IMA Journal of Numerical Analysis - current issue

Andrew Ronald Mitchell

Griffiths, D. F., Sanz-Serna, C. — Thu, 21 Jan 2010 13:06:40 PST

Numerical approximation of gradient flows for closed curves in Rd

Barrett, J. W., Garcke, H., Nurnberg, R. — Thu, 21 Jan 2010 13:06:40 PST

We present parametric finite-element approximations of curvature flows for curves in R^d, where d ≥ 2, as well as for curves on two-dimensional manifolds in R³. Here we consider the curve shortening flow, the curve diffusion and the elastic flow. It is demonstrated that the curve shortening and the elastic flows on manifolds can be used to compute nontrivial geodesics and that the corresponding geodesic curve diffusion flow leads to solutions of partitioning problems on two-dimensional manifolds in R³. In addition, we extend these schemes to anisotropic surface energy densities. The presented schemes have very good properties with respect to stability and the distribution of mesh points, and hence no remeshing is needed in practice.

A new finite-element discretization of the Stokes problem coupled with the Darcy equations

Bernardi, C., Hecht, F., Nouri, F. Z. — Thu, 21 Jan 2010 13:06:40 PST

The flow in a rigid porous medium with a crack is usually modelled by the Darcy equations coupled with the Stokes problem. We first propose a new variational formulation of the Stokes system, where the unknowns are the vorticity, the velocity and the pressure, and describe the corresponding finite-element discretization. We extend this discretization to the case where the Darcy and the Stokes equations are coupled and prove optimal a priori and a posteriori error estimates. We conclude with some numerical experiments.

A preconditioned Newton algorithm for the nearest correlation matrix

Borsdorf, R., Higham, N. J. — Thu, 21 Jan 2010 13:06:40 PST

Various methods have been developed for computing the correlation matrix nearest in the Frobenius norm to a given matrix. We focus on a quadratically convergent Newton algorithm recently derived by Qi and Sun. Various improvements to the efficiency and reliability of the algorithm are introduced. Several of these relate to the linear algebra: the Newton equations are solved by minres instead of the conjugate gradient method, as it more quickly satisfies the inexact Newton condition; we apply a Jacobi preconditioner, which can be computed efficiently even though the coefficient matrix is not explicitly available; an efficient choice of eigensolver is identified; and a final scaling step is introduced to ensure that the returned matrix has unit diagonal. Potential difficulties caused by rounding errors in the Armijo line search are avoided by altering the step selection strategy. These and other improvements lead to a significant speed-up over the original algorithm and allow the solution of problems of dimension a few thousand in a few tens of minutes.

The spectral problem for a class of highly oscillatory Fredholm integral operators

Brunner, H., Iserles, A., Norsett, S. P. — Thu, 21 Jan 2010 13:06:40 PST

Let be a linear, complex-symmetric Fredholm integral operator with highly oscillatory kernel K₀(x, y)e^i|x–y|. We study the spectral problem for large , showing that the spectrum consists of infinitely many discrete (complex) eigenvalues and give a precise description of the way in which they converge to the origin. In addition, we investigate the asymptotic properties of the solutions f = f(x;) to the associated Fredholm integral equation f = µ f + a as $$\omega \to \mathrm{\infty }$$, thus refining a classical result by Ursell. Possible extensions of these results to highly oscillatory Fredholm integral operators with more general highly oscillating kernels are also discussed.

Trees, B-series and exponential integrators

Butcher, J. C. — Thu, 21 Jan 2010 13:06:40 PST

Questions concerning the accuracy of numerical methods for differential equations are often analysed using B-series and other formulations based on rooted trees. The analysis of numerical methods, such as Rosenbrock and certain exponential methods, requires an additional algebraic structure to represent the direct use of Jacobian matrices in the computation. It is shown how this can be done within a context containing a broad review of the existing B-series formulation.

Optimal stability for trapezoidal-backward difference split-steps

Dharmaraja, S., Wang, Y., Strang, G. — Thu, 21 Jan 2010 13:06:40 PST

The marginal stability of the trapezoidal method makes it dangerous to use for highly non-linear oscillations. Damping is provided by backward differences. The split-step combination (t trapezoidal, (1 – )t for BDF2) retains second-order accuracy. The ‘magic choice’ allows the same Jacobian for both steps, when Newton's method solves these implicit difference equations. That choice is known to give the smallest error constant, and we prove that also gives the largest region of linearized stability.

Comparisons between pseudospectral and radial basis function derivative approximations

Fornberg, B., Flyer, N., Russell, J. M. — Thu, 21 Jan 2010 13:06:40 PST

Fourier-based pseudospectral (PS) methods have been used since the 1970s for obtaining spectrally accurate solutions to partial differential equations (PDEs) in periodic geometries. Radial basis functions (RBFs) were introduced about the same time for interpolation on scattered nodes in irregular geometries. As was later recognized, they can also be used for accurate numerical solution of PDEs. Although the main strength of RBFs lies in their outstanding geometric flexibility, also offering possibilities of spectral accuracy over irregularly shaped finite domains, it is still of interest to compare them against Fourier-based PS methods in the extremely simple geometries (infinite or periodic domains) where the latter can also be used. Mostly by means of heuristic arguments and graphical illustrations based on Fourier analysis and numerical experiments, we show that there are notable differences (more pronounced in increasing numbers of dimensions) in how the two spectral approaches approximate derivatives.

On the approximation and efficient evaluation of integral terms in PDE models of cell adhesion

Gerisch, A. — Thu, 21 Jan 2010 13:06:40 PST

Recently, a nonlocal term has been introduced in time-dependent partial differential equation (PDE) models of cell migration in tissue. This term is used to model adhesive effects between cells and also between cells and the extracellular matrix. We assume periodic boundary conditions for the model and that the PDE system is discretized following the method of lines and using a finite-volume scheme on a uniform grid in space. For high-resolution simulations of the PDE system an efficient evaluation of the approximation of the nonlocal term is crucial. For one and two spatial dimensions we develop suitable approximations of the nonlocal term and evaluate these using fast Fourier transform (FFT) techniques. Comprehensive numerical tests show the accuracy and efficiency of our approach. We also demonstrate the impact of the proposed scheme for the treatment of the nonlocal term on simulation times for a differential cell adhesion model. We discuss extensions and applicability of our work to systems with nonperiodic boundary conditions and for other nonlocal PDE models from mathematical biology.

Periodic reordering

Grindrod, P., Higham, D. J., Kalna, G. — Thu, 21 Jan 2010 13:06:40 PST

For many networks in nature, science and technology, it is possible to order the nodes so that most links are short-range, connecting near-neighbours, and relatively few long-range links, or shortcuts, are present. Given a network as a set of observed links (interactions), the task of finding an ordering of the nodes that reveals such a range-dependent structure is closely related to some sparse matrix reordering problems arising in scientific computation. The spectral, or Fiedler vector, approach for sparse matrix reordering has successfully been applied to biological data sets, revealing useful structures and subpatterns. In this work we argue that a periodic analogue of the standard reordering task is also highly relevant. Here, rather than encouraging nonzeros only to lie close to the diagonal of a suitably ordered adjacency matrix, we also allow them to inhabit the off-diagonal corners. Indeed, for the classic small-world model of Watts & Strogatz (1998, Collective dynamics of ‘small-world’ networks. Nature, 393, 440–442) this type of periodic structure is inherent. We therefore devise and test a new spectral algorithm for periodic reordering. By generalizing the range-dependent random graph class of Grindrod (2002, Range-dependent random graphs and their application to modeling large small-world proteome datasets. Phys. Rev. E, 66, 066702-1–066702-7) to the periodic case, we can also construct a computable likelihood ratio that suggests whether a given network is inherently linear or periodic. Tests on synthetic data show that the new algorithm can detect periodic structure, even in the presence of noise. Further experiments on real biological data sets then show that some networks are better regarded as periodic than linear. Hence, we find both qualitative (reordered networks plots) and quantitative (likelihood ratios) evidence of periodicity in biological networks.

Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional-order evolution equation

McLean, W., Thomee, V. — Thu, 21 Jan 2010 13:06:40 PST

In a previous paper, McLean & Thomée (2009, J. Integr. Equ. Appl. (to appear)), we studied three numerical methods for the discretization in time of a fractional-order evolution equation in a Banach space framework. Each of the methods applied a quadrature rule to a contour integral representation of the solution in the complex plane, where for each quadrature point an elliptic boundary-value problem had to be solved to determine the value of the integrand. The first two methods involved the Laplace transform of the forcing term, but the third did not. We analysed both the quadrature error and the error arising from a spatial discretization by finite elements, measured in the L₂-norm. The present work extends our earlier results by proving error bounds in the technically more complicated case of the maximum norm. We also establish new regularity properties for the exact solution that are needed for our analysis.

The convection-diffusion Petrov-Galerkin story

Morton, K. W. — Thu, 21 Jan 2010 13:06:40 PST

The term ‘Petrov–Galerkin method’ is probably due to Ron Mitchell and his collaborators, and he was certainly very active in studying the application of finite-element methods to second-order partial differential equations ‘with significant first derivatives’. Our aim in the present paper is to trace links between such early methods and the more recent discontinuous Galerkin methods—not only the methods but also their analysis. Also as Mitchell, like the author, was initially steeped in finite-difference methods, we shall sometimes use their manipulation in our analysis.

Asymptotic behaviour in linear least squares problems

Osborne, M. R. — Thu, 21 Jan 2010 13:06:40 PST

The asymptotic behaviour of a class of least squares problems when subjected to structured perturbations is considered. It is permitted that the number of rows (observations) in the design matrix can be unbounded while the number of degrees of freedom (variables) is fixed. It is shown that for certain classes of random data the solution sensitivity depends asymptotically on the condition number of the design matrix rather than on its square, which is the generic result for inconsistent systems when the norm of the residual is not small. Extension of these results to the case where the perturbations are due to rounding errors is considered.

ADI orthogonal spline collocation methods for parabolic partial integro-differential equations

Pani, A. K., Fairweather, G., Fernandes, R. I. — Thu, 21 Jan 2010 13:06:41 PST

Alternating direction implicit (ADI) orthogonal spline collocation schemes are formulated and analysed for a class of partial integro-differential equations of parabolic type. These techniques are based on the -method, for [1/2, 1], (where = 1 yields the backward Euler method and = 1/2 yields the Crank–Nicolson method) and the second-order backward differentiation formula (BDF) method. For each method, optimal estimates in various norms at each time step are derived and confirmed by results of numerical experiments.

A survey of results on the q-Bernstein polynomials

Phillips, G. M. — Thu, 21 Jan 2010 13:06:41 PST

It is now nearly a century since S. N. Bernstein introduced his well-known polynomials. This paper is concerned with generalizations of the Bernstein polynomials, mainly with the so called q-Bernstein polynomials. These are due to the author of this paper and are based on the q integers. They reduce to the Bernstein polynomials when we put q = 1 and share the shape-preserving properties of the Bernstein polynomials when q (0, 1). This paper also describes another earlier generalization of the Bernstein polynomials, a sequence of rational functions that are also based on the q-integers, proposed by A. Lupas, and two even earlier generalizations due to D. D. Stancu. The present author summarizes various results, due to a number of authors, that are concerned with the q-Bernstein polynomials and with Stancu's two generalizations.

On the convergence of a wide range of trust region methods for unconstrained optimization

Powell, M. J. D. — Thu, 21 Jan 2010 13:06:41 PST

We consider trust region methods for seeking the unconstrained minimum of an objective function F( ), , when the gradient ( ), , is available. The methods are iterative with ₁ being given. The new vector of variables _k+1 is derived from a quadratic approximation to F that interpolates F( _k) and , where k is the iteration number. The second derivative matrix of the quadratic approximation, B_k say, can be indefinite, because the approximation is employed only if the vector of variables satisfies , where _k is a "trust region radius" that is adjusted automatically. Thus the approximation is useful if is sufficiently large and if ||B_k|| and _k are sufficiently small. It is proved under mild assumptions that the condition is achieved after a finite number of iterations, where is any given positive constant, and then it is usual to end the calculation. The assumptions include a Lipschitz condition on and also F has to be bounded below. The termination property is established in a single theorem that applies to a wide range of trust region methods that force the sequence F( _k), k = 1, 2, 3, ..., to decrease monotonically. Any choice of each symmetric matrix B_k is allowed, provided that ||B_k|| is bounded above by a constant multiple of k.

Discretization error and modelling error in the context of the rapid inflation of hyperelastic membranes

Shaw, S., Warby, M. K., Whiteman, J. R. — Thu, 21 Jan 2010 13:06:41 PST

The computational modelling of the rapid large inflation of hyperelastic circular sheets modelled as axisymmetric membranes is treated, with the aim of estimating engineering quantities of interest and their errors. Fine (involving inertia terms) and coarse (quasi-static) models of the inflation are considered and, using goal-oriented techniques, both modelling and discretization errors are estimated. Numerical results involving only discretization errors for the quasi-static problem and both modelling and discretization errors for the dynamic problem are presented.

Improved contour integral methods for parabolic PDEs

Weideman, J. A. C. — Thu, 21 Jan 2010 13:06:41 PST

One way of computing the matrix exponential that arises in semidiscrete parabolic partial differential equations is via the Dunford–Cauchy integral formula. The integral is approximated by the trapezoidal or midpoint rules on a Hankel contour defined by a suitable change of variables. In a recent paper by Weideman & Trefethen (2007, Math. Comput., 76, 1341–1356) two widely used contours were analysed. Estimates for the optimal parameters that define these contours were proposed. In this paper this analysis is extended in two directions. First, the effect of roundoff error is now included in the error model. Second, we extend the results to the case of a model convection–diffusion equation, where a large convective term causes the matrix to be highly non-normal.