Past Computational Mathematics and Applications Seminar

21 June 2001
14:00
Prof Gilbert Strang
Abstract
Tridiagonal matrices and three term recurrences and second order equations appear amazingly often, throughout all of mathematics. We won't try to review this subject. Instead we look in two less familiar directions. \\ \\ Here is a tridiagonal matrix problem that waited surprisingly long for a solution. Forward elimination factors T into LDU, with the pivots in D as usual. Backward elimination, from row n to row 1, factors T into U_D_L_. Parlett asked for a proof that diag(D + D_) = diag(T) + diag(T^-1).^-1. In an excellent paper (Lin Alg Appl 1997) Dhillon and Parlett extended this four-diagonal identity to block tridiagonal matrices, and also applied it to their "Holy Grail" algorithm for the eigenproblem. I would like to make a different connection, to the Kalman filter. \\ \\ The second topic is a generalization of tridiagonal to "tree-diagonal". Unlike the interval, the tree can branch. In the matrix T, each vertex is connected only to its neighbors (but a branch point has more than two neighbors). The continuous analogue is a second order differential equation on a tree. The "non-jump" conditions at a meeting of N edges are continuity of the potential (N-1 equations) and Kirchhoff's Current Law (1 equation). Several important properties of tridiagonal matrices, including O(N) algorithms, survive on trees.
  • Computational Mathematics and Applications Seminar
Prof Mike J D Powell
Abstract
Let the thin plate spline radial basis function method be applied to interpolate values of a smooth function $f(x)$, $x \!\in\! {\cal R}^d$. It is known that, if the data are the values $f(jh)$, $j \in {\cal Z}^d$, where $h$ is the spacing between data points and ${\cal Z}^d$ is the set of points in $d$ dimensions with integer coordinates, then the accuracy of the interpolant is of magnitude $h^{d+2}$. This beautiful result, due to Buhmann, will be explained briefly. We will also survey some recent findings of Bejancu on Lagrange functions in two dimensions when interpolating at the integer points of the half-plane ${\cal Z}^2 \cap \{ x : x_2 \!\geq\! 0 \}$. Most of our attention, however, will be given to the current research of the author on interpolation in one dimension at the points $h {\cal Z} \cap [0,1]$, the purpose of the work being to establish theoretically the apparent deterioration in accuracy at the ends of the range from ${\cal O} ( h^3 )$ to ${\cal O} ( h^{3/2} )$ that has been observed in practice. The analysis includes a study of the Lagrange functions of the semi-infinite grid ${\cal Z} \cap \{ x : x \!\geq\! 0 \}$ in one dimension.
  • Computational Mathematics and Applications Seminar
Dr Lawrence Daniels and Dr Iain Strachan
Abstract
In this talk we review experiences of using the Harwell Subroutine Library and other numerical software codes in implementing large scale solvers for commercial industrial process simulation packages. Such packages are required to solve problems in an efficient and robust manner. A core requirement is the solution of sparse systems of linear equations; various HSL routines have been used and are compared. Additionally, the requirement for fast small dense matrix solvers is examined.
  • Computational Mathematics and Applications Seminar
15 March 2001
14:00
Abstract
The sphere is an important setting for applied mathematics, yet the underlying approximation theory and numerical analysis needed for serious applications (such as, for example, global weather models) is much less developed than, for example, for the cube. \\ \\ This lecture will apply recent developments in approximation theory on the sphere to two different problems in scientific computing. \\ \\ First, in geodesy there is often the need to evaluate integrals using data selected from the vast amount collected by orbiting satellites. Sometimes the need is for quadrature rules that integrate exactly all spherical polynomials up to a specified degree $n$ (or equivalently, that integrate exactly all spherical harmonies $Y_{\ell ,k}(\theta ,\phi)$ with $\ell \le n).$ We shall demonstrate (using results of M. Reimer, I. Sloan and R. Womersley in collaboration with W. Freeden) that excellent quadrature rules of this kind can be obtained from recent results on polynomial interpolation on the sphere, if the interpolation points (and thus the quadrature points) are chosen to be points of a so-called extremal fundamental system. \\ \\ The second application is to the scattering of sound by smooth three-dimensional objects, and to the inverse problem of finding the shape of a scattering object by observing the pattern of the scattered sound waves. For these problems a methodology has been developed, in joint work with I.G. Graham, M. Ganesh and R. Womersley, by applying recent results on constructive polynomial approximation on the sphere. (The scattering object is treated as a deformed sphere.)
  • Computational Mathematics and Applications Seminar

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