Date

Speaker

Title and abstract

09/19/11

Monday

3:30PM-4:30PM

ACE 6.304

Arieh Iserles

Cambridge University

Rapid Expansion in Orthogonal Polynomials

The computation of the first n terms of an expansion into orthogonal polynomials in O(n log n) operations is a long-standing challenge in computational mathematics. In the first part of the talk we describe such algorithms for the computation of expansions in ultrasperical (a.k.a. Gegenbauer) polynomials. This proceeds in three somewhat counterintuitive steps. Firstly, expansion coefficients are expressed as an infinite linear combination of derivatives. Secondly, using the Cauchy theorem, this is converted into an integral transform with a hypergeometric kernel. Finally, using a serendipitous transformation of the kernel into a rapidly-convergent function, the integral transform is computed as a finite linear combination of a discrete Fourier transform of the underlying function along a Bernstein ellipse. In the second part of the talk we generalize the first two stages to arbitrary orthogonal polynomial systems supported by compact real intervals, as well as to polynomials and Laurent polynomials orthogonal on the unit circle. This is joint work with María José Cantero.

10/07/11

3:00-4:00PM

ACE 6.304

Lin Lin

Lawrence Berkeley National Lab

Host: Lexing Ying

Towards the optimal basis set in the Kohn-Sham density functional theory

Kohn-Sham density functional theory (KSDFT) is by far the most widely used electronic structure theory for condensed matter systems.  However, the discretization cost, i.e. the number of basis functions to discretize the Kohn-Sham Hamiltonian operator is generally large, which is an important factor that hinders the application of KSDFT to systems of large size.  Recently we have developed two techniques to reduce the discretization cost effectively and systematically.  In the first part of the talk, we propose the adaptive local basis functions, which achieve high accuracy in the total energy calculation with a small number of basis functions.  The adaptive local basis functions are strictly localized in the real space, and the Kohn-Sham orbitals are reconstructed from the adaptive local basis functions under the discontinuous Galerkin (DG) framework.  In the second part of the talk, we propose the optimized local basis functions which further improve the adaptive local basis functions, and can be used for the force calculation with application to structure optimization and molecular dynamics.  We show that using the optimized local basis functions, the atomic force can be accurately evaluated by the Hellman-Feynman force with systematically controlled Pulay force. (Joint work with Weinan E, Jianfeng Lu and Lexing Ying).

11/04/11

3:00PM-4:00PM

ACE 6.304

George Biros

ICES, UT Austin

A Fast Algorithm for Inverse Medium Problems with Multiple Excitations

We consider the inverse medium problem for the time-harmonic wave equation with broadband and multi-point illumination in the low frequency regime. Such a problem finds many applications in geosciences (e.g. ground penetrating radar), non-destructive evaluation (acoustics), and medicine (optical tomography). We use an integral-equation (Lippmann-Schwinger) formulation, which we discretize using a quadrature method. We consider only small perturbations (Born approximation). To solve this inverse problem, we use a least-squares formulation. We present a new fast algorithm for the efficient solution of this particular least-squares problem. If $N_{\fr}$ is the number of excitation frequencies, $N_{s}$ the number of different source locations for the point illuminations, $N_{d}$ the number of detectors, and the parametrization for the scatterer, a dense singular value decomposition for the overall input-output map will have $ [\min(N_{s} N_{\fr}N_{d}, N)]^{2} \times \max(N_{s} N_{\fr}N_{d}, N) $ cost. We have developed a fast SVD-based preconditioner that brings the cost down to $O(N_{s}N_{\fr} N_{d} N)$ thus, providing orders of magnitude improvements over a black-box dense SVD and an unpreconditioned linear iterative solver.

11/15/11

Tuesday

3:30PM-5:00PM

ACE 6.304

Ben Leimkuhler

University of Edinburgh

Statistical Distributions as Stationary Measures of Stochastic-Dynamical Systems: Formulation, Numerical Integrators, and Applications

I will discuss degenerate diffusions for sampling Gibbs (i.e smooth) measures. While familiar methods such as Langevin dynamics have proven to be reliable performers in some types of applications, they can be less desirable in treating problems with a complicated structure. I will show that it is often possible to use Ornstein-Uhlenbeck processes (avoiding the introduction of multiplicative noise) and to obtain in this way simplified results on the invariant measure of numerical methods. I will also discuss the treatment of constrained measures, including isoenergetic, isokinetic and other cases. Applications will include a study of the Hamiltonian vortex method in which the energetic interactions with a bath of weak vortices are treated as thermal fluctuations.

11/18/11

Friday

1:00PM-2:00PM

RLM 10.176

Jianfeng Lu

Courant, NYU

Multiscale analysis of solid materials: From electronic structure models to continuum theories

Modern material sciences focus on studies on the microscopic scale. This calls for mathematical understanding of electronic structure and atomistic models, and also their connections to continuum theories. In this talk, we will discuss some recent works where we develop and generalize ideas and tools from mathematical analysis of continuum theories to these microscopic models. We will focus on macroscopic limit and microstructure pattern formation of electronic structure models.

11/18/11

Friday

3:00PM-4:00PM

ACE 6.304

Georg Stadler

ICES, UT Austin

Hessian-based approximate solution of large-scale statistical inverse
seismic wave propagation

We are interested in inferring local wave speeds in acoustic and elastic media from waveforms, as well as in estimating the uncertainty in the inferred solution.  This inverse problem is cast as a large-scale statistical inverse problem in the framework of Bayesian inference. The complicating factors are the high-dimensional parameter space (due to discretization of the infinite-dimensional parameter fields for the local wave speeds) and the expensive forward problems given by the time-dependent elastic-acoustic wave equation.

We exploit that at the maximum likelihood point, the covariance matrix for the Gaussian approximation of the posterior probability density is the inverse of the Hessian.  A low rank approximation of the Hessian that exploits the compact nature of the data misfit operator for problems with limited measurements is constructed by the Lanczos method. The covariance operator can then be computed using the Woodbury identity. This allows, for instance, to compute the pointwise variance field, or samples for the inferred local wave speed field.

We use a first-order system formulation of the wave equation, which allows to treat the elastic and the acoustic equation in the same framework.  This system is discretized using a high-order spectral discontinuous Galerkin (dG) method, which uses an upwind numerical flux. To obtain accurate gradients and Hessians, derivatives of the misfit are computed for the dG-discretized equation. The resulting discretization for the adjoint wave equation is a dG scheme with a downwind numerical flux. The method is applied to infer global earth models from synthetic measurements.

This work is joint with: Tan Bui-Thanh, Carsten Burstedde, Omar
Ghattas, James Martin and Lucas C. Wilcox

11/25/11

Thanksgiving Holiday

12/02/11