# Math Colloquia: Fall 2010

#### Friday September 03

Title |
Numerical Methods for Large-Scale Ill-Posed Inverse Problems in Imaging Applications |

Speaker |
Dr. Julianne Chung |

Department of Computer Science | |

University of Maryland, College Park | |

http://www.cs.umd.edu/~jmchung/Home/Welcome.html |

**Abstract:**

Many scientific and engineering applications require numerical methods to compute efficient and reliable solutions to inverse problems. The basic goal of an inverse problem is to compute an approximation of the original model, given observed data and knowledge about the forward model. Physical systems that require reconstruction of an unknown input signal from the measured output signal are natural examples of inverse problems. In other systems, the internal structure of an object is desired, but only measured output data is provided. The difficulty with ill-posed inverse problems is that small errors may give rise to significant errors in the computed approximations, so regularization must be used to compute stable solution approximations. Furthermore, real-life applications often require the computer to process extremely large amounts of data, and previously proposed methods for solving inverse problems are not adequate for these large-scale problems.

In this talk, we investigate hybrid methods for regularization of linear and nonlinear least squares problems and describe an efficient parallel implementation based on the Message Passing Interface (MPI) library for use on state-of-the-art computer architectures. Numerical experiments illustrate the effectiveness and efficiency of the proposed methods on problems from image reconstruction, super-resolution imaging, cryo-electron microscopy reconstruction, and digital tomosynthesis.

**Speaker Biography:** After graduating from Emory University in 2004 with a BA in mathematics and dance and movement studies, Julianne Chung received her Ph.D. in Computational Mathematics at Emory University in 2009. Her advisor was James Nagy. As a graduate student, she received a Department of Energy Computational Science Graduate Fellowship. She conducted a summer research practicum at the Lawrence Berkeley National Laboratory, working with Chao Yang. Julianne was awarded a National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship, and currently she is in the Department of Computer Science at the University of Maryland, working with Dianne O’Leary. Her research interests include numerical methods for ill-posed inverse problems, image processing for biomedical applications, and high-performance computing for large-scale problems.

#### Friday September 10

Title |
Simulation of stochastically modeled biochemical networks |

Speaker |
Dr. David F. Anderson |

Department of Mathematics | |

University of Wisconsin, Madison | |

http://www.math.wisc.edu/~anderson/ |

**Abstract:**

This talk will focus on stochastically modeled chemical reaction networks, and will have a special emphasis on a pathwise representation for such models usually termed the “random time change representation.” Such a representation is incredibly useful in that it allows us to (i) see the natural algorithms that can be used to simulate the processes and (ii) perform error analyses that can, in a rigorous manner, tell us what the algorithms are doing.

While exact simulation methods exist for discrete-stochastic models of biochemical reaction networks, they are oftentimes too inefficient for use because the number of computations scales linearly with the number of reaction events; thus, approximate algorithms have been developed. However, stochastically modeled reaction networks often have natural scales and it is crucial that these be accounted for when developing and analyzing numerical approximation methods. I will show that conducting such a non-standard error analysis leads to fundamentally different conclusions than previous analyses.

#### Friday September 17

No talk today

#### Friday September 24

No talk today

#### Friday October 01

Title |
Revolutionizing Climate Modeling: Advantages of Dedicated High Performance Computing |

Speaker |
Dr. James Kinter |

Center for Ocean-Land-Atmosphere Studies (COLA) |

**Abstract:**

A collaboration, bringing together an international team of over 30 people, from six institutions on three continents, including climate and weather scientists and modelers and experts in high-performance computing (HPC), has demonstrated the feasibility of using dedicated HPC resources to rapidly accelerate progress in addressing one of the most critical problems facing the global community, namely, global climate change. The scientific basis for undertaking this project was established in the May 2008 World Modeling Summit. In this project, two types of computationally-intensive experiments used the entire 18,048-core Athena Cray XT-4 supercomputer at the University of Tennessee’s National Institute for Computational Sciences (NICS) for the period October 2009 — March 2010. The numerical experiments were intended to determine whether increasing weather and climate model resolution to accurately resolve mesoscale phenomena in the atmosphere can improve the fidelity of the models in simulating the mean climate and the distribution of variances and covariances. Explicitly resolving cloud processes in the atmosphere without approximation by parameterization was examined as well. The effect of increasing greenhouse gas concentrations, associated with global warming, on the regional aspects of extreme temperature and precipitation, storminess, floods and droughts in key regions of the world also was evaluated in these experiments.

The two sets of numerical experiments were conducted with two different models. One was an experimental version of the European Centre for Medium-range Weather Forecasts (ECMWF) Integrated Forecast System (IFS), a global atmospheric general circulation model, which is used operationally every day to produce 10-day weather forecasts. The IFS was run at several resolutions down to 10-km grid spacing to evaluate the statistical distribution and nature of high-impact and extreme events in 20th and 21st century simulations. The other was the NICAM global atmospheric model from the Japan Agency for Marine-Earth Science and Technology (JAMSTEC), which was run at 7-km, cloud-system-resolving grid resolution to simulate the boreal summer climate, over many years, focusing on tropical cyclones, monsoon systems, and summer flood and drought situations. Both models were run in long simulations for the first time in the U.S.

The project has stretched the limits of CPU, disk, I/O, metadata management and tape archive resources. The data generated by this project will be made available to the communities of climate scientists interested in analyzing high-resolution climate simulations and computational scientists who can learn about operational considerations of running dedicated production at nearly petascale.

#### Friday October 08

Title |
Nonlinear Stochastic Perturbations of Dynamical Systems |

Speaker |
Dr. Leonid Koralov |

Department of Mathematics | |

University of Maryland, College Park | |

http://www-users.math.umd.edu/~koralov/ |

**Abstract:**

We will describe the asymptotic behavior of solutions to quasi-linear parabolic equations with a small parameter at the second order term and the long time behavior of corresponding diffusion processes. In particular, we discuss the exit problem and metastability for the processes corresponding to quasi-linear initial-boundary value problems.

#### Friday October 15

Title |
Spectral Approach to Uncertainty Quantification in Complex Dynamical Systems |

Speaker |
Dr. Omar M. Knio |

Department of Mechanical Engineering | |

Johns Hopkins University | |

http://tinyurl.com/23br9vm |

**Abstract:**

This talk outlines recent developments in spectral methods for uncertainty quantification in model-based simulations, and selected applications to dynamical systems. The fundamental principle of stochastic spectral methods is to parametrize model uncertainty in terms of a finite set of random variables with known probability law, and to express the solution in terms of orthogonal basis functions that are typically polynomials in these random variables. The unknown coefficients in the expansion are determined using a weighted residual formalism, which provides quantitative estimate of the dependence of the solution on random model inputs. Elementary examples will first be discussed in order to highlight fundamental concepts. Attention will then be focused on the development and implementation of robust methods that can accommodate challenging situations where the solution exhibits steep or discontinuous dependence on the random in puts. We conclude with a brief discussion of new directions that are anticipated to result in substantial benefits.

#### Friday October 22

Title |
A comparison of two domain decomposition methods for a linearized contact problem |

Speaker |
Dr.Jungho Lee |

Computer Science and Mathematics Division | |

Oak Ridge National Lab | |

http://cims.nyu.edu/~jungho/ |

**Abstract:**

Linear systems of algebraic equations which arise from the discretization of partial differential equations are often huge and ill-conditioned. Such systems are solved by a preconditioned Krylov subspace method. Research in domain decomposition methods (DDM) has been motivated primarily out of the need to solve such systems. Domain decomposition methods can be viewed as a class of preconditioning techniques, utilizing massively parallel computers. The main challenge in designing a DDM is to ensure that the performance of the algorithm does not deteriorate as more and more processors are used (i.e., scalability).

In this talk, we compare two domain decomposition methods for a linearized contact problem. The first method we consider has been used in an engineering community; we provide theoretical and numerical evidence that this method is not scalable with respect to the number of subdomains (processors). We propose a scalable alternative and analyze its properties, both theoretically and numerically. We also solve a model problem using a combination of a primal-dual active set method, viewed as a semismooth Newton method, and the scalable alternative.

#### Friday October 29

*11:00am–noon in MP 401A Joint Math/Stat Colloquium*

Title |
Convexity of the inverse and Moore-Penrose inverse |

Speaker |
Dr. Kenneth Nordstrom |

Department of Mathematical Sciences | |

University of Oulu, Finland | |

http://stat.oulu.fi/nordstrom/ |

**Abstract:**

The topic of this talk is the convexity of the inverse of positive definite matrices and of the Moore-Penrose inverse of nonnegative definite matrices w.r.t. the partial ordering induced by nonnegative-definiteness. A survey of such convexity properties will be presented, and several extensions and generalizations of earlier work will be described. In particular, a concept of strong convexity will be introduced, extending the conventional concept of strict convexity. Connections with reproducing kernel Hilbert space theory, matrix means, and indefinite inner product space theory are briefly indicated.

#### Friday November 05

Title |
Title TBA |

Speaker |
Dr. Thomas Seidman |

Department of Mathematics and Statistics | |

UMBC | |

http://www.math.umbc.edu/~seidman/ |

**Abstract:**

#### Friday November 12

Title |
Title TBA |

Speaker |
Dr. Russell K. Jackson |

Mathematics Department | |

U. S. Naval Academy | |

http://www.usna.edu/Users/math/rkjackso/ |

**Abstract:**

#### Friday November 19

Title |
Title TBA |

Speaker |
Dr. Tamás Terlaky |

Department of Industrial and Systems Engineering | |

Lehigh University | |

http://www.cas.mcmaster.ca/~terlaky/ |

**Abstract:**

#### Friday November 26

Thanskgiving

#### Friday December 03

*11:00am–noonA Joint Math/Stat Colloquium*

Title |
Title TBA |

Speaker |
Dr. George Ostrouchov |

http://www.csm.ornl.gov/~ost/ | |

Statistics and Data Sciences Group | |

Oak Ridge National Laboratory |

**Abstract:**

#### Friday December 10

Title |
Title TBA |

Speaker |
Dr. Lois Curfman McInnes |

Mathematics and Computer Science Division | |

Argonne National Laboratory | |

http://press.mcs.anl.gov/curfman/ |

**Abstract:**