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41
Recent computational developments in Krylov subspace methods for linear systems
 NUMER. LINEAR ALGEBRA APPL
, 2007
"... Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are metho ..."
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Cited by 51 (12 self)
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Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.
On the Approximate Cyclic Reduction Preconditioner
 SIAM J. Sci. Comput
, 2000
"... We present a preconditioning method for the iterative solution of large sparse systems of equations. The preconditioner is based on ideas both from ILU preconditioning and from multigrid. The resulting preconditioning technique requires the matrix only. A multilevel structure is obtained by using ma ..."
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Cited by 16 (3 self)
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We present a preconditioning method for the iterative solution of large sparse systems of equations. The preconditioner is based on ideas both from ILU preconditioning and from multigrid. The resulting preconditioning technique requires the matrix only. A multilevel structure is obtained by using maximal independent sets for graph coarsening. A Schur complement approximation is constructed using a sequence of point Gaussian elimination steps. The resulting preconditioner has a transparant modular structure similar to the algoritmic structure of a multigrid Vcycle.
Differences in the effects of rounding errors in Krylov solvers for symmetric indefinite linear systems
, 1999
"... The 3term Lanczos process leads, for a symmetric matrix, to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of symmetric indefinite linear systems, by solving the reduced system in one way or another. Thi ..."
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Cited by 15 (0 self)
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The 3term Lanczos process leads, for a symmetric matrix, to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of symmetric indefinite linear systems, by solving the reduced system in one way or another. This leads to wellknown methods: MINRES, GMRES, and SYMMLQ. We will discuss in what way and to what extent these approaches differ in their sensitivity to rounding errors. In our analysis we will assume that the Lanczos basis is generated in exactly the same way for the different methods, and we will not consider the errors in the Lanczos process itself. We will show that the method of solution may lead, under certain circumstances, to large additional errors, that are not corrected by continuing the iteration process. Our findings are supported and illustrated by numerical examples. 1 Introduction We will consider iterative methods for the construction of approximate solutions, starting with...
An Analysis Of A Preconditioner For The Discretized Pressure Equation Arising In Reservoir Simulation
, 1995
"... We analyze the use of fast solvers as preconditioners for the discretized pressure equation arising in reservoir simulation. Under proper conditions on the permeability functions and the source term, we show that the number of iterations for the Conjugate Gradient method is bounded independently of ..."
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Cited by 8 (2 self)
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We analyze the use of fast solvers as preconditioners for the discretized pressure equation arising in reservoir simulation. Under proper conditions on the permeability functions and the source term, we show that the number of iterations for the Conjugate Gradient method is bounded independently of both the lower bound ffi of the permeability and the discretization parameter h. Such results are obtained for a special class of selfadjoint second order elliptic problems with discontinuous coefficients. We also discuss how fast solvers can be utilized in the presence of nonrectangular domains by applying a domain imbedding procedure. The theoretical results are illustrated and supplemented by a series of numerical experiments.
Efficient numerical methods for simulation of highfrequency active devices
 IEEE Trans. Microw. Theory Tech
, 2006
"... Abstract—We present two new numerical approaches for physical modeling of highfrequency semiconductor devices using filterbank transforms and the alternatingdirection implicit finitedifference timedomain method. In the first proposed approach, a preconditioner based on the filterbank and wavele ..."
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Cited by 6 (6 self)
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Abstract—We present two new numerical approaches for physical modeling of highfrequency semiconductor devices using filterbank transforms and the alternatingdirection implicit finitedifference timedomain method. In the first proposed approach, a preconditioner based on the filterbank and wavelet transforms is used to facilitate the iterative solution of Poisson’s equation and the other semiconductor equations discretized using implicit schemes. The second approach solves Maxwell’s equations which, in conjunction with the semiconductor equations, describe the complete behavior of highfrequency active devices, with larger timestep size. These approaches lead to the significant reduction of the fullwave simulation time. For the first time, we can reach over 95 % reduction in the simulation time by using these two techniques while maintaining the same degree of accuracy achieved using the conventional approach. Index Terms—Alternatingdirection implicit finitedifference timedomain (ADIFDTD) method, filterbank transforms, fullwave analysis, global modeling, highfrequency devices, preconditioning. I.
A Finite Element Method for Fully Nonlinear Water Waves
, 1996
"... We introduce a numerical method for fully nonlinear, threedimensional water surface waves, described by standard potential theory. The method is based on a transformation of the dynamic water volume onto a fixed domain. Regridding at each time step is thereby avoided. The transformation introduces ..."
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Cited by 5 (0 self)
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We introduce a numerical method for fully nonlinear, threedimensional water surface waves, described by standard potential theory. The method is based on a transformation of the dynamic water volume onto a fixed domain. Regridding at each time step is thereby avoided. The transformation introduces an elliptic boundary value problem which is solved by a preconditioned conjugate gradient method. Moreover, a simple domain imbedding precedure is introduced to solve problems with an obstacle in the water volume. Numerical experiments are presented and they show nice convergence properties of the iterative solver as well as convergence of the entire solution towards a reference solution computed by another scheme.
Variations on Richardson's method and acceleration
, 1996
"... The aim of this paper is to present an acceleration procedure based on projection and preconditioning for iterative methods for solving systems of linear equations. A cycling strategy leads to a new iterative method. These procedures are closely related to Richardson's method and acceleratio ..."
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Cited by 5 (4 self)
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The aim of this paper is to present an acceleration procedure based on projection and preconditioning for iterative methods for solving systems of linear equations. A cycling strategy leads to a new iterative method. These procedures are closely related to Richardson's method and acceleration. Numerical examples illustrate the purpose.
Penalty methods for the numerical solution of American multiasset option problems
, 2000
"... We derive and analyse a penalty method for solving American multiasset option problems. A small, nonlinear penalty term is added to the BlackScholes equation. This approach gives a xed solution domain, removing the free and moving boundary imposed by the early exercise feature of the contract. Ex ..."
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Cited by 5 (1 self)
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We derive and analyse a penalty method for solving American multiasset option problems. A small, nonlinear penalty term is added to the BlackScholes equation. This approach gives a xed solution domain, removing the free and moving boundary imposed by the early exercise feature of the contract. Explicit, implicit and semiimplicit nite dierence schemes are derived, and in the case of independent assets, we prove that the approximate option prices satisfy some basic properties of the American option problem. Several numerical experiments are carried out in order to investigate the performance of the schemes. We give examples indicating that our results are sharp. Finally, experiments indicate that in the case of correlated underlying assets, the same properties are valid as in the independent case.