Algorithm For Saddle Point Problems Research Articles

Unified linear convergence of first-order primal-dual algorithms for saddle point problems

Unified linear convergence of first-order primal-dual algorithms for saddle point problems

Approximate first-order primal-dual algorithms for saddle point problems

Approximate first-order primal-dual algorithms for saddle point problems

Inexact first-order primal\u2013dual algorithms

We investigate the convergence of a recently popular class of first-order primal–dual algorithms for saddle point problems under the presence of errors in the proximal maps and gradients. We study several types of errors and show that, provided a sufficient decay of these errors, the same convergence rates as for the error-free algorithm can be established. More precisely, we prove the (optimal) Oleft( 1/Nright) convergence to a saddle point in finite dimensions for the class of non-smooth problems considered in this paper, and prove a Oleft( 1/N^2right) or even linear Oleft( theta ^Nright) convergence rate if either the primal or dual objective respectively both are strongly convex. Moreover we show that also under a slower decay of errors we can establish rates, however slower and directly depending on the decay of the errors. We demonstrate the performance and practical use of the algorithms on the example of nested algorithms and show how they can be used to split the global objective more efficiently.

Analysis of BDDC algorithms for Stokes problems with hybridizable discontinuous Galerkin discretizations

The BDDC (balancing domain decomposition by constraints) methods have been applied to solve the saddle point problem arising from a hybridizable discontinuous Galerkin (HDG) discretization of the incompressible Stokes problem. In the BDDC algorithms, the coarse problem is composed by the edge/face constraints across the subdomain interface for each velocity component. As for the standard approaches of the BDDC algorithms for saddle point problems, these constraints ensure that the BDDC preconditioned conjugate gradient (CG) iterations stay in a subspace where the preconditioned operator is positive definite. However, there are several popular choices of the local stabilization parameters used in the HDG discretizations. Different stabilization parameters change the properties of the resulting discretized operators, and some special observations and tools are needed in the analysis of the condition numbers of the BDDC preconditioned Stokes operators. In this paper, condition number estimates for different choices of stabilization parameters are provided. Numerical experiments confirm the theory.

A class of accelerated Uzawa algorithms for saddle point problems

In this paper, we establish a class of accelerated Uzawa (AU) algorithms for solving the large sparse nonsingular saddle point problems by making use of the extrapolation technique. This extrapolation technique is based on the eigenvalues of the iterative matrix. These AU algorithms involve two iteration parameters whose special choices can cover the known classical Uzawa method, as well as yield new ones. Firstly, the accelerated model for the Uzawa algorithm is established and the detail algorithm description of AU method is presented. Then the convergence analyse of the AU method is given. Moreover, theoretical analyses show that the AU algorithm converges faster than some Uzawa-type methods (the Uzawa method is also included in) when the eigenvalues of the iterative matrix and the parameter τ satisfy some conditions. Numerical experiments on a few model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of the AU method.

On the iterative algorithm for saddle point problems

In this paper, we point out that the generalized iteration method presented in a recent paper by Ling and Hu [X.-F. Ling, X.-Z. Hu, On the iterative algorithm for large sparse saddle point problems, Appl. Math. Comput. 178 (2006) 372–379] belongs to the inexact Uzawa algorithm, and an improper statement in a theorem in that paper is corrected.

A convergence analysis of the nonlinear Uzawa algorithm for saddle point problems

In this paper we discuss the convergence behavior of the nonlinear inexact Uzawa algorithm for solving saddle point problems presented in a recent paper by Cao [Z.H. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math. 46 (2003) 157–171]. We show that this algorithm converges under a condition weaker than that stated in this paper.

A Unified Approach for Uzawa Algorithms

We present a unified approach in analyzing Uzawa iterative algorithms for saddle point problems. We study the classical Uzawa method, the augmented Lagrangian method, and two versions of inexact Uzawa algorithms. The target application is the Stokes system, but other saddle point systems, e.g., arising from mortar methods or Lagrange multipliers methods, can benefit from our study. We prove convergence of Uzawa algorithms and find optimal rates of convergence in an abstract setting on finite‐ or infinite‐dimensional Hilbert spaces. The results can be used to design multilevel or adaptive algorithms for solving saddle point problems. The discrete spaces do not have to satisfy the LBB stability condition.

A new nonlinear Uzawa algorithm for generalized saddle point problems

In [J.H. Bramble, I.E. Pasciak, A.T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal. 34 (1997) 1072–1092], a nonlinear Uzawa algorithm for solving generalized saddle point problems iteratively was considered. In [Z. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math. 46 (2003) 157–171], Cao gave another nonlinear Uzawa algorithm in order to accelerate convergence. In this paper, a new nonlinear Uzawa method, which is defined by two nonlinear approximate inverses, is proposed and its convergence result is deduced. At the same time, we analyze convergence results of three nonlinear Uzawa methods. The results of numerical experiments are presented when we apply them to Stokes equations discretized by mixed finite element.

Adaptive wavelet schemes for an elliptic control problem with Dirichlet boundary control

An adaptive algorithm based on wavelets is proposed for the fast numerical solution of control problems governed by elliptic boundary value problems with Dirichlet boundary control. A quadratic cost functional representing Sobolev norms of the state and a regularization in terms of the control is to be minimized subject to linear constraints in weak form. In particular, the constraints are formulated as a saddle point problem that allows to handle the varying boundary conditions explicitly. In the framework of (biorthogonal) wavelets, a representer for the functional is derived in terms of l2-norms of wavelet expansion coefficients and the constraints are written in form of an l2 automorphism. Standard techniques from optimization are then used to deduce the resulting first order necessary conditions as a (still infinite) system in l2. Applying the machinery developed in [8,9] which has been extended to control problems in [14], an adaptive method is proposed which can be interpreted as an inexact gradient method for the control. In each iteration step, in turn the primal and the adjoint saddle point system are solved up to a prescribed accuracy by an adaptive iterative Uzawa algorithm for saddle point problems which has been proposed in [10]. Under these premises, it can be shown that the adaptive algorithm containing now three layers of iterations is asymptotically optimal. This means that the convergence rate achieved for computing the solution up to a desired target tolerance is asymptotically the same as the wavelet-best N-term approximation of the solution, and the total computational work is proportional to the number of computational unknowns.

Optimization of one class of nonsymmetrizable algorithms for saddle point problems

To solve the nondegenerate system of linear special-type equations we consider a threeparameter algorithm which is a generalization of the known Arrow-Hurwicz algorithm. For the proposed algorithm we solve an asymptotic optimization problem in two different statements. We obtain analytic formulae for asymptotically optimal parameters and the convergence factor. The results obtained are of an unimprovable character for the studied class of algorithms.

A sufficient condition for the convergence of the inexact Uzawa algorithm for saddle point problems

In this paper, the convergence property of the inexact Uzawa algorithm for solving symmetric indefinite linear systems is studied. A simple sufficient condition for the convergence of the inexact Uzawa algorithm is obtained. Two examples and numerical experiments illustrating the conclusion are provided.

Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems

In this paper, we consider the so-called "inexact Uzawa" algorithm for iteratively solving linear block saddle point problems. Such saddle point problems arise, for example, in finite element and finite difference discretizations of Stokes equations, the equations of elasticity, and mixed finite element discretization of second-order problems. We consider both the linear and nonlinear variants of the inexact Uzawa iteration. We show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled. Bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned Uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left-hand block. In the case of nonlinear iteration, the inexact Uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left-hand block is of sufficient accuracy. Bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear Uzawa algorithm. Applications to the Stokes equations and mixed finite element discretization of second-order elliptic problems are discussed and, finally, the results of numerical experiments involving the algorithms are presented.

On the convergence of the proximal algorithm for saddle-point problems

The proximal algorithm for saddle-point problems min subx∊X max suby∊Y L(x,y), where $si:X$esi:,$si:Y$esi: are Hilbert spaces and is a proper, closed convex-concave function in X×Y is considered. Under a minimal assumption on L,δ:= inf sup L= sup inf L, it is proved that the proximmal point algorithm (PPA), with positive parameters {λ n }, converges in general provided that .

Inexact and Preconditioned Uzawa Algorithms for Saddle Point Problems

Variants of the Uzawa algorithm for solving symmetric indefinite linear systems are developed and analyzed. Each step of this algorithm requires the solution of a symmetric positive-definite system of linear equations. It is shown that if this computation is replaced by an approximate solution produced by an arbitrary iterative method, then with relatively modest requirements on the accuracy of the approximate solution, the resulting inexact Uzawa algorithm is convergent, with a convergence rate close to that of the exact algorithm. In addition, it is shown that preconditioning can be used to improve performance. The analysis is illustrated and supplemented using several examples derived from mixed finite element discretization of the Stokes equations.