Decomposition Method For Problems Research Articles

Parallel decomposition of large-scale stochastic nonlinear programs

Many practical decision problems involve both nonlinear relationships and uncertainties. The resulting stochastic nonlinear programs become quite difficult to solve as the number of possible scenarios increases. In this paper, we provide a decomposition method for problems in which nonlinear constraints appear within periods. We also show how the method extends to lower bounding refinements of the set of scenarios when the random data are independent from period to period. We then apply the method to a stochastic model of the U.S. economy based on the Global 2100 method developed by Manne and Richels.

Overlapping Domain Decomposition Algorithms for General Sparse Matrices

Domain decomposition methods for finite element problems using a partition based on the underlying finite element mesh have been extensively studied. In this paper, we discuss algebraic extensions of the class of overlapping domain decomposition algorithms for general sparse matrices. The subproblems are created with an overlapping partition of the graph corresponding to the sparsity structure of the matrix. These algebraic domain decomposition methods are especially useful for unstructured mesh problems. We also discuss some difficulties encountered in the algebraic extension, particularly the issues related to the coarse solver.

Overlapping Domain Decomposition Algorithms for General Sparse Matrices

Domain decomposition methods for finite element problems using a partition based on the underlying finite element mesh have been extensively studied. In this paper, we discuss algebraic extensions of the class of overlapping domain decomposition algorithms for general sparse matrices. The subproblems are created with an overlapping partition of the graph corresponding to the sparsity structure of the matrix. These algebraic domain decomposition methods are especially useful for unstructured mesh problems. We also discuss some difficulties encountered in the algebraic extension, particularly the issues related to the coarse solver.

A preconditioner for domain decomposition methods for advection-dominated problems

Abstract We present a new preconditioner for the Schur complement in domain decomposition methods. The new preconditioner seems to be particularly suited for dealing with advection-dominated elliptic problems.

Thevenin decomposition and large-scale optimization

The Thevenin theorem, one of the most celebrated results of electric circuit theory, provides a two-parameter characterization of the behavior of an arbitrarily large circuit, as seen from two of its terminals. We interpret the theorem as a sensitivity result in an associated minimum energy/network flow problem, and we abstract its main idea to develop a decomposition method for convex quadratic programming problems with linear equality constraints, of the type arising in a variety of contexts such as the Newton method, interior point methods, and least squares estimation. Like the Thevenin theorem, our method is particularly useful in problems involving a system consisting of several subssystems, connected to each other with a small number of coupling variables.

A convergence theory of multilevel additive Schwarz methods on unstructured meshes

We develop a convergence theory for two level and multilevel additive Schwarz domain decomposition methods for elliptic and parabolic problems on general unstructured meshes in two and three dimensions. The coarse and fine grids are assumed only to be shape regular, and the domains formed by the coarse and fine grids need not be identical. In this general setting, our convergence theory leads to completely local bounds for the condition numbers of two level additive Schwarz methods, which imply that these condition numbers are optimal, or independent of fine and coarse mesh sizes and subdomain sizes if the overlap amount of a subdomain with its neighbors varies proportionally to the subdomain size. In particular, we will show that additive Schwarz algorithms are still very efficient for nonselfadjoint parabolic problems with only symmetric, positive definite solvers both for local subproblems and for the global coarse problem. These conclusions for elliptic and parabolic problems improve our earlier results in [12, 15, 16]. Finally, the convergence theory is applied to multilevel additive Schwarz algorithms. Under some very weak assumptions on the fine mesh and coarser meshes, e.g., no requirements on the relation between neighboring coarse level meshes, we are able to derive a condition number bound of the orderO(ρ2 L 2), whereρ = max1≤l≤L(h l +l− 1)/δ l,h l is the element size of thelth level mesh,δ l the overlap of subdomains on thelth level mesh, andL the number of mesh levels.

Erratum to “Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes”

Erratum to “Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes”

On Convergence of an Augmented Lagrangian Decomposition Method for Sparse Convex Optimization

A decomposition method for large-scale convex optimization problems with block-angular structure and many linking constraints is analysed. The method is based on a separable approximation of the au...

On Convergence of an Augmented Lagrangian Decomposition Method for Sparse Convex Optimization

A decomposition method for large-scale convex optimization problems with block-angular structure and many linking constraints is analysed. The method is based on a separable approximation of the augmented Lagrangian function. Weak global convergence of the method is proved and speed of convergence analysed. It is shown that convergence properties of the method are heavily dependent on sparsity of the linking constraints. Application to large-scale linear programming and stochastic programming is discussed.

PRIMAL DECOMPOSITION METHOD FOR MULTIOBJECTIVE STRUCTURED NONLINEAR PROGRAMS WITH FUZZY GOALS

In this paper, we present a primal decomposition method for multiobjective nonlinear programming problems with the block angular structure by incorporating a fuzzy goal of a decision maker for each of the objective functions. In the proposed method, after eliciting the membership functions through the interaction with the decision maker, if we adopt the add-operator for combining them, it is shown that the original problem can be decomposed into several subproblems by introducing a right-hand-side allocation vector. Then a two-level optimization algorithm to derive the satisficing solution for the decision maker is proposed. An illustrative numerical example is provided to indicate the feasibility of the proposed method.

Adaptive Pseudo-Spectral Domain Decomposition and the Approximation of Multiple Layers

When Chebyshev pseudo-spectral methods are used with domain decomposition procedures in the numerical solution of partial differential equations, the use of multiple domains can significantly affect the accuracy of the approximation. This is particularly true when the solution exhibits layer type behavior, i.e., there are narrow regions of rapid variation. Accuracy may be enhanced if the interfaces between adjacent subdomains are such that large gradients occur near the interfaces, while accuracy can be degraded if the rapid variations occur in the interior of the subdomains. The use of appropriate mappings within each subdomain can improve the accuracy of the approximation by choosing mappings so that the transformed function is more readily approximated by a low order polynomial. The particular choice of mappings, however, depends critically on whether the solution exhibits boundary layer or interior layer behavior within each subdomain. We analyze the relationship between interface location and mappings required to obtain an efficient approximation of such functions. We compare two strategies, both based on constructing subdomains so that each subdomain contains only one layer. In the first strategy interface locations are chosen so that the rapid variations occur as interior layers and mappings are employed which enhance the resolution of such layers (strategy I for interior). In the second strategy interface locations are chosen so that rapid variations occur as boundary layers and mappings are employed which enhance resolution of boundary layers (strategy B for boundary). Both strategies lead to adaptive domain decomposition procedures based entirely on the locations of the layers. We demonstrate that strategy B offers superior accuracy for a given computational effort and employ this strategy in developing an adaptive domain decomposition method for problems with multiple layers. Both strategies are comparable regarding spectral radii of the resulting matrices, and we conclude that domain decomposition itself cannot result in larger stable time steps when accuracy of the approximation of the layer is considered. The adaptive domain decomposition method is illustrated by the computation of both axisymmetric and cellular flames with a sequential reaction mechanism involving two reaction zones.

Different domain decompositions at different times for capturing moving local phenomena

A conservative Galerkin domain decomposition method for time-dependent problems is given and analyzed. This method allows one to apply different domain decompositions at different time levels when necessary, in order to capture time-changing local phenomena, such as, propagating fronts or moving layers. Error estimates in the energy and L 2 norms are established. Numerical results are presented.

A simple modification of dantzig-wolfe decomposition

We describe a simple modification of the Dintzig-wolfe decomposition method for nonlinear programming problems, consisting of including an additional column in the master problem, which immediately improves the bound obtained. We investigate the consequences of this modification and can directly give the optimal primal solution and one optimal dual solution of the modified master problem. Furthermore we show that the dual solution is always degenerated and give an explicit expression of all the optimal dual solutions. Finally we propose a new hybrid method combining Dantzig-Wolfe decomposition with an improvement step of subgradient type, that generates an additional cut for the master problem in each interation

A fuzzy dual decomposition method for large-scale multiobjective nonlinear programming problems

Abstract In this paper, we propose a fuzzy dual decomposition method for large-scale multiobjective nonlinear programming problems (LS-MONLPs) with the block angular structure. By considering the vague nature of human judgements, we assume that the decision maker (DM) may have a fuzzy goal for each of the objective functions in the LS-MONLP. After eliciting the corresponding membership function for each of the objective functions through the interaction with the DM, an extended primal problem and the corresponding extended dual problem are formulated. Then a two-level optimization algorithm for the extended dual problem is proposed for deriving the compromise solution for the DM to the LS-MONLP. Based on the proposed algorithm, FORTRAN programs are developed and an illustrative numerical example is demonstrated.

Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes

We give several additive Schwarz domain decomposition methods for solving finite element problems which arise from the discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Our theory requires no assumption (for the main results) on the substructures which constitute the whole domain, so each substructure can be of arbitrary shape and of different size. The global coarse mesh is allowed to be non-nested to the fine grid on which the discrete problem is to be solved and both the coarse meshes and the fine meshes need not be quasi-uniform. In this general setting, our algorithms have the same optimal convergence rate of the usual domain decomposition methods on structured meshes. The condition numbers of the preconditioned systems depend only on the (possibly small) overlap of the substructures and the size of the coares grid, but is independent of the sizes of the subdomains.

A convergence proof for linear mean value cross decomposition

The mean value cross decomposition method for linear programming problems is a modification of ordinary cross decomposition that eliminates the need for using the Benders or Dantzig-Wolfe master problem. It is a generalization of the Brown-Robinson method for a finite matrix game and can also be considered as a generalization of the Kornai-Liptak method. It is based on the subproblem phase in cross decomposition, where we iterate between the dual subproblem and the primal subproblem. As input to the dual subproblem we use the average of a part of all dual solutions of the primal subproblem, and as input to the primal subproblem we use the average of a part of all primal solutions of the dual subproblem. In this paper we give a new proof of convergence for this procedure. Previously convergence has only been shown for the application to a special separable case (which covers the Kornai-Liptak method), by showing equivalence to the Brown-Robinson method.

A proximal-based decomposition method for convex minimization problems

This paper presents a decomposition method for solving convex minimization problems. At each iteration, the algorithm computes two proximal steps in the dual variables and one proximal step in the ...

A proximal-based decomposition method for convex minimization problems

This paper presents a decomposition method for solving convex minimization problems. At each iteration, the algorithm computes two proximal steps in the dual variables and one proximal step in the primal variables. We derive this algorithm from Rockafellar's proximal method of multipliers, which involves an augmented Lagrangian with an additional quadratic proximal term. The algorithm preserves the good features of the proximal method of multipliers, with the additional advantage that it leads to a decoupling of the constraints, and is thus suitable for parallel implementation. We allow for computing approximately the proximal minimization steps and we prove that under mild assumptions on the problem's data, the method is globally convergent and at a linear rate. The method is compared with alternating direction type methods and applied to the particular case of minimizing a convex function over a finite intersection of closed convex sets.

Multiplicative Schwarz Algorithms for Some Nonsymmetric and Indefinite Problems

The classical Schwarz alternating method has recently been generalized in several directions. This effort has resulted in a number of new powerful domain decomposition methods for elliptic problems, in new insight into multigrid methods, and in the development of a very useful framework for the analysis of a variety of iterative methods. Most of this work has focused on positive definite, symmetric problems. In this paper, a general framework is developed for multiplicative Schwarz algorithms for nonsymmetric and indefinite problems. Several applications are then discussed including two- and multilevel Schwarz methods and iterative substructuring algorithms. Some new results on additive Schwarz methods are also presented.

An Optimal Two-Level Overlapping Domain Decomposition Method for Elliptic Problems in Two and Three Dimensions

The solution of linear systems of algebraic equations that arise from elliptic finite element problems is considered. A two-level overlapping domain decomposition method that can be viewed as a combination of the additive and multiplicative Schwarz methods is studied. This method combines the advantages of the two methods. It converges faster than the additive Schwarz algorithm and is more parallelizable than the multiplicative Schwarz algorithm, and works for general, not necessarily self-adjoint, linear, second-order, elliptic equations. The GMRES method is used to solve the resulting preconditioned linear system of equations and it is shown that the algorithm is optimal in the sense that the rate of convergence is independent of the mesh size and the number of subregions in both $R^2 $ and $R^3 $. A numerical comparison with the additive and multiplicative Schwarz preconditioned GMRES is reported.