921 research outputs found
An iterative algorithm for computing the cycle mean of a Toeplitz matrix in special form
summary:The paper presents an iterative algorithm for computing the maximum cycle mean (or eigenvalue) of triangular Toeplitz matrix in max-plus algebra. The problem is solved by an iterative algorithm which is applied to special cycles. These cycles of triangular Toeplitz matrices are characterized by sub-partitions of
Multilevel Approach For Signal Restoration Problems With Toeplitz Matrices
We present a multilevel method for discrete ill-posed problems arising from the discretization of Fredholm integral equations of the first kind. In this method, we use the Haar wavelet transform to define restriction and prolongation operators within a multigrid-type iteration. The choice of the Haar wavelet operator has the advantage of preserving matrix structure, such as Toeplitz, between grids, which can be exploited to obtain faster solvers on each level where an edge-preserving Tikhonov regularization is applied. Finally, we present results that indicate the promise of this approach for restoration of signals and images with edges
Multigrid waveform relaxation for the time-fractional heat equation
In this work, we propose an efficient and robust multigrid method for solving
the time-fractional heat equation. Due to the nonlocal property of fractional
differential operators, numerical methods usually generate systems of equations
for which the coefficient matrix is dense. Therefore, the design of efficient
solvers for the numerical simulation of these problems is a difficult task. We
develop a parallel-in-time multigrid algorithm based on the waveform relaxation
approach, whose application to time-fractional problems seems very natural due
to the fact that the fractional derivative at each spatial point depends on the
values of the function at this point at all earlier times. Exploiting the
Toeplitz-like structure of the coefficient matrix, the proposed multigrid
waveform relaxation method has a computational cost of
operations, where is the number of time steps and is the number of
spatial grid points. A semi-algebraic mode analysis is also developed to
theoretically confirm the good results obtained. Several numerical experiments,
including examples with non-smooth solutions and a nonlinear problem with
applications in porous media, are presented
Uniform convergence of V-cycle multigrid algorithms for two-dimensional fractional Feynman-Kac equation
In this paper we derive new uniform convergence estimates for the V-cycle MGM
applied to symmetric positive definite Toeplitz block tridiagonal matrices, by
also discussing few connections with previous results. More concretely, the
contributions of this paper are as follows: (1) It tackles the Toeplitz systems
directly for the elliptic PDEs. (2) Simple (traditional) restriction operator
and prolongation operator are employed in order to handle general Toeplitz
systems at each level of the recursion. Such a technique is then applied to
systems of algebraic equations generated by the difference scheme of the
two-dimensional fractional Feynman-Kac equation, which describes the joint
probability density function of non-Brownian motion. In particular, we consider
the two coarsening strategies, i.e., doubling the mesh size (geometric MGM) and
Galerkin approach (algebraic MGM), which lead to the distinct coarsening
stiffness matrices in the general case: however, several numerical experiments
show that the two algorithms produce almost the same error behaviour.Comment: 26 page
Multigrid Waveform Relaxation on Spatial Finite Element Meshes: The Discrete-Time Case
The efficiency of numerically solving time-dependent partial differential equations on parallel computers can be greatly improved by computing the solution on many time levels simultaneously. The theoretical properties of one such method, namely the discrete-time multigrid waveform relaxation method, are investigated for systems of ordinary differential equations obtained by spatial finite-element discretisation of linear parabolic initial-boundary value problems. The results are compared to the corresponding continuous-time results. The theory is illustrated for a one-dimensional and a two-dimensional model problem and checked against results obtained by numerical experiments
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