Schedule of Events
New England Numerical Analysis Day 2011

"Rank-Deficient Nonlinear Least Squares Problems and Subset Selection"


We examine the local convergence of the Levenberg-Marquardt method for the solution of nonlinear least squares problems that are rank deficient and have nonzero residual. We show that replacing the Jacobian by a truncated singular value decomposition can be numerically unstable. We recommend instead the use of subset selection. We corroborate our recommendations by perturbation analyses and numerical experiments. This is joint work with Tim Kelley and Scott Pope.

"Stabilized Finite Element Solutions to the Advection-Dominated Optimal Control Problems"


Standard Galerkin finite element discretizations applied to advection-dominated, elliptic PDEs can lead to highly oscillatory solutions, unless the grid is sufficiently fine. Over the years a number of stabilized methods, such as streamline upwind/Petrov Galerkin (SUPG) or Discontinuous Galerkin (DG) methods, were developed. These methods are frequently applied to advection-dominated elliptic PDEs. Local and global error estimates for these methods are well known.

In this talk we are interested in local and global error estimates of the SUPG and DG solutions of optimal control problems. We show that the discretization error for these methods in the optimal control context behave differently as a function of the mesh size than it does for scalar advection-dominated elliptic problems. This is true even for local error estimates in the regions of smoothness, away from interior or boundary layers. We also show that for local error estimates it is also essential how the Dirichlet boundary conditions are enforced. We will provide error estimates for the computed solution of the optimal control problem and we will present numerical results to illustrate our findings.

"Certified Reduced Basis Method for a Pacman Scattering Problem"


The reduced basis method (RBM) is indispensable in scenarios where a large number of numerical solutions to a parametrized partial differential equation are desired in a fast/real-time fashion. RBM can improve efficiency by several orders of magnitudes.
In this talk, we consider the scattering of TM-polarized electromagnetic waves by a perfectly conducting 2D cylinder with a cut-out wedge. The parameter of this problem is the size of the cut-out, that is, the angle of the wedge. RBM is applied to this problem with exponential convergence achieved. We also study the monostatic scattering as a function of the wedge angle. This reveals the capability of RBM to capture the critical wedge angle for the optimal reduction of the backscattering, obviously important information for the corresponding design problems.

"Approximation of Smoluchowski Diffusion-Limited Reaction Models"


We will give an introduction to the classical diffusion limited reaction model of Smoluchowski, leading to a coupled system of partial differential equations. We will then discuss how another common stochastic reaction-diffusion model, the reaction-diffusion master equation, may be interpreted as a spatial discretization of this model. We will show that the discretization is in fact divergent for systems containing bimolecular reactions, but may be interpreted as an asymptotic approximation for small, but nonzero, mesh sizes. Time-permitting, we will discuss recent work on the relation between a third, spatially continuous, stochastic reaction-diffusion PDE model and the Smoluchowski model.

"Fourier Continuation, Fast Algorithms and Applications in Data Analysis and PDE Solution"


Fourier Continuation (or Fourier Extension) techniques that allow for highly accurate Fourier representation of smooth but non-periodic data will be discussed. Recently developed algorithms for the fast calculation of these Continuations (at or near FFT speeds) will be presented. Applications in data analysis and PDE solution in complex domains will be discussed including recent advances allowing for even greater accuracies than previously published through an unconditionally stable Fourier based PDE solver.

"Anderson acceleration for fixed-point iterations"


Fixed-point iterations occur naturally and are commonly used in a broad variety of computational science and engineering applications. In practice, fixed-point iterates often converge undesirably slowly, if at all, and procedures for accelerating the convergence are desirable. This talk will focus on a particular acceleration method that originated in work of D. G. Anderson [J. Assoc. Comput. Machinery, 12 (1965), 547-560]. This method has enjoyed considerable success in electronic-structure computations but seems to have been untried or underexploited in many other important applications. Moreover, while other acceleration methods have been extensively studied by mathematicians and numerical analysts, Anderson acceleration has received relatively little attention from them, despite there being many significant unanswered mathematical questions. In this talk, I will outline Anderson acceleration, discuss some of its theoretical properties, and demonstrate its performance in several applications. This work is joint in part with Peng Ni.

"Computational Practices in the Pre- Electronic Computer Days"

"Banded Matrices with Banded Inverses and A = LPU" (Click for abstract)

"Third Order Tensors as Operators on Matrices, Revisited"


Joint work with Karen Braman (SDSMT), Ning Hao (Tufts)

Abstract:
Recent work by Kilmer and Martin (LAA, in press) and Braman (LAA,2010) provides a setting which the familiar tools of linear algebra can be extended to better understand third order tensors (i.e. 3D arrays). In this talk we give further theoretical results and explore their implications in terms of extending basic algorithms such as the power method, QR iteration, and Krylov subspaces. We conclude with an example from image deblurring where the deblurred image is recovered via a conjugate gradient type of algorithm.

"Local boundary condition based spectral collocation methods for 2D and 3D Naiver-Stokes equations"


We present a simple approach to accurately and efficiently computing local boundary conditions in spectral collocation schemes for the Navier-Stokes equations 2D and 3D dimensions. These locally computed boundary values makes possible the decoupling of the computation of the primary flow variables, resulting in highly efficient schemes. In 2D the local vorticity boundary values are employed in the vorticity-stream function formulation, while in 3D local values of a Neumann boundary condition for the pressure in the velocity-pressure formulation are used. The straightforward extension of the approach to the Boussinesq system is also discussed. The resulting schemes are well suited for the simulation of moderate to high Reynolds and Rayleigh number flows. Simulations of Rayleigh-Bernard convection problem for Rayleigh number up to $10^{10}$ are presented to demonstrate that the schemes are capable of producing accurate results at a reasonable computational cost. This is joint work with Cheng Wang and Jian-Guo Liu.

"The Calculus of Moving Surfaces as a Tool for Formulating Computational Problems"


The Calculus of Moving Surfaces (CMS) is an extension of classical differential geometry to moving manifolds. It follows the structure of Ricci and Levi-Civita's absolute differential calculus. Central to the CMS is an invariant time derivative that plays a role analogous to that of the covariant derivative. The CMS is an excellent framework for formulating computational problems with moving surfaces including boundary variation problems as well as problems in shape optimization and dynamics. I will present the fundamental concepts from the CMS and use the framework to reformulate some of the well known problems as well as formulate some new ones that could benefit greatly from attention from the scientific computation community. A particular focus will be the recently formulated dynamic fluid film equations.

"IRBLB: Implicitly Restarted Block Lanczos Bidiagonalization Method with Leja Shifts"


A restarted block Lanczos bidiagonalization method is described for computing, a few of the largest, smallest, or near a specified positive number, singular triplets of a large rectangular matrix. Leja points are used as shifts in the implicitly restarted Lanczos bidiagonalization method. The new method is often computational faster than other implicitly restarted Lanczos bidiagonalization methods, requires a much smaller storage space, and can be used to find interior singular values. Computed examples show the new method to be competitive with available schemes.

"Diffraction gratings and photonic crystals: new integral representations for periodic scattering and eigenvalue problems"


Many numerical problems arising in modern photonic and electromagnetic applications involve the interaction of linear waves with periodic, piecewise-homogeneous media. Boundary integral equations are an efficient approach to solving such boundary-value problems with high-order convergence. In the case of plane-wave scattering from an array (grating), the standard way to periodize is then to replace the free-space Green's function kernel with its quasi-periodic cousin. However, a major drawback is that the quasi-periodic Green's function fails to exist for parameter families known as Wood's anomalies, even though the underlying scattering problem remains well-posed.

We bypass this problem with a new integral representation that relies on the *free-space* Green's function alone, adding auxiliary layer potentials on the boundary of the unit cell strip, while enforcing quasi-periodicity with an expanded linear system. The result is a 2nd kind scheme that achieves spectral accuracy, is immune to Wood's anomalies, avoids lattice sums, and enables fast multipole acceleration. A doubly-periodic version provides similar benefits for the robust solution of the eigenvalue (band structure) problem for Bloch waves in a photonic crystal. We show two-dimensional examples achieving 10-digit accuracy with only a couple of hundred unknowns.

Joint work with Leslie Greengard (NYU).