A capacitance matrix method for Dirichlet problem on polygon region

Dryja, Maksymilian

doi:10.1007/bf01399311

Cited by 110 publications

(55 citation statements)

References 4 publications

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“…Among them, Dryja's preconditioner [6] based on the square root of the finite difference analogue of −(d 2 /ds 2 ) along the boundary was proved to be spectrally equivalent to the capacitance matrix, which means that a conjugate-gradient-type algorithm applied to solve this linear system can converge in a constant number of iterations. The hierarchical basis preconditioner [19] has been proved to reduce the condition number of the transformed capacitance matrix to O(log 2 n).…”

Section: The Surface Reconstruction Problemmentioning

confidence: 99%

“…Then the matrix UV T only contains the differences of rows corresponding to the boundary condition, i.e., data constraints, between K andK. Dryja's spectrally equivalent preconditioner [6] for the capacitance matrix arising from the elliptic boundary value problem can be directly applied to this problem, thus making the BCG method converge in a constant number of iterations. Thus, our generalized capacitance matrix algorithm can solve this problem in O(n) operations.…”

Section: Surface Interpolation Problemmentioning

confidence: 99%

“…Since the structure of the capacitance matrix is similar to that of the surface interpolation problem with data points given along a closed contour, we can apply a preconditioning technique similar to that in [6] to obtain an O(n) solution for the shape from orientation problem. Existing algorithms usually take O(n 2 ) (conjugate gradient) or O(n log n) (hierarchical basis conjugate gradient) operations.…”

Section: Shape From Orientation Problemmentioning

confidence: 99%

See 2 more Smart Citations

Generalized Capacitance Matrix Theorems and Algorithm for Solving Linear Systems

Lai¹,

Vemuri²

1998

SIAM J. Sci. Comput.

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Abstract. The capacitance matrix method has been widely used as an efficient numerical tool for solving the boundary value problems on irregular regions. Initially, this method was based on the Sherman-Morrison-Woodbury formula, an expression for the inverse of the matrix (A + UV T ) with A ∈ n×n and U, V ∈ n×p . Extensions of this method reported in literature have made restrictive assumptions on the matrices A and (A + UV T ). In this paper, we present several theorems which are generalizations of the capacitance matrix theorem in [4] and are suited for very general matrices A and (A + UV T ). A generalized capacitance matrix algorithm is developed from these theorems and holds the promise of being applicable to more general problems; in addition, it gives ample freedom to choose the matrix A for developing very efficient numerical algorithms. We demonstrate the usefulness of our generalized capacitance matrix algorithm by applying it to efficiently solve large sparse linear systems in several computer vision problems, namely, the surface reconstruction and interpolation, and the shape from orientation problem.

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Section: The Surface Reconstruction Problemmentioning

confidence: 99%

Section: Surface Interpolation Problemmentioning

confidence: 99%

Section: Shape From Orientation Problemmentioning

confidence: 99%

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Generalized Capacitance Matrix Theorems and Algorithm for Solving Linear Systems

Lai¹,

Vemuri²

1998

SIAM J. Sci. Comput.

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“…[7]. A typical one is Dryja's preconditioner [9], which is defined to be the square root of the negative one-dimensional Laplacian and which can be inverted by the use of FFTs in O(n log n) time, where n is the number of unknowns on the interface. Recently, Smith and Widlund [19] proposed a hierarchical basis preconditioner for S which is cheaper than Dryja's preconditioner, requiring only O(n) work per iteration.…”

mentioning

confidence: 99%

Multilevel Filtering Preconditioners: Extensions to More General Elliptic Problems

Tong¹,

Chan²,

Kuo³

1992

SIAM J. Sci. and Stat. Comput.

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Abstract. The concept of multilevel filtering (MF) preconditioning applied to second-order selfadjoint elliptic problems is briefly reviewed. It is then shown how to effectively apply this concept to other elliptic problems such as the second-order anisotropic problem, biharmonic equation, equations on locally refined grids and interface operators arising from domain decomposition methods.Numerical results are given to show the effectiveness of the MF preconditioners on these problems.

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“…Thus the conjugate gradient method may be applied to (6) with respect to (7). The importance of making a "good" choice for B is well known.…”

mentioning

confidence: 99%

An Iterative Method for Elliptic Problems on Regions Partitioned into Substructures

Bramble¹,

Pasciak²,

Schatz³

1986

Mathematics of Computation

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Abstract. Some new preconditioners for discretizations of elliptic boundary problems are studied. With these preconditioners, the domain under consideration is broken into subdomains and preconditioners are defined which only require the solution of matrix problems on the subdomains. Analytic estimates are given which guarantee that under appropriate hypotheses, the preconditioned iterative procedure converges to the solution of the discrete equations with a rate per iteration that is independent of the number of unknowns. Numerical examples are presented which illustrate the theoretically predicted iterative convergence rates.

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A capacitance matrix method for Dirichlet problem on polygon region

Cited by 110 publications

References 4 publications

Generalized Capacitance Matrix Theorems and Algorithm for Solving Linear Systems

Generalized Capacitance Matrix Theorems and Algorithm for Solving Linear Systems

Multilevel Filtering Preconditioners: Extensions to More General Elliptic Problems

An Iterative Method for Elliptic Problems on Regions Partitioned into Substructures

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