30,179 research outputs found
A variational principle for fluid sloshing with vorticity, dynamically coupled to vessel motion
A variational principle is derived for two-dimensional incompressible
rotational fluid flow with a free surface in a moving vessel when both the
vessel and fluid motion are to be determined. The fluid is represented by a
stream function and the vessel motion is represented by a path in the planar
Euclidean group. Novelties in the formulation include how the pressure boundary
condition is treated, the introduction of a stream function into the
Euler-Poincar\'e variations, the derivation of free surface variations, and how
the equations for the vessel path in the Euclidean group, coupled to the fluid
motion, are generated automatically.Comment: 19 pages, 3 figure
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
Constraint interface preconditioning for topology optimization problems
The discretization of constrained nonlinear optimization problems arising in
the field of topology optimization yields algebraic systems which are
challenging to solve in practice, due to pathological ill-conditioning, strong
nonlinearity and size. In this work we propose a methodology which brings
together existing fast algorithms, namely, interior-point for the optimization
problem and a novel substructuring domain decomposition method for the ensuing
large-scale linear systems. The main contribution is the choice of interface
preconditioner which allows for the acceleration of the domain decomposition
method, leading to performance independent of problem size.Comment: To be published in SIAM J. Sci. Com
Interior-point solver for convex separable block-angular problems
Constraints matrices with block-angular structures are pervasive in Optimization. Interior-point methods have shown to be competitive for these structured problems by exploiting the linear algebra. One of these approaches solved the normal equations using sparse Cholesky factorizations for the block constraints, and a preconditioned conjugate gradient (PCG) for the linking constraints. The preconditioner is based on a power series expansion which approximates the inverse of the matrix of the linking constraints system. In this work we present an efficient solver based on this algorithm. Some of its features are: it solves linearly constrained convex separable problems (linear, quadratic or nonlinear); both Newton and second-order predictor-corrector directions can be used, either with the Cholesky+PCG scheme or with a Cholesky factorization of normal equations; the preconditioner
may include any number of terms of the power series; for any number of these terms, it estimates the spectral radius of the matrix in the power series (which is instrumental for the quality of the precondi-
tioner). The solver has been hooked to SML, a structure-conveying modelling language based on the popular AMPL modeling language. Computational results are reported for some large and/or difficult instances in the literature: (1) multicommodity flow problems; (2) minimum congestion problems; (3) statistical data protection problems using l1 and l2 distances (which are linear and quadratic problems, respectively), and the pseudo-Huber function, a nonlinear approximation to l1 which improves the preconditioner. In the largest instances, of up to 25 millions of variables and 300000 constraints, this approach is from two to three orders of magnitude faster than state-of-the-art linear and quadratic optimization solvers.Preprin
Some Preconditioning Techniques for Saddle Point Problems
Saddle point problems arise frequently in many applications in science and engineering, including constrained optimization, mixed finite element formulations of partial differential equations, circuit analysis, and so forth. Indeed the formulation of most problems with constraints gives rise to saddle point systems. This paper provides a concise overview of iterative approaches for the solution of such systems which are of particular importance in the context of large scale computation. In particular we describe some of the most useful preconditioning techniques for Krylov subspace solvers applied to saddle point problems, including block and constrained preconditioners.\ud
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The work of Michele Benzi was supported in part by the National Science Foundation grant DMS-0511336
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