New Relaxation Modulus Based Iterative Method for Large and Sparse Implicit Complementarity Problem

This article presents a class of new relaxation modulus-based iterative methods to process the large and sparse implicit complementarity problem (ICP). Using two positive diagonal matrices, we formulate a fixed-point equation and prove that it is equivalent to ICP. Also, we provide sufficient convergence conditions for the proposed methods when the system matrix is a $P$-matrix or an $H_+$-matrix. Keyword: Implicit complementarity problem, $H_{+}$-matrix, $P$-matrix, matrix splitting, convergenceComment: arXiv admin note: substantial text overlap with arXiv:2303.1251

Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming

Cover title.Includes bibliographical references.Partially supported by the U.S. Army Research Office (Center for Intelligent Control Systems) DAAL03-86-K-0171 Partially supported by the National Science Foundation. NSF-ECS-8519058Paul Tseng

The Reduced Order Method for Solving the Linear Complementarity Problem with an M-Matrix

In this paper, by seeking the zero and the positive entry positions of the solution, we provide a direct method, called the reduced order method, for solving the linear complementarity problem with an M-matrix. By this method, the linear complementarity problem is transformed into a low order linear complementarity problem with some low order linear equations and the solution is constructed by the solution of the low order linear complementarity problem and the solutions of these low order linear equations in the transformations. In order to show the accuracy and the effectiveness of the method, the corresponding numerical experiments are performed

Newton-type Alternating Minimization Algorithm for Convex Optimization

We propose NAMA (Newton-type Alternating Minimization Algorithm) for solving structured nonsmooth convex optimization problems where the sum of two functions is to be minimized, one being strongly convex and the other composed with a linear mapping. The proposed algorithm is a line-search method over a continuous, real-valued, exact penalty function for the corresponding dual problem, which is computed by evaluating the augmented Lagrangian at the primal points obtained by alternating minimizations. As a consequence, NAMA relies on exactly the same computations as the classical alternating minimization algorithm (AMA), also known as the dual proximal gradient method. Under standard assumptions the proposed algorithm possesses strong convergence properties, while under mild additional assumptions the asymptotic convergence is superlinear, provided that the search directions are chosen according to quasi-Newton formulas. Due to its simplicity, the proposed method is well suited for embedded applications and large-scale problems. Experiments show that using limited-memory directions in NAMA greatly improves the convergence speed over AMA and its accelerated variant

A numerical method for fluid-structure interactions of slender rods in turbulent flow

This thesis presents a numerical method for the simulation of fluid-structure interaction (FSI) problems on high-performance computers. The proposed method is specifically tailored to interactions between Newtonian fluids and a large number of slender viscoelastic structures, the latter being modeled as Cosserat rods. From a numerical point of view, such kind of FSI requires special techniques to reach numerical stability. When using a partitioned fluid-structure coupling approach this is usually achieved by an iterative procedure, which drastically increases the computational effort. In the present work, an alternative coupling approach is developed based on an immersed boundary method (IBM). It is unconditionally stable and exempt from any global iteration between the fluid part and the structure part. The proposed FSI solver is employed to simulate the flow over a dense layer of vegetation elements, usually designated as canopy flow. The abstracted canopy model used in the simulation consists of 800 strip-shaped blades, which is the largest canopy-resolving simulation of this type done so far. To gain a deeper understanding of the physics of aquatic canopy flows the simulation data obtained are analyzed, e.g., concerning the existence and shape of coherent structures

Exploring novel designs of NLP solvers: Architecture and Implementation of WORHP

Mathematical Optimization in general and Nonlinear Programming in particular, are applied by many scientific disciplines, such as the automotive sector, the aerospace industry, or the space agencies. With some established NLP solvers having been available for decades, and with the mathematical community being rather conservative in this respect, many of their programming standards are severely outdated. It is safe to assume that such usability shortcomings impede the wider use of NLP methods; a representative example is the use of static workspaces by legacy FORTRAN codes. This dissertation gives an account of the construction of the European NLP solver WORHP by using and combining software standards and techniques that have not previously been applied to mathematical software to this extent. Examples include automatic code generation, a consistent reverse communication architecture and the elimination of static workspaces. The result is a novel, industrial-grade NLP solver that overcomes many technical weaknesses of established NLP solvers and other mathematical software