459 research outputs found
Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of Hamilton--Jacobi--Bellman equations
We analyse a class of nonoverlapping domain decomposition preconditioners for
nonsymmetric linear systems arising from discontinuous Galerkin finite element
approximation of fully nonlinear Hamilton--Jacobi--Bellman (HJB) partial
differential equations. These nonsymmetric linear systems are uniformly bounded
and coercive with respect to a related symmetric bilinear form, that is
associated to a matrix . In this work, we construct a
nonoverlapping domain decomposition preconditioner , that is based
on , and we then show that the effectiveness of the preconditioner
for solving the} nonsymmetric problems can be studied in terms of the condition
number . In particular, we establish the
bound , where
and are respectively the coarse and fine mesh sizes, and and
are respectively the coarse and fine mesh polynomial degrees. This represents
the first such result for this class of methods that explicitly accounts for
the dependence of the condition number on ; our analysis is founded upon an
original optimal order approximation result between fine and coarse
discontinuous finite element spaces. Numerical experiments demonstrate the
sharpness of this bound. Although the preconditioners are not robust with
respect to the polynomial degree, our bounds quantify the effect of the coarse
and fine space polynomial degrees. Furthermore, we show computationally that
these methods are effective in practical applications to nonsymmetric, fully
nonlinear HJB equations under -refinement for moderate polynomial degrees
A Combined Preconditioning Strategy for Nonsymmetric Systems
We present and analyze a class of nonsymmetric preconditioners within a
normal (weighted least-squares) matrix form for use in GMRES to solve
nonsymmetric matrix problems that typically arise in finite element
discretizations. An example of the additive Schwarz method applied to
nonsymmetric but definite matrices is presented for which the abstract
assumptions are verified. A variable preconditioner, combining the original
nonsymmetric one and a weighted least-squares version of it, is shown to be
convergent and provides a viable strategy for using nonsymmetric
preconditioners in practice. Numerical results are included to assess the
theory and the performance of the proposed preconditioners.Comment: 26 pages, 3 figure
A note on the design of hp-version interior penalty discontinuous Galerkin finite element methods for degenerate problems
We consider a variant of the hp-version interior penalty discontinuous Galerkin finite element method (IP-DGFEM) for second order problems of degenerate type. We do not assume uniform ellipticity of the diffusion tensor. Moreover, diffusion tensors or arbitrary form are covered in the theory presented. A new, refined recipe for the choice of the discontinuity-penalisation parameter (that is present in the formlation of the IP-DGFEM) is given. Making use of the recently introduced augmented Sobolev space framework, we prove an hp-optimal error bound in the energy norm and an h-optimal and slightly p-suboptimal (by only half an order of p) bound in the L2 norm, provided that the solution belongs to an augmented Sobolev space
Efficient numerical calculation of drift and diffusion coefficients in the diffusion approximation of kinetic equations
In this paper we study the diffusion approximation of a swarming model given
by a system of interacting Langevin equations with nonlinear friction. The
diffusion approximation requires the calculation of the drift and diffusion
coefficients that are given as averages of solutions to appropriate Poisson
equations. We present a new numerical method for computing these coefficients
that is based on the calculation of the eigenvalues and eigenfunctions of a
Schr\"odinger operator. These theoretical results are supported by numerical
simulations showcasing the efficiency of the method
Numerical Methods and Algorithms for High Frequency Wave Scattering Problems in Homogeneous and Random Media
This dissertation consists of four integral parts with a unified objective of developing efficient numerical methods for high frequency time-harmonic wave equations defined on both homogeneous and random media. The first part investigates the generalized weak coercivity of the acoustic Helmholtz, elastic Helmholtz, and time-harmonic Maxwell wave operators. We prove that such a weak coercivity holds for these wave operators on a class of more general domains called generalized star-shape domains. As a by-product, solution estimates for the corresponding Helmholtz-type problems are obtained.
The second part of the dissertation develops an absolutely stable (i.e. stable in all mesh regimes) interior penalty discontinuous Galerkin (IP-DG) method for the elastic Helmholtz equations. A special mesh-dependent sesquilinear form is proposed and is shown to be weakly coercive in all mesh regimes. We prove that the proposed IP-DG method converges with optimal rate with respect to the mesh size. Numerical experiments are carried out to demonstrate the theoretical results and compare this method to the standard finite element method.
The third part of the dissertation develops a Monte Carlo interior penalty discontinuous Galerkin (MCIP-DG) method for the acoustic Helmholtz equation defined on weakly random media. We prove that the solution to the random Helmholtz problem has a multi-modes expansion (i.e., a power series in a medium- related small parameter). Using this multi-modes expansion an efficient and accurate numerical method for computing moments of the solution to the random Helmholtz problem is proposed. The proposed method is also shown to converge optimally. Numerical experiments are carried out to compare the new multi-modes MCIP-DG method to a classical Monte Carlo method.
The last part of the dissertation develops a theoretical framework for Schwarz pre- conditioning methods for general nonsymmetric and indefinite variational problems which are discretized by Galerkin-type discretization methods. Such a framework has been missing in the literature and is of great theoretical and practical importance for solving convection-diffusion equations and Helmholtz-type wave equations. Condition number estimates for the additive and hybrid Schwarz preconditioners are established under some structure assumptions. Numerical experiments are carried out to test the new framework
A Fast Alternating Minimization Algorithm for Total Variation Deblurring Without Boundary Artifacts
Recently, a fast alternating minimization algorithm for total variation image
deblurring (FTVd) has been presented by Wang, Yang, Yin, and Zhang [{\em SIAM
J. Imaging Sci.}, 1 (2008), pp. 248--272]. The method in a nutshell consists of
a discrete Fourier transform-based alternating minimization algorithm with
periodic boundary conditions and in which two fast Fourier transforms (FFTs)
are required per iteration. In this paper, we propose an alternating
minimization algorithm for the continuous version of the total variation image
deblurring problem. We establish convergence of the proposed continuous
alternating minimization algorithm. The continuous setting is very useful to
have a unifying representation of the algorithm, independently of the discrete
approximation of the deconvolution problem, in particular concerning the
strategies for dealing with boundary artifacts. Indeed, an accurate restoration
of blurred and noisy images requires a proper treatment of the boundary. A
discrete version of our continuous alternating minimization algorithm is
obtained following two different strategies: the imposition of appropriate
boundary conditions and the enlargement of the domain. The first one is
computationally useful in the case of a symmetric blur, while the second one
can be efficiently applied for a nonsymmetric blur. Numerical tests show that
our algorithm generates higher quality images in comparable running times with
respect to the Fast Total Variation deconvolution algorithm
- …