153 research outputs found
Model order reduction for neutral systems by moment matching
published_or_final_versio
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
Solving variational inequalities and cone complementarity problems in nonsmooth dynamics using the alternating direction method of multipliers
This work presents a numerical method for the solution of variational inequalities arising in nonsmooth flexible multibody problems that involve set-valued forces. For the special case of hard frictional contacts, the method solves a second order cone complementarity problem. We ground our algorithm on the Alternating Direction Method of Multipliers (ADMM), an efficient and robust optimization method that draws on few computational primitives. In order to improve computational performance, we reformulated the original ADMM scheme in order to exploit the sparsity of constraint jacobians and we added optimizations such as warm starting and adaptive step scaling. The proposed method can be used in scenarios that pose major difficulties to other methods available in literature for complementarity in contact dynamics, namely when using very stiff finite elements and when simulating articulated mechanisms with odd mass ratios. The method can have applications in the fields of robotics, vehicle dynamics, virtual reality, and multiphysics simulation in general
GMRES-Accelerated ADMM for Quadratic Objectives
We consider the sequence acceleration problem for the alternating direction
method-of-multipliers (ADMM) applied to a class of equality-constrained
problems with strongly convex quadratic objectives, which frequently arise as
the Newton subproblem of interior-point methods. Within this context, the ADMM
update equations are linear, the iterates are confined within a Krylov
subspace, and the General Minimum RESidual (GMRES) algorithm is optimal in its
ability to accelerate convergence. The basic ADMM method solves a
-conditioned problem in iterations. We give
theoretical justification and numerical evidence that the GMRES-accelerated
variant consistently solves the same problem in iterations
for an order-of-magnitude reduction in iterations, despite a worst-case bound
of iterations. The method is shown to be competitive against
standard preconditioned Krylov subspace methods for saddle-point problems. The
method is embedded within SeDuMi, a popular open-source solver for conic
optimization written in MATLAB, and used to solve many large-scale semidefinite
programs with error that decreases like , instead of ,
where is the iteration index.Comment: 31 pages, 7 figures. Accepted for publication in SIAM Journal on
Optimization (SIOPT
Numerical Methods for Mixed-Integer Optimal Control with Combinatorial Constraints
This thesis is concerned with numerical methods for Mixed-Integer Optimal Control Problems with Combinatorial Constraints. We establish an approximation theorem
relating a Mixed-Integer Optimal Control Problem with Combinatorial Constraints to a continuous relaxed convexified Optimal Control Problems with Vanishing Constraints that provides the basis for numerical computations. We develop a a Vanishing-
Constraint respecting rounding algorithm to exploit this correspondence computationally.
Direct Discretization of the Optimal Control Problem with Vanishing Constraints yield a subclass of Mathematical Programs with Equilibrium Constraints. Mathematical Programs with Equilibrium Constraint constitute a class of challenging problems due to their inherent non-convexity and non-smoothness. We develop an active-set
algorithm for Mathematical Programs with Equilibrium Constraints and prove global convergence to Bouligand stationary points of this algorithm under suitable technical conditions.
For efficient computation of Newton-type steps of Optimal Control Problems, we establish the Generalized Lanczos Method for trust region problems in a Hilbert space
context. To ensure real-time feasibility in Online Optimal Control Applications with tracking-type Lagrangian objective, we develop a Gauß-Newton preconditioner for
the iterative solution method of the trust region problem.
We implement the proposed methods and demonstrate their applicability and efficacy on several benchmark problems
Convex relaxations of a continuum aggregation model, and their efficient numerical solution
In this dissertation, the global minimization of a large deviations rate function (the Helmholtz free energy functional) for the Boltzmann distribution is discussed. The Helmholtz functional arises in large systems of interacting particles — which are widely used as models in computational chemistry and molecular dynamics. Global minimizers of the rate function (Helmholtz functional) characterize the asymptotics of the partition function and thereby determine many important physical properties such as self-assembly, or phase transitions. Finding and verifying local minima to the Helmholtz free energy functional is relatively straightforward. However, finding and verifying global minima is much more difficult since the Helmholtz energy is nonconvex and nonlocal. Instead of minimizing the original nonconvex functional, the approach in this dissertation is to find minimizers to a convex lower bound functional. The so-called relaxed problem consists of a linear variational problem with an infinite number of Fourier constraints, leading to a variety of computational challenges. A fast solver (for the relaxed problem) based on matrix-free interior-point algorithms is developed by exploiting the Fourier structure in the problem in conjunction with a new preconditioner
- …