27 research outputs found
Preconditioning of wavelet BEM by the incomplete Cholesky factorization
The present paper is dedicated to the preconditioning of boundary element matrices which are given in wavelet coordinates. We investigate the incomplete Cholesky factorization (ICF) for a pattern which includes also the coefficients of all off-diagonal bands associated with the level-level-interactions. The pattern is chosen in such a way that the ICF is computable in log-linear complexity. Numerical experiments are performed to quantify the effects of the proposed preconditionin
A fast direct solver for nonlocal operators in wavelet coordinates
In this article, we consider fast direct solvers for nonlocal operators. The
pivotal idea is to combine a wavelet representation of the system matrix,
yielding a quasi-sparse matrix, with the nested dissection ordering scheme. The
latter drastically reduces the fill-in during the factorization of the system
matrix by means of a Cholesky decomposition or an LU decomposition,
respectively. This way, we end up with the exact inverse of the compressed
system matrix with only a moderate increase of the number of nonzero entries in
the matrix.
To illustrate the efficacy of the approach, we conduct numerical experiments
for different highly relevant applications of nonlocal operators: We consider
(i) the direct solution of boundary integral equations in three spatial
dimensions, issuing from the polarizable continuum model, (ii) a parabolic
problem for the fractional Laplacian in integral form and (iii) the fast
simulation of Gaussian random fields
Efficient numerical methods for capacitance extraction based on boundary element method
Fast and accurate solvers for capacitance extraction are needed by the VLSI industry
in order to achieve good design quality in feasible time. With the development
of technology, this demand is increasing dramatically. Three-dimensional capacitance
extraction algorithms are desired due to their high accuracy. However, the present
3D algorithms are slow and thus their application is limited. In this dissertation, we
present several novel techniques to significantly speed up capacitance extraction algorithms
based on boundary element methods (BEM) and to compute the capacitance
extraction in the presence of floating dummy conductors.
We propose the PHiCap algorithm, which is based on a hierarchical refinement
algorithm and the wavelet transform. Unlike traditional algorithms which result in
dense linear systems, PHiCap converts the coefficient matrix in capacitance extraction
problems to a sparse linear system. PHiCap solves the sparse linear system iteratively,
with much faster convergence, using an efficient preconditioning technique. We also
propose a variant of PHiCap in which the capacitances are solved for directly from a
very small linear system. This small system is derived from the original large linear
system by reordering the wavelet basis functions and computing an approximate LU
factorization. We named the algorithm RedCap. To our knowledge, RedCap is the
first capacitance extraction algorithm based on BEM that uses a direct method to solve a reduced linear system.
In the presence of floating dummy conductors, the equivalent capacitances among
regular conductors are required. For floating dummy conductors, the potential is unknown
and the total charge is zero. We embed these requirements into the extraction
linear system. Thus, the equivalent capacitance matrix is solved directly. The number
of system solves needed is equal to the number of regular conductors.
Based on a sensitivity analysis, we propose the selective coefficient enhancement
method for increasing the accuracy of selected coupling or self-capacitances with
only a small increase in the overall computation time. This method is desirable
for applications, such as crosstalk and signal integrity analysis, where the coupling
capacitances between some conductors needs high accuracy. We also propose the
variable order multipole method which enhances the overall accuracy without raising
the overall multipole expansion order. Finally, we apply the multigrid method to
capacitance extraction to solve the linear system faster.
We present experimental results to show that the techniques are significantly
more efficient in comparison to existing techniques
Fast solution methods
International audienceThe standard boundary element method applied to the time harmonic Helmholtz equation yields a numerical method with complexity when using a direct solution of the fully populated system of linear equations. Strategies to reduce this complexity are discussed in this paper. The complexity issuing from the direct solution is first reduced to by using iterative solvers. Krylov subspace methods as well as strategies of preconditioning are reviewed. Based on numerical examples the influence of different parameters on the convergence behavior of the iterative solvers is investigated. It is shown that preconditioned Krylov subspace methods yields a boundary element method of complexity. A further advantage of these iterative solvers is that they do not require the dense matrix to be set up. Only matrix–vector products need to be evaluated which can be done efficiently using a multilevel fast multipole method. Based on real life problems it is shown that the computational complexity of the boundary element method can be reduced to for a problem with unknowns
Fast Numerical Methods for Non-local Operators
[no abstract available
Preconditioners for iterative solutions of large-scale linear systems arising from Biot's consolidation equations
Ph.DDOCTOR OF PHILOSOPH
Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity
Dense kernel matrices Θ∈R^(N×N) obtained from point evaluations of a covariance function G at locations {x_i}1≤i≤N arise in statistics, machine learning, and numerical analysis. For covariance functions that are Green's functions elliptic boundary value problems and approximately equally spaced sampling points, we show how to identify a subset S⊂{1,…,N}×{1,…,N}, with #S=O(Nlog(N)log^d(N/ϵ)), such that the zero fill-in block-incomplete Cholesky decomposition of Θ_(i,j)1_((i,j)∈S) is an ϵ-approximation of Θ. This block-factorisation can provably be obtained in O(Nlog^2(N)(log(1/ϵ)+log^2(N))^(4d+1)) complexity in time. Numerical evidence further suggests that element-wise Cholesky decomposition with the same ordering constitutes an O(Nlog^2(N)log^(2d)(N/ϵ)) solver. The algorithm only needs to know the spatial configuration of the x_i and does not require an analytic representation of G. Furthermore, an approximate PCA with optimal rate of convergence in the operator norm can be easily read off from this decomposition. Hence, by using only subsampling and the incomplete Cholesky decomposition, we obtain at nearly linear complexity the compression, inversion and approximate PCA of a large class of covariance matrices. By inverting the order of the Cholesky decomposition we also obtain a near-linear-time solver for elliptic PDEs