(February 3) Matthias K. Gobbert: Parallel Computing for Long-Time Simulations of Calcium Waves in a Heart Cell — The release of calcium ions in a heart cell can lead to self-organized diffusion waves, which play a role in controlling the beating of the heart. This process is modeled mathematically by a system of transient reaction-diffusion equations. The large number of point sources to model the injection of calcium ions and the requirement for long-time simulations on high resolution meshes are among the challenges for convergent and efficient numerical methods for this problem. The combination of these challenges is tackled by a special purpose code that has enabled successful long-time simulations that capture the crucial physical effect of self-organized wave initiation. However, these simulations also highlight shortcomings of the underlying model. We will also show performance results from the new distributed-memory cluster hpc with InfiniBand interconnect in the UMBC High Performance Computing Facility (HPCF) that demonstrate the excellent scalability of the code when using all available cores on the compute nodes with two dual-core processors. This work is joint with Bradford E. Peercy, also at UMBC.

(February 10) Kate Evans: Fully Implicit solution Framework for Scalable Earth System Models — Major algorithmic challenges exist for the solution of the newest generation of Earth system problems including coupled nonlinear physics, multiple disparate time scales, and scalability requirements. To access a range of solver capabilities needed to address these issues, a Fortran interface package of the Trilinos project is implemented within two components of the Community Climate System Model, the High-order Method Modeling Environment (HOMME) option of the Community Atmosphere Model (CAM), and the Parallel Ocean Program (POP) global ocean model component. HOMME uses a cubed sphere grid and spectral element spatial discretization to achieve maximum scalability. A fully-implicit Jacobian-Free Newton-Krylov (JFNK) solution method is used to provide a coherent nonlinear solution to all dependent variables for several test cases of the model components. Enhanced accuracy is achieved using a second order temporal discretization for a given time step size and increased efficiency is attained for a steady state test case where the solution is not hindered by time step size constraints. Further, the JFNK solution framework maintains the same scaling as the explicit method currently used in the model. The key to efficient integration for the more complete model using JFNK is a good preconditioner. Several options already under investigation will be discussed.

(February 17) Benjamin Stamm: Stabilization Strategies for Discontinuous Galerkin Methods — Most DG-methods require penalization of the solution jumps to improve continuity of the solution. The added stability leads to optimal convergence with respect to interpolation in the energy-norm. However, although the stabilization terms are consistent they may perturb physical properties such as local mass conservation and a~@ect the conditioning of the linear system. The aim of this talk is to provide techniques that may be used to increase the understanding of stabilization mechanisms in DG-methods and provide more natural schemes. Examples on how stabilization may be reduced in order to not influence local mass conservation, while keeping optimality will be given. Both elliptic and hyperbolic problems are discussed for different DG-formulations and approximation spaces. All theoretical explanations will be followed by numerical illustrations.

(February 26) R. B. Kellogg: A Singularly Perturbed Semilinear reaction-diffusion problem in a Polygonal Domain — The semilinear reaction-diffusion equation $-\e^2\D u+b(x,u)=0$ with Dirichlet boundary conditions is considered in a convex polygonal domain. The diffusion parameter $\eps^2$ is arbitrarily small, and the ``reduced equation'' $b(x,u_0(x))=0$ may have multiple solutions. An asymptotic expansion for $u$ is constructed that involves boundary and corner layer functions. By perturbing this asymptotic expansion, we obtain certain sub- and super-solutions and thus show the existence of a solution $u$ that is close to the constructed asymptotic expansion. The polygonal boundary forces the study of the nonlinear autonomous elliptic equation $-\D z+f(z)=0$ posed in an infinite sector, and the well-posedness of the corresponding linearized problem. The arguments that lead to this well-posedness may have an independent interest.

The material in the talk is joint work with N. Kopteva (Limerick, Ireland).

(March 3) Brian Sutton: The CS Decomposition: Random Matrix Theory and Computation — The CS decomposition (CSD) is analogous to the eigenvalue and singular value decompositions but is specifically designed for partitioned unitary matrices. We will consider the CSD of a random matrix drawn from the unitary group's Haar measure. The result not only solves a problem in random matrix theory but also motivates a new algorithm for computing the CSD. This new algorithm has the benefit of computing both forms of the CSD: the 2-by-2 CSD for square unitary matrices, as originally described by G. W. Stewart, and the 2-by-1 CSD for tall and skinny matrices with orthonormal columns, with application to the generalized singular value decomposition.

(March 10) Karsten Urban: Numerical Optimization in Engineering Applications — Several processes in engineering are subject to optimization, e.g. reducing cost, enhancing effectivity, increasing speed etc. Typically, such optimization problems are quite complex, nonlinear, high dimensional and often involve stochastical influences. Even though the mathematical theory of optimization is quite advanced, not so much is known in such `realistic situations.

We show some industrial optimization problems and indicate the mathematical structure of these problems. We show that relatively simple numerical optimization schemes already give rise to enormous improvements, but we also indicate that the use of such simple methods is limited. In order to overcome such limitations, we describe three techniques, namely automatic differentiation, reduced basis methods and parameter-dependent optimization.

(March 24) Yuesheng Xu: Fast methods for optimal kernel selection in machine learning — Optimal kernel methods are popular in machine learning. However, they suffer from their huge computational costs in finding the optimal kernels because they lead to full matrices of large orders. To tackle this bottleneck problem, we propose fast algorithms for selecting the optimal kernel in the context of machine learning. Results on convergence and computational complexity will be presented. Numerical examples will be shown to demonstrate the approximation accuracy and computational efficiency of the proposed methods.

(March 31) Radu Balan: A Nonlinear Reconstruction Algorithm from Absolute Value of Frame Coefficients for Low Redundancy Frames — In this talk we present a signal reconstruction algorithm from absolute value of frame coefficients that requires a relatively low redundancy. The basic idea is to use a nonlinear embeding of the input signal Hilbert space into a higher dimensional Hilbert space of sesquilinear functionals so that absolute values of frame coefficients are associated to relevant inner products in that space. In that space the reconstruction becomes linear and can be performed in a polynomial number of steps.

(April 14) Tobias von Petersdorff: Exponential convergence of hp quadrature for integral operators — Integral equations occur in many applications. For the numerical approximation with a Galerkin method the elements of the stiffness matrix have to be computed. These are integrals of the type

\int_{S₁} \int_{S₂} g(x,y) dy dx

The computation of these integrals is the main difficulty in the numerical solution of an integral equation: The singular integrals have to be computed with high accuracy in order to maintain the convergence rate of the Galerkin method, but using too many function evaluations increases the complexity of the resulting algorithm.

Most methods in the literature rely on a very specific form of the kernel function. We present an algorithm which

• works for most kernel functions commonly occurring in integral equations; even in the case where the kernel is not analytic but only in a Gevrey class away from the diagonal

• works for simplices or parallelotopes in arbitrary dimensions, and all possible cases where simplices share vertices, edges, faces, etc.; also for isotropic mesh refinement

• uses only function evaluations of the integrand g(x,y)

• the error is exponentially convergent. Hence only |log(h)|^a function evaluations are required for each integral to achieve a convergence rate h^b for the Galerkin method

Preprint (ETH Zurich)

(April 21) Malena Espanol: A Multilevel, Modified Regularized Total Least Norm Approach to Signal Deblurring — In this talk, we present a multilevel method for discrete ill-posed problems formulated as total least norm problems. We will focus on the signal deblurring problem where both the blurring operator and the blurred signal contain noise. Regularized total least norm (R-TLN) approaches which have beed developed to solve this problem require the minimization of a functional with respect to the unknown perturbation in the blurring operator and the desired image. Much of the work to date in solving R-TLN has required the perturbation operator to have special structure (e.g. sparsity structure or Toeplitz type structure) in order to make the minimization problem more computationally tractable. Our goal is to gain additional efficiency by means of a multilevel approach. Therefore, we present a multilevel method that uses the Haar wavelets as restriction and prolongation operators. We show that the choice of the Haar wavelet operator has the advantage of preserving matrix structure, such as Toeplitz, among grids, and we discuss how this can be incorporated into intermediate R-TLN problems on each level. Finally, we present results that indicate the promise of this approach on deblurring signals with edges.

This is joint work with Misha Kilmer and Dianne O'Leary.

(April 28) Rosemary Renaut: Statistical Properties of the Regularized Least Squares Functional and a Hybrid LSQR Newton method for Finding the Regularization Parameter: Application in Image Deblurring and Signal Restoration — Image deblurring or signal restoration can be formulated as a data fitting least squares problem, but the problem is severely ill-posed and regularization is needed, hence introducing the need to find a regularization parameter. I will review the background on finding the regularization parameter dependent on the properties of the regularized least squares functional

||Ax-b||^2_{W_b} + ||D(x-x0)||_{W_x}^2

for the solution of discretely ill-posed systems of equations. It was recently shown to follow a chi-squared distribution when the a priori information x0 on the solution is assumed to represent the mean of the solution x. But of course for image deblurring, we don't wish to assume knowledge of a prior image to obtain the image. On the other hand, it is possible to obtain statistical properties of the given image, hence given the mean value of the right hand side, b, the functional is again a chi-squared distribution, but one that is non-central. These results can be used to design a Newton method, using a hybrid LSQR approach, for the determination of the optimal regularization parameter lambda when the weight matrix is W_x=lambda^2 I. Numerical results using test problems demonstrate the efficiency of the method, particularly for the hybrid LSQR implementation. Results are compared to another statistical method, the unbiased predictive risk (UPRE) algorithm. The method has potential for efficient image deblurring, and current work is aimed at extending the method for determining local regularization parameters. Results are illustrated for image deblurring.

(April 30) Leslie Greengard: A New Formalism for Electromagnetic Scattering in Complex Geometry — We will describe some recent, elementary results in the theory of electromagnetic scattering. There are two classical approaches that we will review - one based on the vector and scalar potential and applicable in arbitrary geometry, and one based on two scalar potentials (due to Lorenz, Debye and Mie), valid only in the exterior of a sphere. In trying to extend the Lorenz-Debye-Mie approach to arbitrary geometry, we have encountered some new mathematical questions involving differential geometry, partial differential equations and numerical analysis. This is joint work with Charlie Epstein.