1,093 research outputs found
Multigrid methods for two-player zero-sum stochastic games
We present a fast numerical algorithm for large scale zero-sum stochastic
games with perfect information, which combines policy iteration and algebraic
multigrid methods. This algorithm can be applied either to a true finite state
space zero-sum two player game or to the discretization of an Isaacs equation.
We present numerical tests on discretizations of Isaacs equations or
variational inequalities. We also present a full multi-level policy iteration,
similar to FMG, which allows to improve substantially the computation time for
solving some variational inequalities.Comment: 31 page
Boundary Treatment and Multigrid Preconditioning for Semi-Lagrangian Schemes Applied to Hamilton-Jacobi-Bellman Equations
We analyse two practical aspects that arise in the numerical solution of
Hamilton-Jacobi-Bellman (HJB) equations by a particular class of monotone
approximation schemes known as semi-Lagrangian schemes. These schemes make use
of a wide stencil to achieve convergence and result in discretization matrices
that are less sparse and less local than those coming from standard finite
difference schemes. This leads to computational difficulties not encountered
there. In particular, we consider the overstepping of the domain boundary and
analyse the accuracy and stability of stencil truncation. This truncation
imposes a stricter CFL condition for explicit schemes in the vicinity of
boundaries than in the interior, such that implicit schemes become attractive.
We then study the use of geometric, algebraic and aggregation-based multigrid
preconditioners to solve the resulting discretised systems from implicit time
stepping schemes efficiently. Finally, we illustrate the performance of these
techniques numerically for benchmark test cases from the literature
A Multigrid Optimization Algorithm for the Numerical Solution of Quasilinear Variational Inequalities Involving the -Laplacian
In this paper we propose a multigrid optimization algorithm (MG/OPT) for the
numerical solution of a class of quasilinear variational inequalities of the
second kind. This approach is enabled by the fact that the solution of the
variational inequality is given by the minimizer of a nonsmooth energy
functional, involving the -Laplace operator. We propose a Huber
regularization of the functional and a finite element discretization for the
problem. Further, we analyze the regularity of the discretized energy
functional, and we are able to prove that its Jacobian is slantly
differentiable. This regularity property is useful to analyze the convergence
of the MG/OPT algorithm. In fact, we demostrate that the algorithm is globally
convergent by using a mean value theorem for semismooth functions. Finally, we
apply the MG/OPT algorithm to the numerical simulation of the viscoplastic flow
of Bingham, Casson and Herschel-Bulkley fluids in a pipe. Several experiments
are carried out to show the efficiency of the proposed algorithm when solving
this kind of fluid mechanics problems
Adaptive monotone multigrid methods for nonlinear variational problems
A wide range of problems occurring in engineering and industry is characterized by the presence of a free (i.e. a priori unknown) boundary where the underlying physical situation is changing in a discontinuous way. Mathematically, such phenomena can be often reformulated as variational inequalities or related non–smooth minimization problems.
In these research notes, we will describe a new and promising way of constructing fast solvers for the corresponding discretized problems providing globally convergent iterative schemes with (asymptotic) multigrid
convergence speed. The presentation covers physical modelling, existence and uniqueness results, finite element approximation and adaptive mesh–refinement based on a posteriori error estimation. The numerical properties
of the resulting adaptive multilevel algorithm are illustrated by typical applications, such as semiconductor device simulation or continuous casting
A full approximation scheme multilevel method for nonlinear variational inequalities
We present the full approximation scheme constraint decomposition (FASCD)
multilevel method for solving variational inequalities (VIs). FASCD is a common
extension of both the full approximation scheme (FAS) multigrid technique for
nonlinear partial differential equations, due to A.~Brandt, and the constraint
decomposition (CD) method introduced by X.-C.~Tai for VIs arising in
optimization. We extend the CD idea by exploiting the telescoping nature of
certain function space subset decompositions arising from multilevel mesh
hierarchies. When a reduced-space (active set) Newton method is applied as a
smoother, with work proportional to the number of unknowns on a given mesh
level, FASCD V-cycles exhibit nearly mesh-independent convergence rates, and
full multigrid cycles are optimal solvers. The example problems include
differential operators which are symmetric linear, nonsymmetric linear, and
nonlinear, in unilateral and bilateral VI problems.Comment: 25 pages, 9 figure
Multigrid methods for obstacle problems
In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which is closely related to truncated multigrid. The numerical properties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set
Monotone multigrid methods for elliptic variational inequalities II
We derive globally convergent multigrid methods for discrete elliptic variational inequalities of the second kind as obtained from the approximation of related continuous problems by piecewise linear finite elements. The coarse grid corrections are computed from certain obstacle problems. The actual constraints are fixed by the preceding nonlinear fine grid smoothing. This new approach allows the implementation as a classical V-cycle and preserves the usual multigrid efficiency. We give 1−O(j−3) estimates for the asymptotic convergence rates. The numerical results indicate a significant improvement as compared with previous multigrid approaches
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