Application of hierarchical matrices for solving multiscale problems (original) (raw)

Hierarchical Matrices for Convection-Dominated Problems

Lecture Notes in Computational Science and Engineering

Hierarchical matrices provide a technique to efficiently compute and store explicit approximations to the inverses of stiffness matrices computed in the discretization of partial differential equations. In a previous paper, Le Borne [2003], it was shown how standard H-matrices must be modified in order to obtain good approximations in the case of a convection dominant equation with a constant convection direction. This paper deals with a generalization to arbitrary (non-constant) convection directions. We will show how these H-matrix approximations to the inverse can be used as preconditioners in iterative methods.

ℋ2-matrices – Multilevel methods for the approximation of integral operators

Computing and Visualization in Science

Multigrid methods are typically used to solve partial differential equations, i.e., they approximate the inverse of the corresponding partial differential operators. At least for elliptic PDEs, this inverse can be expressed in the form of an integral operator by Green's theorem. This implies that multigrid methods approximate certain integral operators, so it is straightforward to look for variants of multigrid methods that can be used to approximate more general integral operators. H 2-matrices combine a multigrid-like structure with ideas from panel clustering algorithms in order to provide a very efficient method for discretizing and evaluating the integral operators found, e.g., in boundary element applications.

ℋ-Matrices for the Convection-Diffusion Equation

PAMM

H-matrices for the convection-diffusion equation Hierarchical matrices (H-matrices) provide a technique for the sparse approximation of large, fully populated matrices. This technique has been shown to be applicable to stiffness matrices arising in boundary element method applications where the kernel function displays certain smoothness properties. The error estimates for an approximation of the kernel function by a separable function can be carried over directly to error estimates for an approximation of the stiffness matrix by an H-matrix, using a certain standard partitioning and admissibility condition for matrix blocks. Similarly, H-matrix techniques can be applied in the finite element context where it is the inverse of the stiffness matrix that is fully populated. Here one needs a separable approximation of Green's function of the underlying boundary value problem in order to prove approximability by matrix blocks of low rank. Unfortunately, Green's function for the convection-diffusion equation does not satisfy the required smoothness properties, hence prohibiting a straightforward generalization of the separable approximation through Taylor polynomials. We will use Green's function to motivate a modification in the (hierarchical) partitioning of the index set and as a consequence the resulting hierarchy of block partitionings as well as the admissibility condition. We will illustrate the effect of the proposed modifications by numerical results.

ℋ-Matrix Approximation for Elliptic Solution Operators in Cylinder Domains

Journal of Numerical Mathematics, 2001

We develop a data-sparse and accurate approximation of the normalised hyperbolic operator sine family generated by a strongly P-positive elliptic operator defined in [4, 7]. In the preceding papers [14]-[18], a class of H-matrices has been analysed which are data-sparse and allow an approximate matrix arithmetic with almost linear complexity. An H-matrix approximation to the operator exponent with a strongly P-positive operator was proposed in [5]. In the present paper, we apply the H-matrix techniques to approximate the elliptic solution operator on cylindric domains Ω × [a, b] associated with the elliptic operator d 2 dx 2 − L, x ∈ [a, b]. It is explicitly presented by the operator-valued normalised hyperbolic sine function sinh −1 (√ L) sinh(x √ L) of an elliptic operator L defined in Ω. Starting with the Dunford-Cauchy representation for the hyperbolic sine operator, we then discretise the integral by the exponentially convergent quadrature rule involving a short sum of resolvents. The latter are approximated by the H-matrix techniques. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [a, b]. The approach is applied to elliptic partial differential equations in domains composed of rectangles or cylinders. In particular, we consider the H-matrix approximation to the interface Poincaré-Steklov operators with application in the Schur-complement domain decomposition method.

( BV , L 2 ) Multiscale Hierarchical Decomposition : Modes and Rates of Convergence

2014

Tadmor, Nezzar and Vese [Eitan Tadmor, Suzanne Nezzar, and Luminita Vese. A multiscale image representation using hierarchical (BV, L2) decompositions. Multiscale Model. Simul., 2(4):554–579, 2004.] developed a total variation based multiscale method for decomposing a function f ∈ BV into a countable set of features {uk : k = 0, 1, 2 . . .} associated to a sequence of dyadic scales {λk = λ02−k : k = 0, 1, 2, . . .} such that for each k, [uk+1, vk+1] = arg min{λk|Du| + ‖v‖L2 : u + v = f }. They showed that f = ∑∞ k=0 uk in L 2(Ω) and strongly in W−1,∞(Ω). In this paper, we study the convergence of the series ∞ ∑ k=0 uk in the weak*, strict and normed topologies of the space of functions with bounded variation. We show that in general, the convergence of the series f = ∞ ∑ k=0 uk in any of the three topologies of BV is conditioned by its rate of convergence in L2, and prove that the convergence in L2 is geometric.

Data-sparse approximation to the operator-valued functions of elliptic operator

Mathematics of Computation, 2004

In previous papers the arithmetic of hierarchical matrices has been described, which allows to compute the inverse, for instance, of finite element stiffness matrices discretising an elliptic operator L. The required computing time is up to logarithmic factors linear in the dimension of the matrix. In particular, this technique can be used for the computation of the discrete analogue of a resolvent (zI − L) −1 , z ∈ C.

H-Matrix Approximation for Elliptic Solution Operators in Cylindric Domains

2001

We develop a data-sparse and accurate approximation of the normalised hyperbolic operator sine family generated by a strongly P-positive elliptic operator defined in (4, 7). In the preceding papers (14)-(18), a class of H-matrices has been analysed which are data-sparse and allow an approximate matrix arithmetic with almost linear complexity. An H-matrix approximation to the operator exponent with a strongly