Zdenek Dostál

Box Constrained Quadratic Programming with Proportioning and Projections

Two new closely related concepts are introduced that depend on a positive constant

Augmented Lagrangians with Adaptive Precision Control for Quadratic Programming with Simple Bounds and Equality Constraints

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Solution of contact problems by FETI domain decomposition with natural coarse space projections

. An iteration is proportional if the norm of violation of the Kuhn--Tucker conditions at active variables does not excessively exceed the norm of the part of the gradient that corresponds to free variables, while a progressive direction determines a descent direction that enables the released variables to move far enough from the boundary in a step called proportioning. An algorithm that uses the conjugate gradient method to explore the face of the region defined by the current iterate until a disproportional iteration is generated is proposed. It then changes the face by means of the progressive direction. It is proved that for strictly convex problems, the proportioning is a spacer iteration so that the algorithm converges to the solution. If the solution is nondegenerate then the algorithm finds the solution in a finite number of steps. Moreover, a simple lower bound on

An Optimal Algorithm for Bound and Equality Constrained Quadratic Programming Problems with Bounded Spectrum

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A scalable TFETI algorithm for two-dimensional multibody contact problems with friction

is given to ensure finite termination even for problems with degenerate solutions. The theory covers a class of algorithms, allowing many constraints to be added or dropped at a time and accepting approximate solutions of auxiliary problems. Preliminary numerical results are promising.

Total FETI domain decomposition method and its massively parallel implementation

In this paper we discuss a specialization of the augmented Lagrangian-type algorithm of Conn, Gould, and Toint to the solution of strictly convex quadratic programming problems with simple bounds and equality constraints. The new feature of the presented algorithm is the adaptive precision control of the solution of auxiliary problems in the inner loop of the basic algorithm which yields a rate of convergence that does not have any term that accounts for inexact solution of auxiliary problems. Moreover, boundedness of the penalty parameter is achieved for the precision control used. Numerical experiments illustrate the efficiency of the presented algorithm and encourage its usage.

A scalable FETI-DP algorithm with non-penetration mortar conditions on contact interface

Abstract An efficient non-overlapping domain decomposition algorithm of the Neumann–Neumann type for solving both coercive and semicoercive contact problems is presented. The discretized problem is first turned by the duality theory of convex programming to the quadratic programming problem with bound and equality constraints and the latter is further modified by means of orthogonal projectors to the natural coarse space introduced by Farhat and Roux in the framework of their FETI method. The resulting problem is then solved by an augmented Lagrangian type algorithm with an outer loop for the Lagrange multipliers for the equality constraints and an inner loop for the solution of the bound constrained quadratic programming problems. The projectors are shown to guarantee fast convergence of iterative solution of auxiliary linear problems and to comply with efficient quadratic programming algorithms proposed earlier. Reported theoretical results and numerical experiments indicate high numerical scalability of the algorithm which preserves the parallelism of the FETI methods.

Scalable Total BETI based algorithm for 3D coercive contact problems of linear elastostatics

An implementation of the recently proposed semi-monotonic augmented Lagrangian algorithm for solving the large convex bound and equality constrained quadratic programming problems is considered. It is proved that if the algorithm is applied to the class of problems with uniformly bounded spectrum of the Hessian matrix, then the algorithm finds an approximate solution at O(1) matrix-vector multiplications. The optimality results are presented that do not depend on conditioning of the matrix which defines the equality constraints. Theory covers also the problems with dependent constraints. Theoretical results are illustrated by numerical experiments.

Separable Spherical Constraints and the Decrease of a Quadratic Function in the Gradient Projection Step

A Total FETI (TFETI) based domain decomposition algorithm with preconditioning by a natural coarse grid of rigid body motions is adapted to the solution of two-dimensional multibody contact problems of elasticity with the Coulomb friction and proved to be scalable for the Tresca friction. The algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition parameter and the discretization parameter is bounded. The analysis is based on the classical results by Farhat, Mandel, and Roux on scalability of FETI with a natural coarse grid for linear problems and on our development of optimal quadratic programming algorithms for bound and equality constrained problems. The algorithm preserves parallel scalability of the classical FETI method. Both theoretical results and numerical experiments indicate a high efficiency of our algorithm. In addition, its performance is illustrated on analysis of the yielding clamp connection with the Coulomb friction.

Augmented Lagrangians with Adaptive Precision Control for Quadratic Programming with Equality Constraints: Corrigendum and Addendum

We describe an efficient massively parallel implementation of our variant of the FETI type domain decomposition method called Total FETI with a lumped preconditioner. A special attention is paid to the discussion of several variants of parallelization of the action of the projections to the natural coarse grid and to the effective regularization of the stiffness matrices of the subdomains. Both numerical and parallel scalability of the proposed TFETI method are demonstrated on a 2D elastostatic benchmark up to 314,505,600 unknowns and 4800cores. The results are also important for implementation of scalable algorithms for the solution of nonlinear contact problems of elasticity by TFETI based domain decomposition method.