10 research outputs found
Fast cubature of high dimensional biharmonic potential based on Approximate Approximations
We derive new formulas for the high dimensional biharmonic potential acting
on Gaussians or Gaussians times special polynomials. These formulas can be used
to construct accurate cubature formulas of an arbitrary high order which are
fast and effective also in very high dimensions. Numerical tests show that the
formulas are accurate and provide the predicted approximation rate (O(h^8)) up
to the dimension 10^7
Approximation of solutions to multidimensional parabolic equations by approximate approximations
We propose a fast method for high order approximations of the solution of n-dimensional parabolic problems over hyper-rectangular domains in the framework of the method of approximate approximations. This approach, combined with separated representations, makes our method effective also in very high dimensions. We report on numerical results illustrating that our formulas are accurate and provide the predicted approximation rate 6 also in high dimensions
A high-order scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator
ArticleThis is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA Journal of Numerical Analysis following peer review. The version of record IMA J Numer Anal (2015) is available online at http://imajna.oxfordjournals.org/content/early/2015/06/16/imanum.drv021The manuscript presents a technique for efficiently solving the classical wave equation, the shallow water equations, and, more generally, equations of the form âu/ât=Luâu/ât=Lu, where LL is a skew-Hermitian differential operator. The idea is to explicitly construct an approximation to the time-evolution operator exp(ÏL)expâĄ(ÏL) for a relatively large time-step ÏÏ. Recently developed techniques for approximating oscillatory scalar functions by rational functions, and accelerated algorithms for computing functions of discretized differential operators are exploited. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many characteristic wavelengths and large speed-ups over existing methods in situations where simulation over long times are required. Numerical examples involving the 2D rotating shallow water equations and the 2D wave equation in an inhomogenous medium are presented, and the method is compared to the 4th order RungeâKutta (RK4) method and to the use of Chebyshev polynomials. The new method achieved high accuracy over long-time intervals, and with speeds that are orders of magnitude faster than both RK4 and the use of Chebyshev polynomials
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
Numerical solution of fractional elliptic stochastic PDEs with spatial white noise
The numerical approximation of solutions to stochastic partial differential equations with additive spatial white noise on bounded domains in R-d is considered. The differential operator is given by the fractional power L-beta, beta is an element of (0, 1) of an integer-order elliptic differential operator L and is therefore nonlocal. Its inverse L-beta is represented by a Bochner integral from the Dunford-Taylor functional calculus. By applying a quadrature formula to this integral representation the inverse fractional-order operator L-beta is approximated by a weighted sum of nonfractional resolvents (I + exp(2yl)L)(-1) at certain quadrature nodes t(j) > 0. The resolvents are then discretized in space by a standard finite element method. This approach is combined with an approximation of the white noise, which is based only on the mass matrix of the finite element discretization. In this way an efficient numerical algorithm for computing samples of the approximate solution is obtained. For the resulting approximation the strong mean-square error is analyzed and an explicit rate of convergence is derived. Numerical experiments for L = kappa(2) - Delta, kappa > 0 with homogeneous Dirichlet boundary conditions on the unit cube (0, 1)(d) in d = 1, 2, 3 spatial dimensions for varying beta is an element of (0, 1) attest to the theoretical results
Low-rank updates and a divide-and-conquer method for linear matrix equations
Linear matrix equations, such as the Sylvester and Lyapunov equations, play
an important role in various applications, including the stability analysis and
dimensionality reduction of linear dynamical control systems and the solution
of partial differential equations. In this work, we present and analyze a new
algorithm, based on tensorized Krylov subspaces, for quickly updating the
solution of such a matrix equation when its coefficients undergo low-rank
changes. We demonstrate how our algorithm can be utilized to accelerate the
Newton method for solving continuous-time algebraic Riccati equations. Our
algorithm also forms the basis of a new divide-and-conquer approach for linear
matrix equations with coefficients that feature hierarchical low-rank
structure, such as HODLR, HSS, and banded matrices. Numerical experiments
demonstrate the advantages of divide-and-conquer over existing approaches, in
terms of computational time and memory consumption
Tensor-Sparsity of Solutions to High-Dimensional Elliptic Partial Differential Equations
A recurring theme in attempts to break the curse of dimensionality in the
numerical approximations of solutions to high-dimensional partial differential
equations (PDEs) is to employ some form of sparse tensor approximation.
Unfortunately, there are only a few results that quantify the possible
advantages of such an approach. This paper introduces a class of
functions, which can be written as a sum of rank-one tensors using a total of
at most parameters and then uses this notion of sparsity to prove a
regularity theorem for certain high-dimensional elliptic PDEs. It is shown,
among other results, that whenever the right-hand side of the elliptic PDE
can be approximated with a certain rate in the norm of
by elements of , then the solution can be
approximated in from to accuracy
for any . Since these results require
knowledge of the eigenbasis of the elliptic operator considered, we propose a
second "basis-free" model of tensor sparsity and prove a regularity theorem for
this second sparsity model as well. We then proceed to address the important
question of the extent such regularity theorems translate into results on
computational complexity. It is shown how this second model can be used to
derive computational algorithms with performance that breaks the curse of
dimensionality on certain model high-dimensional elliptic PDEs with
tensor-sparse data.Comment: 41 pages, 1 figur