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 two-dimensional manifold \Gamma in IR 3 (screen problems). We will focus on the example of a two-dimensional 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...

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 ill-posed 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 worst-case 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)...

We present cubature methods approximating the surface integrals arising by Galerkin discretization of boundary integral equations on surfaces in R 3 . 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. Introduction In this paper, we will consider Fredholm integral equations on two dimensional surfaces in R 3 , which typically arise by applying the boundary integral method to boundary value problems. For the discretization of the integral equation we use the Galerkin method. For the generation of the system matrix, one has to compute integrals over pai...

Abstract. With respect to a wavelet basis, singular integral operators can be well approximated by sparse matrices, and in [Found. Comput. Math., 2 (2002), pp. 203–245] and [SIAM J. Math. Anal., 35 (2004), pp. 1110–1132], this property was used to prove certain optimal complexity results in the context of adaptive wavelet methods. These results, however, were based upon the assumption that, on average, each entry of the approximating sparse matrices can be computed at unit cost. In this paper, we confirm this assumption by carefully distributing computational costs over the matrix entries in combination with choosing efficient quadrature schemes. 1.

We develop a data-sparse and accurate approximation to parabolic solution operators in the case of a rather general elliptic part given by a strongly P-positive operator [4]. In the preceding papers [12]–[17], a class of matrices (H-matrices) has been analysed which are data-sparse and allow an approximate matrix arithmetic withalmost linear complexity. In particular, the matrix-vector/matrixmatrix product withsuchmatrices as well as the computation of the inverse have linear-logarithmic cost. In the present paper, we apply the H-matrix techniques to approximate the exponent of an elliptic operator. Starting with the Dunford-Cauchy 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 H-matrices. Our algorithm inherits a two-level 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 linear-logarithmic complexity of the method.

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

Abstract. We develop a simple, efficient numerical method of boundary integral type for solving an elliptic partial differential equation in a three-dimensional region using the classical formulation of potential theory. Accurate values can be found near the boundary using special corrections to a standard quadrature. We treat the Dirichlet problem for a harmonic function with a prescribed boundary value in a bounded three-dimensional region with a smooth boundary. The solution is a double layer potential, whose strength is found by solving an integral equation of the second kind. The boundary surface is represented by rectangular grids in overlapping coordinate systems, with the boundary value known at the grid points. A discrete form of the integral equation is solved using a regularized form of the kernel. It is proved that the discrete solution converges to the exact solution with accuracy O(h p), p<5, depending on the smoothing parameter. Once the dipole strength is found, the harmonic function can be computed from the double layer potential. For points close to the boundary, the integral is nearly singular, and accurate computation is not routine. We calculate the integral by summing over the boundary grid points and then adding corrections for the smoothing and discretization errors using formulas derived here; they are similar to those in the two-dimensional case given by [J. T. Beale and M.-C. Lai, SIAM J. Numer. Anal., 38 (2001), pp. 1902–1925]. The resulting values of the solution are uniformly of O(h p) accuracy, p<3. With a total of N points, the calculation could be done in essentially O(N) operations if a rapid summation method is used.

In this dissertation, we analyze three classes of iterative methods which are often used as preconditioners for Krylov subspace methods, for the solution of large and sparse linear systems arising from the discretization of partial differential equations. In addition, we propose algorithms for image processing applications and multiple right-hand side problems. The first class is the incomplete LU factorization preconditioners, an intrinsic sequential algorithm. We develop a parallel implementation of ILU(0) and devise a strategy for a priori memory allocation crucial for ILU(k) parallelization. The second class is the sparse approximate inverse (SPAI) preconditioners. We improve and extend its applicability to elliptic PDEs by using wavelets which converts smoothness, often found in the Green's function...

Much of the research on the numerical analysis of Fredholm type integral equations during the past ten years has centered on the solution of boundary integral equations (BIE). A great deal of this research has been on the numerical solution of BIE on simple closed boundary curves S for planar regions. When a BIE is defined on a smooth curve S, there are many numerical methods for solving the equation. The numerical analysis of most such problems is now well-understood, for both BIE of the first and second kind, with many people having contributed to the area. For the case with the BIE defined on a curve S which is only piecewise smooth, new numerical methods have been developed during the past decade. Such methods for BIE of the second kind were developed in the mid to late 80s; and more recently, high order collocation methods have been given and analyzed for BIE of the first kind. The numerical analysis of BIE on surfaces S in R 3 has become more active during the past decade, and...

Model reduction is a common theme within the simulation, control and optimization of complex dynamical systems. For instance, in control problems for partial differential equations, the associated large-scale systems have to be solved very often. To attack these problems in reasonable time it is absolutely necessary to reduce the dimension of the underlying system. We focus on model reduction by balanced truncation where a system theoretical background provides some desirable properties of the reduced-order system. The major computational task in balanced truncation is the solution of large-scale Lyapunov equations, thus the method is of limited use for really large-scale applications. We develop an effective implementation of balancing-related model reduction methods in exploiting the structure of the underlying problem. This is done by a data-sparse approximation of the large-scale state matrix A using the hierarchical matrix format. Furthermore, we integrate