Alternating splitting waveform relaxation method and its successive overrelaxation acceleration

For the large sparse implicit linear initial value problem, we present a block successive overrelaxation scheme for the alternating direction implicit waveform relaxation method to further accelerate its convergence speed, and discuss the convergence property of the resulting iteration method in detail. Numerical implementations about several non-Hermitian implicit linear initial value problems show that the alternating direction implicit waveform relaxation method is very effective, and the block successive overrelaxation technique really accelerates its convergence speed.     

The generalized set-valued strongly nonlinear implicit variational inequalities

In this paper, we introduce and study a new class of generalized set-valued strongly nonlinear implicit variational inequalities in Hilbert spaces. We construct the algorithm for solving this kind of generalized set-valued strongly nonlinear implicit variational inequality problem by using the auxiliary principle technique of Glowinski, Lions, and Tremolieres, prove the existence of solutions for this class of generalized set-valued strongly nonlinear implicit variational inequalities and the convergence of iterative sequences generated by the algorithm.     

Optimally Convergent HDG Method for Third-Order Korteweg–de Vries Type Equations

We develop and analyze a new hybridizable discontinuous Galerkin method for solving third-order Korteweg–de Vries type equations. The approximate solutions are defined by a discrete version of a characterization of the exact solution in terms of the solutions to local problems on each element which are patched together through transmission conditions on element interfaces. We prove that the semi-discrete scheme is stable with proper choices of stabilization function in the numerical traces. For the linearized equation, we carry out error analysis and show that the approximations to the exact solution and its derivatives have optimal convergence rates. In numerical experiments, we use an implicit scheme for time discretization and the Newton–Raphson method for solving systems of nonlinear equations, and observe optimal convergence rates for both the linear and the nonlinear third-order equations.     

Incorporating illumination constraints in deformable models for shape from shading and light direction estimation

We present a method for the integration of nonlinear holonomic constraints in deformable models and its application to the problems of shape and illuminant direction estimation from shading. Experimental results demonstrate that our method performs better than previous Shape from Shading algorithms applied to images of Lambertian objects under known illumination. It is also more general as it can be applied to non-Lambertian surfaces and it does not require knowledge of the illuminant direction. In this paper, (1) we first develop a theory for the numerically robust integration of nonlinear holonomic constraints within a deformable model framework. In this formulation, we use Lagrange multipliers and a Baumgarte stabilization approach (1972). (2) We also describe a fast new method for the computation of constraint based forces, in the case of high numbers of local parameters. (3) We demonstrate how any type of illumination constraint, from the simple Lambertian model to more complex highly nonlinear models can be incorporated in a deformable model framework. (4) We extend our method to work when the direction of the light source is not known. We couple our shape estimation method with a method for light estimation, in an iterative process, where improved shape estimation results in improved light estimation and vice versa. (5) We perform a series of experiments.     

Pricing American bond options using a penalty method

We develop a novel numerical method to price American options on a discount bond under the Cox–Ingrosll–Ross (CIR) model which is governed by a partial differential complementarity problem. We first propose a penalty approach to this complementarity problem, resulting in a nonlinear partial differential equation (PDE). To numerically solve this nonlinear PDE, we develop a novel fitted finite volume method for the spatial discretization, coupled with a fully implicit time-stepping scheme. We show that this full discretization scheme is consistent, stable and monotone, and hence the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed. To solve the discretized nonlinear system, we design an iterative method and prove that the method is convergent. Numerical results are presented to demonstrate the accuracy, efficiency and robustness of our methods.     

Convergence and Optimality of the Adaptive Nonconforming Linear Element Method for the Stokes Problem

In this paper, we analyze the convergence and optimality of a standard adaptive nonconforming linear element method for the Stokes problem. After establishing a special quasi-orthogonality property for both the velocity and the pressure in this saddle point problem, we introduce a new prolongation operator to carry through the discrete reliability analysis for the error estimator. We then use a specially defined interpolation operator to prove that, up to oscillation, the error can be bounded by the approximation error within a properly defined nonlinear approximate class. Finally, by introducing a new parameter-dependent error estimator, we prove the convergence and optimality estimates.     

Operator-splitting methods for the 2D convective Cahn–Hilliard equation

In this work, we present operator-splitting methods for the two-dimensional nonlinear fourth-order convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. The full problem is split into hyperbolic, nonlinear diffusion and linear fourth-order problems. We prove that the semi-discrete approximate solution obtained from the operator-splitting method converges to the weak solution. Numerical methods are then constructed to solve each sub equations sequentially. The hyperbolic conservation law is solved by efficient finite volume methods and dimensional splitting method, while the one-dimensional hyperbolic conservation laws are solved using front tracking algorithm. The front tracking method is based on the exact solution and hence has no stability restriction on the size of the time step. The nonlinear diffusion problem is solved by a linearized implicit finite volume method, which is unconditionally stable. The linear fourth-order equation is solved using a pseudo-spectral method, which is based on an exact solution. Finally, some numerical experiments are carried out to test the performance of the proposed numerical methods.     

Deformable Objects Collision Handling with Fast Convergence

We present a stable and efficient simulator for deformable objects with collisions and contacts. For stability, an optimization derived from the implicit time integrator is solved in each timestep under the inequality constraints coming from collisions. To achieve fast convergence, we extend the MPRGP based solver from handling box constraints only to handling general linear constraints and prove its convergence. This generalization introduces a cost of solving dense linear systems in each step, but these systems can be reduced into diagonal ones for efficiency without affecting the general stability via pruning redundant collisions. Our solver is an order of magnitude faster, especially for elastic objects under large deformation compared with iterative constraint anticipation method (ICA), a typical method for stability. The efficiency, robustness and stability are further verified by our results.     

An efficient split-step quasi-compact finite difference method for the nonlinear fractional Ginzburg–Landau equations

In this paper, we propose a split-step quasi-compact finite difference method to solve the nonlinear fractional Ginzburg–Landau equations both in one and two dimensions. The original equations are split into linear and nonlinear subproblems. The Riesz space fractional derivative is approximated by a fourth-order fractional quasi-compact method. Furthermore, an alternating direction implicit scheme is constructed for the two dimensional linear subproblem. The unconditional stability and convergence of the schemes are proved rigorously in the linear case. Numerical experiments are performed to confirm our theoretical findings and the efficiency of the proposed method.     

A general iterative method with strongly positive operators for general variational inequalities

In this paper, we introduce and study a general iterative method with strongly positive operators for finding solutions of a general variational inequality problem with inverse-strongly monotone mapping in a real Hilbert space. The explicit and implicit iterative algorithms are proposed by virtue of the general iterative method with strongly positive operators. Under two sets of quite mild conditions, we prove the strong convergence of these explicit and implicit iterative algorithms to the unique common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality problem, respectively.