the random problem.
Veli B Shakhmurov - One of the best experts on this subject based on the ideXlab platform.
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embedding Operators and maximal regular Differential Operator equations in banach valued function spaces
Journal of Inequalities and Applications, 2005Co-Authors: Veli B ShakhmurovAbstract:This study focuses on anisotropic Sobolev type spaces associated with Banach spaces , . Several conditions are found that ensure the continuity and compactness of embedding Operators that are optimal regular in these spaces in terms of interpolations of and . In particular, the most regular class of interpolation spaces between , , depending of and order of spaces are found that mixed derivatives belong with values; the boundedness and compactness of Differential Operators from this space to -valued spaces are proved. These results are applied to Partial Differential-Operator equations with parameters to obtain conditions that guarantee the maximal regularity uniformly with respect to these parameters.
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maximal regular boundary value problems in banach valued weighted space
Boundary Value Problems, 2005Co-Authors: Ravi P. Agarwal, Martin Bohner, Veli B ShakhmurovAbstract:This study focuses on nonlocal boundary value problems for elliptic ordinary and Partial Differential-Operator equations of arbitrary order, defined in Banach-valued function spaces. The region considered here has a varying bound and depends on a certain parameter. Several conditions are obtained that guarantee the maximal regularity and Fredholmness, estimates for the resolvent, and the completeness of the root elements of Differential Operators generated by the corresponding boundary value problems in Banach-valued weighted spaces. These results are applied to nonlocal boundary value problems for regular elliptic Partial Differential equations and systems of anisotropic Partial Differential equations on cylindrical domain to obtain the algebraic conditions that guarantee the same properties.