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SIAM J. on Numerical Analysis

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1993

Volume 30, Issue 2, pp. 305-607


On the Sensitivity of Solutions of Parameterized Equations

Werner C. Rheinboldt

SIAM J. Numer. Anal. 30, pp. 305-320 (16 pages) | Cited 2 times

Online Publication Date: July 14, 2006

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The sensitivity of a solution of a parameterized equation $F(z,\lambda ) = 0$ with respect to the parameter vector $\lambda $ is usually defined as the change of the state $z$ in dependence of $\lambda $. In other words, for any solution expressible in the form $(z(\lambda ),\lambda )$ with some smooth function $z = z(\lambda )$, the sensitivity is the derivative $Dz(\lambda )$. Typically the solutions form a manifold $M$ in the product of 1the state space and the parameter space and this sensitivity is available only at those points of $M$ where the parameters can be used to define a local coordinate system. This paper introduces a general sensitivity concept which applies at all solutions on $M$ and which includes the earlier definition. Some general geometric interpretations of the new measure are presented and it is shown that the sensitivity analysis can be easily integrated into the solution process. The theory also suggests the introduction of a readily computable second-order sensitivity measure reflecting the curvature behavior of $M$. Two numerical examples illustrate the discussion.

Legendre Pseudospectral Viscosity Method for Nonlinear Conservation Laws

Yvon Maday, Sidi M. Ould Kaber, and Eitan Tadmor

SIAM J. Numer. Anal. 30, pp. 321-342 (22 pages) | Cited 45 times

Online Publication Date: July 14, 2006

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In this paper, the Legendre spectral viscosity (SV) method for the approximate solution of initial boundary value problems associated with nonlinear conservation laws is studied. The authors prove that by adding a small amount of SV, bounded solutions of the Legendre SV method converge to the exact scalar entropy solution. The convergence proof is based on compensated compactness arguments, and therefore applies to certain $2 \times 2$ systems. Finally, numerical experiments for scalar as well as the one-dimensional system of gas dynamics equations are presented, which confirm the convergence of the Legendre SV method. Moreover, these numerical experiments indicate that by post-processing the SV approximation, one can recover the entropy solution within spectral accuracy.

Preconditioning and Boundary Conditions without $H_2$ Estimates: $L_2$ Condition Numbers and the Distribution of the Singular Values

C. I. Goldstein, Thomas A. Manteuffel, and Seymour V. Parter

SIAM J. Numer. Anal. 30, pp. 343-376 (34 pages) | Cited 6 times

Online Publication Date: July 14, 2006

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This work deals with the behavior—in the $L_2$ norm—of the condition number and distribution of the $L_2$ singular values of the preconditioned operators $B_h^{ - 1} A_h $ and $A_h B_h^{ - 1} $, where $A_h$ and $B_h$ are finite element discretizations of second-order elliptic operators, $A$ and $B$. In an earlier work, Manteuflel and Parter [SIAM J. Numer. Anal., 27 (1989), pp. 656–694] proved that $B_h^{ - 1} A_h (A_h B_h^{ - 1} )$ have a uniformly bounded $L_2$ condition number if and only if $A^ * $ and $B^ * $ ($A$ and $B$) have the same boundary conditions. This earlier work used the $H_2$ regularity of $A$ and $B$, as well as optimal $L_2$ error estimates and a quasi-uniform grid for the finite element spaces. In the present paper, we first extend these condition number results to the case in which neither $H_2$ regularity (and hence optimal $L_2$ error estimates) nor the quasi-uniformity assumption need be satisfied. Instead, it is assumed that the principal part of the preconditioning operator $B$ is a scalar multiple, ${1 / \mu }$, of the principal part of $A$. It is also proven in this case that the operators $Q = B^{ - 1} A - \mu I$ and $\tilde Q = AB^{ - 1} \mu I$ are compact, and the corresponding discrete operators are collectively compact and consistent approximations to $Q$ and $\tilde Q$. Using this, it is shown that the $L_2$ singular values of $B_h^{ - 1} A_h $ and $A_h B_h^{ - 1} $ “fill” the interval $[\mu _0 ,\mu _1 ]$ where $\mu _0 > 0$ and $\mu _1 $ are the minimum and maximum values of $\mu $. Moreover, for any $\varepsilon > 0$ there are (at most) a finite number, $n(\varepsilon )$, of singular values outside the interval $[\mu _0 - \varepsilon ,\mu _1 + \varepsilon ]$. Analogous results are also proven for the case when $B_h^{ - 1} $ is replaced by a more practical preconditioner, say $\dot B_h^{ - 1} $, which is equivalent to $B_h^{ - 1} $ in the $L_2$ norm. This has important implications for the solution of the preconditioned discrete equations using the conjugate gradient method. In particular, the convergence rate will be better than the usual bound obtained using condition number estimates. Finally, some matrix implementations for both the left and right preconditioned normal equations are discussed in detail. These include implementations that avoid the inversion of the mass matrix.

On Optimal Order Error Estimates for the Nonlinear Schrödinger Equation

Ohannes Karakashian, Georgios D. Akrivis, and Vassilios A. Dougalis

SIAM J. Numer. Anal. 30, pp. 377-400 (24 pages) | Cited 3 times

Online Publication Date: July 14, 2006

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Implicit Runge–Kutta methods in time are used in conjunction with the Galerkin method in space to generate stable and accurate approximations to solutions of the nonlinear (cubic) Schrodinger equation. The temporal component of the discretization error is shown to decrease at the classical rates in some important special cases.

An Error Estimate for a Finite Difference Scheme Approximating a Hyperbolic System of Conservation Laws

Aslak Tveito and Ragnar Winther

SIAM J. Numer. Anal. 30, pp. 401-424 (24 pages) | Cited 5 times

Online Publication Date: July 14, 2006

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A rigorous proof of an error estimate for a finite difference scheme applied to a $2 \times 2$ nonstrictly hyperbolic system of conservation laws is presented. It is established that the error in the $L_1$-norm is of order $(\Delta x)^{{1 / 2}} $, where $\Delta x$ denotes the mesh size. This convergence estimate is optimal since, for special data, the system degenerates to a scalar equation for which a similar estimate is known to be optimal.

A Fast Domain Decomposition Poisson Solver on a Rectangle for Hermite Bicubic Orthogonal Spline Collocation

Bernard Bialecki

SIAM J. Numer. Anal. 30, pp. 425-434 (10 pages) | Cited 5 times

Online Publication Date: July 14, 2006

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A fast domain decomposition solver is presented for the piecewise Hermite bicubic orthogonal spline collocation solution of Poisson's equation on a rectangle. The rectangle is divided into parallel strips and the collocation solution is first obtained on the interfaces by solving a collection of independent tridiagonal linear systems. A recently developed fast Fourier transform solver for piecewise Hermite bicubic orthogonal spline collocation is then used to compute the collocation solution on each strip. On an $N \times N$ uniform partition, the proposed domain decomposition solver requires $O(N^2 \log \log N)$ arithmetic operations, assuming that the strips have the same width and that their number is proportional to ${N / {\log N}}$. The solver is also highly parallel in nature.

Domain Decomposition for Fourth-Order Problems

Wilhelm Heinrichs

SIAM J. Numer. Anal. 30, pp. 435-453 (19 pages) | Cited 2 times

Online Publication Date: July 14, 2006

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Multidomain pseudospectral approximation of fourth-order boundary value problems are considered. Different patching conditions at the interfaces are analyzed. Convergence results are given. An iterative method with interface relaxation is investigated. Numerical results are presented for two or more domains.

Mathematical and Numerical Study of a Magnetostatic Problem Around a Thin Shield

François Rogier

SIAM J. Numer. Anal. 30, pp. 454-477 (24 pages)

Online Publication Date: July 14, 2006

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The purpose of this work is to investigate the behavior of the magnetostatic equations around a ferromagnetic material when its thickness is very small. Several models corresponding to different physical conditions encountered are studied. In each case, a complete mathematical analysis is given that proves the existence of a limit problem. Numerical comparisons with other methods are presented.

A Lagrange Multiplier Method for the Interface Equations from Electromagnetic Applications

Ping Lee

SIAM J. Numer. Anal. 30, pp. 478-506 (29 pages) | Cited 1 time

Online Publication Date: July 14, 2006

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In the vector-scalar potential formulation for three-dimensional electromagnetic field applications, one arises at a system of interface equations in $R^3$ coupling the two potentials. In this paper, a Lagrange multiplier method for solving the coupled equations is presented and it is shown that the resulting Lagrange multiplier operator defined on the interface is not only symmetric positive definite but also spectrally equivalent to the identity operator with respect to an appropriate inner-product. Hence the Lagrange multiplier equation can be solved efficiently using iterative solution schemes such as the conjugate gradient iterations. The exterior problem is approximated by truncating the equation on a bounded domain. Finite element methods are chosen for the approximation of interior and truncated exterior problems as well as for the Lagrange multiplier problem. Error estimates are, proved for the finite element approximations which guarantee convergence as the mesh is refined. An alternative operator representation of the interface equations combining the preconditioned interior problem with the Lagrange multiplier is also presented. Semidiscrete approximations are introduced for the time-dependent problem and error estimates are proved.

Using the Refinement Equation for Evaluating Integrals of Wavelets

Wolfgang Dahmen and Charles A. Micchelli

SIAM J. Numer. Anal. 30, pp. 507-537 (31 pages) | Cited 65 times

Online Publication Date: July 14, 2006

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The Wavelet Galerkin Method for solving partial differential equations leads to the problem of computing integrals of products of derivatives of wavelets. This paper studies the problem from the point of view of stationary subdivision schemes. One of the main results is to identify these integrals as components of the unique solution of a certain eigenvector-moment problem associated with the coefficients of the refinement equation. Asymptotic expansions for the corresponding subdivision schemes form an important ingredient of our approach.

Half-Explicit Runge–Kutta Methods for Sifferential-Algebraic Systems of Index $2$

V. Brasey and E. Hairer

SIAM J. Numer. Anal. 30, pp. 538-552 (15 pages) | Cited 11 times

Online Publication Date: July 14, 2006

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Half-explicit Runge-Kutta methods for differential-algebraic problems of index 2 are investigated. It is shown how the arising order conditions can be solved and a particular method of order 4 is constructed. In addition, this paper simplifies the known convergence theory for such methods and demonstrates by numerical experiments their excellent properties when applied to constrained multibody systems.

Constrained Equations of Motion in Multibody Dynamics as ODEs on Manifolds

Jeng Yen

SIAM J. Numer. Anal. 30, pp. 553-568 (16 pages) | Cited 19 times

Online Publication Date: July 14, 2006

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A convergence theorem for applying linear multistep numerical integration methods to constrained equations of motion in mechanical systems is presented. Using a differential geometric approach, Euler–Lagrange equations are reduced to ordinary differential equations (ODEs) on a local parameter space of the constraint manifold. The reduced ODEs and the algebraic constraints are discretized by applying numerical integration formulas; the resultant nonlinear equations are then solved using Newton’s method. The order and convergence results of numerical integration methods are proven on the local parameter space. Using constant order and fixed stepsize integration methods, numerical solution of the preliminary examples shows that the theorem is valid.

Intermediate Rank Lattice Rules for Multidimensional Integration

S. Joe and S. A. R. Disney

SIAM J. Numer. Anal. 30, pp. 569-582 (14 pages) | Cited 1 time

Online Publication Date: July 14, 2006

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Every lattice rule in $s$ dimensions may be characterised by an integer $m$, lying between 1 and $s$ inclusive, called the rank or index of the rule and a set of $m$ positive integers called the invariants. Earlier work has shown that, in a certain precise sense, lattice rules of rank s are better than the commonly used rank-1 rules. Here this earlier work is extended by showing that a similar result holds for certain lattice rules of intermediate rank $m < s$.

The $GBQ$-Algorithm for Constructing Start Systems of Homotopies for Polynomial Systems

Jan Verschelde and Ann Haegemans

SIAM J. Numer. Anal. 30, pp. 583-594 (12 pages) | Cited 5 times

Online Publication Date: July 14, 2006

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Homotopy methods have become a standard tool for the computation of all solutions of a polynomial system. This paper concerns the solution of deficient polynomial systems which appear to be typical in many engineering applications. The $GBQ$-algorithm presented consists of two parts: the computation of a generalized Bézout number $GB$ and the construction of a multi-homogeneous start system $Q$. The approach generalizes m-homogenization into multihomogenization. It can also be regarded as a generalization “towards” the random product homotopy, however, without making assumptions on the coefficients of the polynomials in the system. As is illustrated in the examples, symmetric polynomial systems also can be solved more efficiently.

Accurate $C^2$ Rational Interpolants in Tension

Roger Delbourgo

SIAM J. Numer. Anal. 30, pp. 595-607 (13 pages) | Cited 7 times

Online Publication Date: July 14, 2006

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A family of rational interpolants is defined for a set of real data points $(x_i ,f_i ),1 \leqq i \leqq n$ , with $x_1 < x_2 < \cdots < x_n $. Each interpolant of the family is a $C^2$ spline with a cubic numerator and a quadratic denominator which is identified uniquely by the value of a positive tension parameter $C_0$. The solution constructed is nonparametric. A value of $C_0$ close to zero will produce piecewise linear behaviour while a large value of $C_0$ will give approximately the cubic spline solution. The method used arises from a study of other rational functional forms for effecting monotonic or convex interpolation. A convergence analysis establishes an error bound in terms of $C_0$, and shows that $O(h^4 )$ accuracy is obtained when exact boundary derivatives are prescribed. Several examples are supplied to support the practical value of the method.
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