126 research outputs found
Pivoting in Linear Complementarity: TwoPolynomial-Time Cases
We study the behavior of simple principal pivoting methods for the P-matrix linear complementarity problem (P-LCP). We solve an open problem of Morris by showing that Murty's least-index pivot rule (under any fixed index order) leads to a quadratic number of iterations on Morris's highly cyclic P-LCP examples. We then show that on K-matrix LCP instances, all pivot rules require only a linear number of iterations. As the main tool, we employ unique-sink orientations of cubes, a useful combinatorial abstraction of the P-LC
Improved Full-Newton-Step Infeasible Interior-Point Method for Linear Complementarity Problems
In this thesis, we present an improved version of Infeasible Interior-Point Method (IIPM) for monotone Linear Complementarity Problem (LCP). One of the most important advantages of this version in compare to old version is that it only requires feasibility steps. In the earlier version, each iteration consisted of one feasibility step and some centering steps (at most three in practice). The improved version guarantees that after one feasibility step, the new iterated point is feasible and close enough to central path. Thus, the centering steps are eliminated. This improvement is based on the Lemma(Roos, 2015). Thanks to this lemma, proximity of the new point after the feasibility step is guaranteed with a more strict upper bound. Another advantage of this method is that it uses full-Newton steps, which means that no calculation of the step size is required at each iteration and that the cost is decreased. The implementation and numerical results demonstrate the reliability of the method
Application of Operator Splitting Methods in Finance
Financial derivatives pricing aims to find the fair value of a financial
contract on an underlying asset. Here we consider option pricing in the partial
differential equations framework. The contemporary models lead to
one-dimensional or multidimensional parabolic problems of the
convection-diffusion type and generalizations thereof. An overview of various
operator splitting methods is presented for the efficient numerical solution of
these problems.
Splitting schemes of the Alternating Direction Implicit (ADI) type are
discussed for multidimensional problems, e.g. given by stochastic volatility
(SV) models. For jump models Implicit-Explicit (IMEX) methods are considered
which efficiently treat the nonlocal jump operator. For American options an
easy-to-implement operator splitting method is described for the resulting
linear complementarity problems.
Numerical experiments are presented to illustrate the actual stability and
convergence of the splitting schemes. Here European and American put options
are considered under four asset price models: the classical Black-Scholes
model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV
model with jumps
A full multigrid method for linear complementarity problems arising from elastic normal contact problems
This paper presents a full multigrid (FMG) technique, which combines
a multigrid method, an active set algorithm and a nested iteration technique, to solve
a linear complementarity problem (LCP) modeling elastic normal contact problems.
The governing system in this LCP is derived from a Fredholm integral of the rst
kind, and its coecient matrix is dense, symmetric and positive denite. One multigrid
cycle is applied to solve this system approximately in each active set iteration.
Moreover, this multigrid solver incorporates a special strategy to handle the complementarity
conditions, including restricting both the defect and the contact area
(active set) to the coarse grid, and setting all quantities outside contact to zero.
The smoother is chosen by some analysis based on the eigenvectors of the iteration
matrix. This method is applied to a Hertzian smooth contact and a rough surface
contact problem
Kinetic Structure Simulations of Nematic Polymers in Plane Couette Cells. II: In-plane structure transitions
Nematic, or liquid crystalline, polymer (LCP) composites are composed of large aspect ratio rod-like or platelet macromolecules. This class of nanocomposites exhibits tremendous potential for high performance material applications, ranging across mechanical, electrical, piezoelectric, thermal, and barrier properties. Fibers made from nematic polymers have set synthetic materials performance standards for decades. The current target is to engineer multifunctional films and molded parts, for which processing flows are shear-dominated. Nematic polymer films inherit anisotropy from collective orientational distributions of the molecular constituents and develop heterogeneity on length scales that are, as yet, not well understood and thereby uncontrollable. Rigid LCPs in viscous solvents have a theoretical and computational foundation from which one can model parallel plate Couette cell experiments and explore anisotropic structure generation arising from nonequilibrium interactions between hydrodynamics, molecular short- and long-range elasticity, and confinement effects. Recent progress on the longwave limit of homogeneous nematic polymers in imposed simple shear and linear planar flows [R. G. Larson and H. Ottinger, Macromolecules, 24 (1991), pp. 6270-6282], [V. Faraoni, M. Grosso, S. Crescitelli, and P. L. Maffettone, J. Rheol., 43 (1999), pp. 829-843], [M. Grosso, R. Keunings, S. Crescitelli, and P. L. Maffettone, Phys. Rev. Lett. 86 (2001), pp. 3184-3187], [M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 43 (2004), pp. 17-37], [M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 44 (2004), pp. 80-93], [M. G. Forest, Q. Wang, R. Zhou, and E. Choate, J. Non-Newtonian Fluid Mech., 118 (2004), pp. 17-31], [M. G. Forest, R. Zhou, and Q. Wang, Phys. Rev. Lett., 93 (2004), 088301] provides resolved kinetic simulations of the molecular orientational distribution. These results characterize anisotropy and dynamic attractors of sheared bulk domains and underscore limitations of mesoscopic models for orientation of the rigid rod or platelet ensembles. In this paper, we apply our resolved kinetic structure code [R. Zhou, M. G. Forest, and Q. Wang, Multiscale Model. Simul., 3 (2005), pp. 853-870] to model onset and saturation of heterogeneity in the orientational distribution by coupling a distortional elasticity potential (with distinct elasticity constants) and anchoring conditions in a plane Couette cell. For this initial study, the flow field is imposed and the orientational distribution is confined to the shear deformation plane, which affords comparison with seminal [T. Tsuji and A. D. Rey, Phys. Rev. E (3), 62 (2000), pp. 8141-8151] as well as our own mesoscopic model simulations [H. Zhou, M. G. Forest, and Q. Wang, J. Non-Newtonian Fluid Mech., submitted], [H. Zhou and M. G. Forest, Discrete Contin. Dyn. Syst. Ser. B, to appear]. Under these controlled conditions, we map out the kinetic phase diagram of spatiotemporal attractors of a Couette cell film in the two-parameter space of Deborah number (normalized shear rate) and Ericksen number (relative strength of elasticity potentials). © 2005 Society for Industrial and Applied Mathematics
Pivoting in Linear Complementarity: Two Polynomial-Time Cases
We study the behavior of simple principal pivoting methods for the P-matrix linear complementarity problem (P-LCP). We solve an open problem of Morris by showing that Murtyâs least-index pivot rule (under any fixed index order) leads to a quadratic number of iterations on Morrisâs highly cyclic P-LCP examples. We then show that on K-matrix LCP instances, all pivot rules require only a linear number of iterations. As the main tool, we employ unique-sink orientations of cubes, a useful combinatorial abstraction of the P-LCP
Mathematical Models and Numerical Methods for Pricing Options on Investment Projects under Uncertainties
In this work, we focus on establishing partial differential equation (PDE) models for pricing flexibility options on investment projects under uncertainties and numerical methods for solving these models. we develop a finite difference method and an advanced fitted finite volume scheme and combine with an interior penalty method, as well as their convergence analyses, to solve the PDE and LCP models developed. The MATLAB program is for implementing testing the models of numerical algorithms developed
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
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