High Order Fast Sweeping Methods for Static Hamilton–Jacobi Equations

We construct high order fast sweeping numerical methods for computing viscosity solutions of static Hamilton–Jacobi equations on rectangular grids. These methods combine high order weighted essentially non-oscillatory (WENO) approximations to derivatives, monotone numerical Hamiltonians and Gauss–Seidel iterations with alternating-direction sweepings. Based on well-developed first order sweeping methods, we design a novel approach to incorporate high order approximations to derivatives into numerical Hamiltonians such that the resulting numerical schemes are formally high order accurate and inherit the fast convergence from the alternating sweeping strategy. Extensive numerical examples verify efficiency, convergence and high order accuracy of the new methods.     

A Dirichlet–Neumann type algorithm for contact problems with friction

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. We introduce a new algorithm for the numerical solution of a nonlinear contact problem with Coulomb friction between linear elastic bodies. The discretization of the nonlinear problem is based on mortar techniques. We use a dual basis Lagrange multiplier space for the coupling of the different bodies. The boundary data transfer at the contact zone is essential for the algorithm. It is realized by a scaled mass matrix which results from the mortar discretization on non-matching triangulations. We apply a nonlinear block Gauss–Seidel method as iterative solver which can be interpreted as a Dirichlet–Neumann algorithm for the nonlinear problem. In each iteration step, we have to solve a linear Neumann problem and a nonlinear Signorini problem. The solution of the Signorini problem is realized in terms of monotone multigrid methods. Numerical results illustrate the performance of our approach in 2D and 3D. Received: 20 March 2001 / Accepted: 1 February 2002 Communicated by P. Deuflhard     

Propagating systems of dense linear integer constraints

In interval propagation approaches to solving nonlinear constraints over reals it is common to build stronger propagators from systems of linear equations. This, as far as we are aware, is not pursued for integer finite domain propagation. In this paper we show how we use interval Gauss–Jordan elimination to build stronger propagators for an integer propagation solver. In a similar fashion we present an interval Fourier elimination preconditioning technique to generate redundant linear constraints from a system of linear inequalities. We show how to convert the resulting interval propagators into integer propagators. This allows us to use existing integer solvers. We give experiments that show how these preconditioning techniques can improve propagation performance on dense systems.     

Gauss–Newton Multilevel Methods for Least-Squares Finite Element Computations of Variably Saturated Subsurface Flow

We apply the least-squares mixed finite element framework to the nonlinear elliptic problems arising in each time-step of an implicit Euler discretization for variably saturated flow. This approach allows the combination of standard piecewise linear H ¹-conforming finite elements for the hydraulic potential with the H(div)-conforming Raviart–Thomas spaces for the flux. It also provides an a posteriori error estimator which may be used in an adaptive mesh refinement strategy. The resulting nonlinear algebraic least-squares problems are solved by an inexact Gauss–Newton method using a stopping criterion for the inner iteration which is based on the change of the linearized least-squares functional relative to the nonlinear least-squares functional. The inner iteration is carried out using an adaptive multilevel method with a block Gauss–Seidel smoothing iteration. For a realistic water table recharge problem, the results of computational experiments are presented. Received January 4, 1999; revised July 19, 1999     

Finite-element Discretization of Static Hamilton-Jacobi Equations based on a Local Variational Principle

We propose a linear finite-element discretization of Dirichlet problems for static Hamilton–Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss–Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.Dedicated to Peter Deuflhard on the occasion of his 60th birthday     

A block-asynchronous relaxation method for graphics processing units

In this paper, we analyze the potential of asynchronous relaxation methods on Graphics Processing Units (GPUs). We develop asynchronous iteration algorithms in CUDA and compare them with parallel implementations of synchronous relaxation methods on CPU- or GPU-based systems. For a set of test matrices from UFMC we investigate convergence behavior, performance and tolerance to hardware failure. We observe that even for our most basic asynchronous relaxation scheme, the method can efficiently leverage the GPUs computing power and is, despite its lower convergence rate compared to the Gauss–Seidel relaxation, still able to provide solution approximations of certain accuracy in considerably shorter time than Gauss–Seidel running on CPUs- or GPU-based Jacobi. Hence, it overcompensates for the slower convergence by exploiting the scalability and the good fit of the asynchronous schemes for the highly parallel GPU architectures. Further, enhancing the most basic asynchronous approach with hybrid schemes–using multiple iterations within the “subdomain” handled by a GPU thread block–we manage to not only recover the loss of global convergence but often accelerate convergence of up to two times, while keeping the execution time of a global iteration practically the same. The combination with the advantageous properties of asynchronous iteration methods with respect to hardware failure identifies the high potential of the asynchronous methods for Exascale computing.     

On Solutions of Linear Ordinary Difference Equations in their Coefficient Field

We extend the notion of monomial extensions of differential fields, i.e. simple transcendental extensions in which the polynomials are closed under differentiation, to difference fields. The structure of such extensions provides an algebraic framework for solving generalized linear difference equations with coefficients in such fields. We then describe algorithms for finding the denominator of any solution of those equations in an important subclass of monomial extensions that includes transcendental indefinite sums and products. This reduces the general problem of finding the solutions of such equations in their coefficient fields to bounding their degrees. In the base case, this yields in particular a new algorithm for computing the rational solutions of q -difference equations with polynomial coefficients.     

Decomposition of arithmetic expressions to improve the behavior of interval iteration for nonlinear systems

Interval iteration can be used, in conjunction with other techniques, for rigorously bounding all solutions to a nonlinear system of equations within a given region, or for verifying approximate solutions. However, because of overestimation which occurs when the interval Jacobian matrix is accumulated and applied, straightforward linearization of the original nonlinear system sometimes leads to nonconvergent iteration. In this paper, we examine interval iterations based on an expanded system obtained from the intermediate quantities in the original system. In this system, there is no overestimation in entries of the interval Jacobi matrix, and nonlinearities can be taken into account to obtain sharp bounds. We present an example in detail, algorithms, and detailed experimental results obtained from applying our algorithms to the example.     

On some new approximate factorization methods for block tridiagonal matrices suitable for vector and parallel processors

In this paper, to obtain an efficient parallel algorithm to solve sparse block-tridiagonal linear systems, stair matrices are used to construct some parallel polynomial approximate inverse preconditioners. These preconditioners are suitable when the desired goal is to maximize parallelism. Moreover, some theoretical results concerning these preconditioners are presented and how to construct preconditioners effectively for any nonsingular block tridiagonal H-matrices is also described. In addition, the validity of these preconditioners is illustrated with some numerical experiments arising from the second order elliptic partial differential equations and oil reservoir simulations.     

Tight and efficient surface bounds in meshless animation

This paper presents a fast approach for computing tight surface bounds in meshless animation, and its application to collision detection. Given a high-resolution surface animated by a comparatively small number of simulation nodes, we are able to compute tight bounding volumes with a cost linear in the number of simulation nodes. Our approach extends concepts about bounds of convex sets to the meshless deformation setting, and we introduce an efficient algorithm for finding extrema of these convex sets. The extrema can be used for efficiently updating bounding volumes such as AABBs or k-DOPs, as we show in our results. The choice of particular bounding volume may depend on the complexity of the contact configurations, but in all cases we can compute surface bound orders of magnitude faster and/or tighter than with previous methods.     

Analysis of validated error bounds of surface-to-surface intersection

In this paper, we address the problem of computing the error bounds of surface-to-surface intersection and propose a novel procedure to reduce them. We formulate the surface-to-surface intersection problem as solving a system of ordinary differential equations and using the validated ODE solver we compute the validated a priori enclosures of an intersection curve, in which the existence and the uniqueness of a solution are guaranteed. Then we use straight line enclosures to reduce the size of the a priori enclosures. These reduced enclosures are again enclosed by bounding curves, which can be used as the reduced error bounds of the intersection curve. We demonstrate our method with tangential and transversal intersections.     

An Iterative Substructuring Method for div-stable Finite Element Approximations of the Oseen Problem

We apply an iterative substructuring algorithm with transmission conditions of Robin–Robin type to the discretized Oseen problem appearing as a linearized variant of the incompressible Navier–Stokes equations. Here we consider finite element approximations using velocity/pressure pairs which satisfy the Babuška–Brezzi stability condition. After proving well-posedness and strong convergence of the method, we derive an a-posteriori error estimate which controls convergence of the discrete subdomain solutions to the global discrete solution by measuring the jumps of the velocities at the interface. Additionally we obtain information how to design a parameter of the Robin interface condition which essentially influences the convergence speed. Numerical experiments confirm the theoretical results and the applicability of the method. Received February 18, 2000; revised February 21, 2001     

Pivot tightening for direct methods for solving symmetric positive definite systems of linear interval equations

The paper considers systems of linear interval equations, i.e., linear systems where the coefficients of the matrix and the right hand side vary between given bounds. We focus on symmetric matrices and consider direct methods for the enclosure of the solution set of such a system. One of these methods is the interval Cholesky method, which is obtained from the ordinary Cholesky decomposition by replacing the real numbers by the related intervals and the real operations by the respective interval operations. We present a method by which the diagonal entries of the interval Cholesky factor can be tightened for positive definite interval matrices, such that a breakdown of the algorithm can be prevented. In the case of positive definite symmetric Toeplitz matrices, a further tightening of the diagonal entries and also of other entries of the Cholesky factor is possible. Finally, we numerically compare the interval Cholesky method with interval variants of two methods which exploit the Toeplitz structure with respect to the computing time and the quality of the enclosure of the solution set.     

Block triangular preconditioners for the discretized time-harmonic Maxwell equations in mixed form

In this paper, we consider the solution of the saddle point linear systems arising from the finite element discretization of the time-harmonic Maxwell equations in mixed form. Two types of block triangular Schur complement-free preconditioners used with Krylov subspace methods are proposed, involving the choice of the parameter. Furthermore, we give the optimal parameter in practice. Theoretical analysis shows that all eigenvalues of the preconditioned matrices are strongly clustered. Finally, numerical experiments that validate the analysis are presented.     

An Effective High-Order Interval Method for Validating Existence and Uniqueness of the Solution of an IVP for an ODE

Validated methods for initial value problems for ordinary differential equations produce bounds that are guaranteed to contain the true solution of a problem. When computing such bounds, these methods verify that a unique solution to the problem exists in the interval of integration and compute a priori bounds for the solution in this interval. A major difficulty in this verification phase is how to take as large a stepsize as possible, subject to some tolerance requirement. We propose a high-order enclosure method for proving existence and uniqueness of the solution and computing a priori bounds.     

Solutions for the Portfolio Selection Problem with Interval and Fuzzy Coefficients

In this paper we consider portfolio selection problem with interval and fuzzy objective function coefficients as a kind of multiple objective problems including uncertainties. For this problem two kinds of efficient solutions are introduced: possibly efficient solution as an optimistic solution, necessarily efficient solution as a pessimistic solution. Investigating the properties of two efficiency conditions by means of preference cones and feasible region, we discuss that the two kinds of solutions can be identified with the sets of combinations of lower or upper bounds of intervals. This revised version was published online in June 2006 with corrections to the Cover Date.     

Finite-dimensional approximation and error bounds for spectralsystems with partially known eigenstructure

An approach is presented for directly computing bounds on the frequency-response error between finite-dimensional modal models and the full infinite-dimensional models of systems described by certain classes of linear hyperbolic and parabolic partial differential equations (PDE's). The models and bounding techniques are developed specifically to be computable when applied to hyperbolic and parabolic systems with spatially variant parameters, complicated boundary shapes, and other cases where the eigenstructure is not available in closed form and must be computed numerically. A controller design example is presented to illustrate the utility of this approach     

Interval Estimations of Solution Sets to Real-Valued Systems of Linear or Non-Linear Equations

This is a second paper devoted to present the Modal Interval Analysis as a framework where the search of formal solutions for a set of simultaneous interval linear or non-linear equations is started on, together with the interval estimations for sets of solutions of real-valued systems in which coefficients and right-hand sides belong to certain intervals. The main purpose of this second paper is to show that the modal intervals are a suitable tool to approach problems where logical references appear, for example, to find interval estimates of a special class of generalized sets of solutions of real-valued linear and non-linear systems, the UE-solution sets.     

Sharp Bounds on Interval Polynomial Roots

An algorithm is developed for computing sharp bounds on the real roots of polynomials with interval coefficients. The procedure is introduced by studying interval quadratic equations.     

Block {ω}-circulant preconditioners¶for the systems of differential equations

The numerical solution of large and sparse nonsymmetric linear systems of algebraic equations is usually the most time consuming part of time-step integrators for differential equations based on implicit formulas. Preconditioned Krylov subspace methods using Strang block circulant preconditioners have been employed to solve such linear systems. However, it has been observed that these block circulant preconditioners can be very ill-conditioned or singular even when the underlying nonpreconditioned matrix is well-conditioned. In this paper we propose the more general class of the block { ω }-circulant preconditioners. For the underlying problems, ω can be chosen so that the condition number of these preconditioners is much smaller than that of the Strang block circulant preconditioner (which belongs to the same class with ω =1) and the related iterations can converge very quickly. Received: January 2002 / Accepted: December 2002 The research of the first author was supported in part by INdAM-GNCS and by a grant from the MURST project “Progetto Giovani Ricercatori anno 2000”. The research of the second author was supported in part by Hong Kong Research Grants Council Grants Nos. HKU 7130/02P and HKU 7132/00P and UK/HK Joint Research Scheme Grant No. 20009819.