Math/ICES Center of Numerical Analysis Seminar (Spring 2016)

Time and Location: Friday, 1:00-2:00PM, POB 6.304 Special time and locations are indicated in red.

If you are interested in meeting a speaker, please contact Kui Ren (

Here are the links to the past seminars: Fall 2015 Spring 2015 Fall 2014 Spring 2014 Spring 2013 Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009


Spekers and Hosts

Title and Abstract



Advances in Kinetic and Fluid Dynamics Transport

Qiang Du
Columbia University
Bridging scales through nonlocal modeling

Nonlocality is ubiquitous in nature. Although partial differential equations (PDEs) remain favored as effective continuum models for many applications, nonlocal equations and nonlocal balanced laws are also becoming acceptable alternatives to model various processes exhibiting anomalies and singularities. They may also serve as effective bridges for multiscale modeling. In this talk, after a brief introduction to the framework of nonlocal vector calculus, we elucidate how it helps us to resolve a few computational issues of particular concerns to nonlocal modeling, including the development of asymptotically compatible schemes for validation and verification, the effective nonlocal gradient recover, and the seamless coupling of local and nonlocal models for efficient and adaptive computation.

Weihua Geng
Southern Methodis University
Accurate and Efficient Interface Methods for Implicitly Solvated Biomolecular Simulation

The Poisson-Boltzmann (PB) model is an effective implicit solvent approach for simulating solvated biomolecular systems. By treating the solvent with a mean field approximation and capturing the mobile ions with the Boltzmann distribution, the PB model largely reduces the degree of freedom and computational cost. However, solving the PB equation suffers many numerical difficulties arising from interface jump conditions, complex geometry, charge singularities, and boundary conditions at infinity. In order to resolve these difficulties, two interface methods with different formulation and discretization are investigated.

The first approach is the matched interface and boundary (MIB) method. This finite difference meshed method repeatedly uses interface jump conditions to capture the non-smoothness of solutions, adaptively applies local interpolation to characterize the complex geometry, and analytically takes Green's function based decomposition to regularize the singularities of the source terms. By computing a series of benchmark tests, the MIB-PB solver shows a solid 2nd order convergence thus stands out among Cartesian grid based PB solvers.

The second approach is the treecode accelerated boundary integral (TABI) method, which adopts a well-conditioned boundary integral formulation to handle aforementioned difficulties while accelerates the Krylov subspace based iterative methods such as GMRES with Cartesian treecode. This treecode is an O(N*logN) scheme with properties of easy implementation, efficient memory usage, infrequent communication, and straightforward parallelization. In addition, the treecode/boundary integral scheme can be conveniently implemented on GPUs. Numerical tests show 100+ times speed-up on a single GPU card, potentially making it possible to run PB model based molecular dynamics simulation for billions of time-steps.



Ming Gu
UC Berkeley
Spectrum-revealing Matrix Factorizations

Low-rank matrix approximations have become of central importance in the era of big data. Efficient and effective methods for such approximations have been proposed in statistics, theoretical computer science, and optimization. In this talk, we establish spectrum-revealing matrix factorizations, a new framework for efficient and effective matrix approximations. These factorizations are variations of the more classical LU, QR, and Cholesky factorizations with row (and/or) column permutations but are competitive with the best matrix approximations in both theory and computational effectiveness. We also discuss extensions of these factorizations for efficient computations of the truncated SVD and solutions of nuclear norm minimization problems. We demonstrate the effectiveness of our approaches with numerical experiments with both synthetic and real data.



Yoonsang Lee
Courant Institute
Stochastic Superparameterization and Multiscale Filtering of Turbulent Tracers

Data assimilation combines a numerical forecast model and observations to provide the best statistical estimate of the state of interest. In this paper we are concerned with data assimilation of the passive tracer advected in turbulent flows using a low-order forecast model. The turbulent flows which contain anisotropic and inhomogeneous structures such as jets are typical in geophysical turbulent flows in atmosphere and ocean sciences. Stochastic superparameterization, which is a seamless multiscale method developed for large-scale models of atmosphere and ocean models without scale-gap between the resolved and unresolved scales, generates large-scale turbulent velocity fields using a significantly smaller degree of freedoms compared to a direct fine resolution numerical simulation. In a large-scale model of the tracer transport, the tracer is advected by the large- scale velocity field generated by the superparameterization with a parameterization of eddies, an additional eddy diffusion given by an anisotropic biharmonic diffusion. To alleviate the problem of mixed observations of the resolved and unresolved scales, we use an ensemble multiscale data assimilation which provides estimates of the resolved scales using mixed observations. The low-order model is 250 times cheaper than the fine resolution solution and thus enables to increase the number of ensembles for accurate predictions of prior distributions. We test the multiscale data assimilation method for the passive tracer model advected by two-layer quasigeostrophic turbulent flows. Numerical experiments show positive results in the estimation of the resolved scales of the tracer. This is joint work with A.J. Majda and D. Qi.

Yingda Cheng
Michigan State University
A Sparse Grid Discontinuous Galerkin Method for High-Dimensional Transport Equations

In this talk, we present a sparse grid discontinuous Galerkin (DG) scheme for  transport equations  and applied it to  kinetic simulations. The method uses the weak formulations of traditional Runge-Kutta DG schemes for hyperbolic problems and is proven to be $L^2$ stable and convergent. A major advantage of the scheme lies in its low computational and storage cost  due to  the employed sparse finite element approximation space. This attractive feature is explored in simulating Vlasov and Boltzmann transport equations. Good performance in accuracy and conservation is  verified by numerical tests in up to four dimensions.

RLM 10.176

Lek-Heng Lim
University of Chicago

Algebraic techniques in numerical computations

We will discuss how techniques from algebra (e.g. multilinear algebra, representation theory, algebraic geometry, algebraic topology) play a role in numerical computations (e.g. numerical linear algebra, convex optimization, statistical computing), drawing on examples from the speaker's recent works.