# Past Computational Mathematics and Applications Seminar

18 October 2001
14:00
Dr Marco Marletta
Abstract
Non-selfadjoint singular differential equation eigenproblems arise in a number of contexts, including scattering theory, the study of quantum-mechanical resonances, and hydrodynamic and magnetohydrodynamic stability theory. \\ \\ It is well known that the spectra of non-selfadjoint operators can be pathologically sensitive to perturbation of the operator. Wilkinson provides matrix examples in his classical text, while Trefethen has studied the phenomenon extensively through pseudospectra, which he argues are often of more physical relevance than the spectrum itself. E.B. Davies has studied the phenomenon particularly in the context of Sturm-Liouville operators and has shown that the eigenfunctions and associated functions of non-selfadjoint singular Sturm-Liouville operators may not even form a complete set in $L^2$. \\ \\ In this work we ask the question: under what conditions can one expect the regularization process used for selfadjoint singular Sturm-Liouville operators to be successful for non-selfadjoint operators? The answer turns out to depend in part on the so-called Sims Classification of the problem. For Sims Case I the process is not guaranteed to work, and indeed Davies has very recently described the way in which spurious eigenvalues may be generated and converge to certain curves in the complex plane. \\ \\ Using the Titchmarsh-Weyl theory we develop a very simple numerical procedure which can be used a-posteriori to distinguish genuine eigenvalues from spurious ones. Numerical results indicate that it is able to detect not only the spurious eigenvalues due to the regularization process, but also spurious eigenvalues due to the numerics on an already-regular problem. We present applications to quantum mechanical resonance calculations and to the Orr-Sommerfeld equation. \\ \\ This work, in collaboration with B.M. Brown in Cardiff, has recently been generalized to Hamiltonian systems.
• Computational Mathematics and Applications Seminar
12 October 2001
14:00
Dr Paul Matthews
Abstract
Stiff systems of ODEs arise commonly when solving PDEs by spectral methods, so conventional explicit time-stepping methods require very small time steps. The stiffness arises predominantly through the linear terms, and these terms can be handled implicitly or exactly, permitting larger time steps. This work develops and investigates a class of methods known as 'exponential time differencing'. These methods are shown to have a number of advantages over the more well-known linearly implicit methods and integrating factor methods.
• Computational Mathematics and Applications Seminar
4 October 2001
14:00
Abstract
The Kestrel interface for submitting optimization problems to the NEOS Server augments the established e-mail, socket, and web interfaces by enabling easy usage of remote solvers from a local modeling environment. \\ \\ Problem generation, including the run-time detection of syntax errors, occurs on the local machine using any available modeling language facilities. Finding a solution to the problem takes place on a remote machine, with the result returned in the native modeling language format for further processing. A byproduct of the Kestrel interface is the ability to solve multiple problems generated by a modeling language in parallel. \\ \\ This mechanism is used, for example, in the GAMS/AMPL solver available through the NEOS Server, which internally translates a submitted GAMS problem into AMPL. The resulting AMPL problem is then solved through the NEOS Server via the Kestrel interface. An advantage of this design is that the GAMS to AMPL translator does not need to be collocated with the AMPL solver used, removing restrictions on solver choice and reducing administrative costs. \\ \\ This talk is joint work with Elizabeth Dolan.
• 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
14 June 2001
14:00
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Abstract
No seminar this week
• 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