3WS SR

Meshless (or meshfree) methods are a relatively new numerical approach for the solution of ordinary- and partial differential equations. They offer the geometrical flexibility of finite elements but without requiring connectivity from the discretization support (ie a mesh). Meshless methods based on the collocation of radial basis functions (RBF methods) are particularly easy to code, and have a number of theoretical advantages as well as practical drawbacks. In this talk, an adaptive RBF scheme is presented for a novel application, namely the solution of (a rather broad class of) delayed- and neutral differential equations.

This talk will focus on highly localized basis functions which exist for certain kernels and spaces associated with these kernels. Such kernels include certain radial basis functions (RBFs), their restrictions to spheres (SBFs), and their restrictions to more general manifolds embeddable in R^d. The first part of the talk will be of an introductory nature. It will discuss radial basis functions and their restriction to manifolds which give rise to various kernels on these manifolds. The talk will then focus on the development (for certain kernels) of highly localized Lagrange functions which serve as effective bases: i.e., bases which are stable and local. Scaled versions of these bases will then be used to establish the stability of the L² minimization operator in L^p, 1 ≤ p ≤ ∞, thus obtaining a multivariate analogue of a result of de Boor. Since these bases are scalable with the data, they have potential uses beyond approximation including meshless methods and, more generally, computations of a multiresolution nature. The talk is primarily based on joint work with T. Hangelbroek, F. J. Narcowich and X. Sun.

We discuss a class of high-order spectral-Galerkin surface integral algorithms with specific focus on simulating the scattering of electromagnetic waves by a collection of three dimensional deterministic and stochastic particles.

We consider saddle point problems arising as (linearized) optimality conditions in elliptic optimal control problems. The efficient solution of such systems is a core ingredient in second-order optimization algorithms. In the spirit of Bramble and Pasciak, the preconditioned systems are symmetric and positive definite with respect to a suitable scalar product. We extend previous work by Schoeberl and Zulehner and consider problems with control and state constraints. It stands out as a particular feature of this approach that an appropriate symmetric indefinite preconditioner can be constructed from standard preconditioners for those matrices which represent the inner products, such as multigrid cycles.

Numerical examples in 2D and 3D are given which illustrate the performance of the method, and limitations and open questions are addressed.

Shifted Laplace preconditioners have attracted considerable attention as

a technique to speed up convergence of iterative solution methods for the

Helmholtz equation. In the talk we present a comprehensive spectral

analysis of the discrete Helmholtz operator preconditioned with a shifted

Laplacian. Our analysis is valid under general conditions. The propagating

medium can be heterogeneous, and the analysis also holds for different types

of damping, including a radiation condition for the boundary of the computational

domain. By combining the results of the spectral analysis of the

preconditioned Helmholtz operator with an upper bound on the GMRES-residual

norm we are able to derive an optimal value for the shift, and to

explain the mesh-depency of the convergence of GMRES preconditioned

with a shifted Laplacian. We will illustrate our results with a seismic test

problem.

Joint work with: Yogi Erlangga (University of British Columbia) and Kees Vuik (TU Delft)

In this talk we describe a general framework for deriving

modified equations for stochastic differential equations with respect to

weak convergence. We will start by quickly recapping of how to derive

modified equations in the case of ODEs and describe how these ideas can

be generalized in the case of SDEs. Results will be presented for first

order methods such as the Euler-Maruyama and the Milstein method. In the

case of linear SDEs, using the Gaussianity of the underlying solutions,

we will derive a SDE that the numerical method solves exactly in the

weak sense. Applications of modified equations in the numerical study

of Langevin equations and in the calculation of effective diffusivities

will also be discussed.

Future discovery and control in biology and medicine will come from

the mathematical modeling of large-scale molecular biological data,

such as DNA microarray data, just as Kepler discovered the laws of

planetary motion by using mathematics to describe trends in

astronomical data. In this talk, I will demonstrate that

mathematical modeling of DNA microarray data can be used to correctly

predict previously unknown mechanisms that govern the activities of

DNA and RNA.

First, I will describe the computational prediction of a mechanism of

regulation, by using the pseudoinverse projection and a higher-order

singular value decomposition to uncover a genome-wide pattern of

correlation between DNA replication initiation and RNA expression

during the cell cycle. Then, I will describe the recent

experimental verification of this computational prediction, by

analyzing global expression in synchronized cultures of yeast under

conditions that prevent DNA replication initiation without delaying

cell cycle progression. Finally, I will describe the use of the

singular value decomposition to uncover "asymmetric Hermite functions,"

a generalization of the eigenfunctions of the quantum harmonic

oscillator, in genome-wide mRNA lengths distribution data.

These patterns might be explained by a previously undiscovered asymmetry

in RNA gel electrophoresis band broadening and hint at two competing

evolutionary forces that determine the lengths of gene transcripts.

Nonlinear eigenvalue problem (NEP) is a class of eigenvalue problems where the matrix depends on the eigenvalue. We will first introduce some NEPs in real applications and some algorithms for general NEPs. Then we introduce our recent advances in NEPs, including second order Arnoldi algorithms for large scale quadratic eigenvalue problem (QEP), analysis and algorithms for symmetric eigenvalue problem with nonlinear rank-one updating, a new linearization for rational eigenvalue problem (REP).