Results 1  10
of
85
A sparse matrix arithmetic based on Hmatrices
 I. Introduction to Hmatrices, Computing
, 1999
"... ..."
(Show Context)
ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS
, 804
"... Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical ..."
Abstract

Cited by 43 (6 self)
 Add to MetaCart
Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random
Efficient automatic quadrature in 3d Galerkin BEM
, 1996
"... We present cubature methods approximating the surface integrals arising by Galerkin discretization of boundary integral equations on surfaces in R³. This numerical integrator does not depend on the explicit form of the kernel function, the trial and test space, or the surface parametrization. Thus, ..."
Abstract

Cited by 32 (10 self)
 Add to MetaCart
We present cubature methods approximating the surface integrals arising by Galerkin discretization of boundary integral equations on surfaces in R³. This numerical integrator does not depend on the explicit form of the kernel function, the trial and test space, or the surface parametrization. Thus, it is possible to generate the system matrix for a broad class of integral equations just by replacing the subroutine for evaluating the kernel function. We will present formulae to determine the minimal order of the cubature methods for a required accuracy. Emphasize is laid on numerical experiments confirming the theoretical results.
Sparse Grids for Boundary Integral Equations
, 1998
"... The potential of sparse grid discretizations for solving boundary integral equations is studied for the screen problem on a square in IR 3 . Theoretical and numerical results on approximation rates, preconditioning, adaptivity and compression for piecewise constant and linear sparse grid spaces ar ..."
Abstract

Cited by 29 (16 self)
 Add to MetaCart
The potential of sparse grid discretizations for solving boundary integral equations is studied for the screen problem on a square in IR 3 . Theoretical and numerical results on approximation rates, preconditioning, adaptivity and compression for piecewise constant and linear sparse grid spaces are obtained. Classification: 45L10, 65N38, 65R20, 65Y20 Keywords: boundary element method, sparse grids, adaptivity, prewavelets, matrix compression 1 Introduction This is a case study for some special boundary integral equations on a twodimensional manifold \Gamma in IR 3 (screen problems). We will focus on the example of a twodimensional unit square in IR 2 embedded into IR 3 where \Gamma = fx : (x 1 ; x 2 ) 2 [0; 1] 2 ; x 3 = 0g : (1) In general, d\Gamma x stands for the surface Lebesgue measure with respect to the variable x, jxj 2 denotes the Euclidean norm of x, and n x is the vector field of normal vectors associated with \Gamma. We specifically have in mind the single lay...
Approximate kernel kmeans: Solution to large scale kernel clustering
 in Proceedings of the International Conference on Knowledge Discovery and Data mining
"... Digital data explosion mandates the development of scalable tools to organize the data in a meaningful and easily accessible form. Clustering is a commonly used tool for data organization. However, many clustering algorithms designed to handle large data sets assume linear separability of data and h ..."
Abstract

Cited by 26 (5 self)
 Add to MetaCart
(Show Context)
Digital data explosion mandates the development of scalable tools to organize the data in a meaningful and easily accessible form. Clustering is a commonly used tool for data organization. However, many clustering algorithms designed to handle large data sets assume linear separability of data and hence do not perform well on real world data sets. While kernelbased clustering algorithms can capture the nonlinear structure in data, they do not scale well in terms of speed and memory requirements when the number of objects to be clustered exceeds tens of thousands. We propose an approximation scheme for kernel kmeans, termed approximate kernel kmeans, that reduces both the computational complexity and the memory requirements by employing a randomized approach. We show both analytically and empirically that the performance of approximate kernel kmeans is similar to that of the kernel kmeans algorithm, but with dramatically reduced runtime complexity and memory requirements.
TwoLevel Preconditioners for Regularized Inverse Problems I: Theory
 Theory, Numer. Math
, 1998
"... We compare additive and multiplicative Schwarz preconditioners for the iterative solution of regularized linear inverse problems, extending and complementing earlier results of Hackbusch, King, and Rieder. Our main findings are that the classical convergence estimates are not useful in this context: ..."
Abstract

Cited by 26 (4 self)
 Add to MetaCart
(Show Context)
We compare additive and multiplicative Schwarz preconditioners for the iterative solution of regularized linear inverse problems, extending and complementing earlier results of Hackbusch, King, and Rieder. Our main findings are that the classical convergence estimates are not useful in this context: rather, we observe that for regularized illposed problems with relevant parameter values the additive Schwarz preconditioner significantly increases the condition number. On the other hand, the multiplicative version greatly improves conditioning, much beyond the existing theoretical worstcase bounds. We present a theoretical analysis to support these results, and include a brief numerical example. More numerical examples with real applications will be given elsewhere. 1 Introduction In this paper we compare two preconditioners for symmetric, positive definite operator equations Au = b (1.1) in a Hilbert space X where A : X ! X takes the particular form A = K K+ ffL ; ff ? 0 ; (1.2)...
Hmatrix approximation for the operator exponential with applications
 NUMER. MATH.
, 2002
"... We develop a datasparse and accurate approximation to parabolic solution operators in the case of a rather general elliptic part given by a strongly Ppositive operator [4]. In the preceding papers [12]–[17], a class of matrices (Hmatrices) has been analysed which are datasparse and allow an app ..."
Abstract

Cited by 21 (12 self)
 Add to MetaCart
We develop a datasparse and accurate approximation to parabolic solution operators in the case of a rather general elliptic part given by a strongly Ppositive operator [4]. In the preceding papers [12]–[17], a class of matrices (Hmatrices) has been analysed which are datasparse and allow an approximate matrix arithmetic withalmost linear complexity. In particular, the matrixvector/matrixmatrix product withsuchmatrices as well as the computation of the inverse have linearlogarithmic cost. In the present paper, we apply the Hmatrix techniques to approximate the exponent of an elliptic operator. Starting with the DunfordCauchy representation for the operator exponent, we then discretise the integral by the exponentially convergent quadrature rule involving a short sum of resolvents. The latter are approximated by the Hmatrices. Our algorithm inherits a twolevel parallelism with respect to both the computation of resolvents and the treatment of different time values. In the case of smooth data (coefficients, boundaries), we prove the linearlogarithmic complexity of the method.
Galerkin boundary element methods for electromagnetic scattering
 in Topics in Computational Wave Propagation. Direct and inverse Problems
, 2003
"... Extended version with new appendix on Scattering from coated dielectric ..."
Abstract

Cited by 21 (3 self)
 Add to MetaCart
Extended version with new appendix on Scattering from coated dielectric
Universal quadratures for boundary integral equations on twodimensional domains with corners
, 2009
"... We describe the construction of a collection of quadrature formulae suitable for the efficient discretization of certain boundary integral equations on a very general class of twodimensional domains with corner points. The resulting quadrature rules allow for the rapid high accuracy solution of Lap ..."
Abstract

Cited by 17 (3 self)
 Add to MetaCart
(Show Context)
We describe the construction of a collection of quadrature formulae suitable for the efficient discretization of certain boundary integral equations on a very general class of twodimensional domains with corner points. The resulting quadrature rules allow for the rapid high accuracy solution of Laplace’s equation and the Helmholtz equation on such domains. Our approach can be adapted to many other boundary value problems as well as to the case of surfaces with singularities in three dimensions. The performance of the quadrature rules is illustrated with several numerical examples.