## Computational and Applied Mathematics Seminar

## Location and Time

University of Wyoming, Ross Hall 247, Fridays from 4:10-5:00 (unless otherwise stated).

## Seminar Chairs

Professors Craig C. Douglas and Man-Chung Yeung.

## Support

The CAM seminar series is currently supported through volunteers and the financial contributions by the UW Mathematics Department, MGNet.org, and and an energy grant from ExxonMobil.

## Schedule

For Fall 2019, the speakers are as follows:

Date Speaker From/Note September 13 Rasika Rajapakshage University of Wyoming September 19* Hiroshi Fujiwara Kyoto University September 27 SIAM Northern States Section Conference University of Wyoming October 4 Stefan Heinz University of Wyoming October 11 Hongjun Guo University of Wyoming October 18** Annie Millet University of Paris 1 November 1 Rasika Rajapakshage University of Wyoming November 8 Fisk lecture: Hari Bercovici University of Indiana November 15 Liqiang Li University of Wyoming November 22*** Cedar Wiseman University of Wyoming December 6 Dan Stanescu University of Wyoming December 13 Clay Carper University of Wyoming * Thursday Colloquium in AG 1032, ** Joint CAM - Analysis seminar, *** Joint CAM - ACNT seminar

We are constantly looking for speakers for the current academic year! The topics can be original research, a survey of an area, or an interesting paper or papers that would interest the CAM community. If you would like to speak, please contact me by email.

The schedule, titles, and abstracts from Spring 2019 are here. The schedule, titles, and abstracts for Spring 2020 are here.

## Titles and Abstracts

## September 13

## Anisotropic Functional Laplace Deconvolution

Dr. Rasika Rajapakshage, Department of Mathematics and Statistcs, University of WyomingIn this presentation we consider the problem of estimating a three dimensional function

fbased on observations from its noisy Laplace convolution. Our study is motivated by the analysis of Dynamic Contrast Enhanced (DCE) imaging data. We construct an adaptive wavelet-Laguerre estimator off, derive minimax lower bounds for theL-risk when^{2}fbelongs to a three dimensional Laguerre-Sobolev ball and demonstrate that the wavelet-Laguerre estimator is adaptive and asymptotically near optimal in a wide range of Laguerre-Sobolev spaces. We carry out a limited simulations study and show that the estimator performs well in a finite sample setting. Finally, we use the technique for the solution of the Laplace deconvolution problem on the basis of DCE Computerized Tomography data.## September 19

## Multiple-Precision Arithmetic and Its Implementation for Matlab

Prof. Hiroshi Fujiwara, Graduate School of Informatics, Kyoto UniversityIn this talk, we discuss rounding errors in numerical computations, and introduce multiple-precision arithmetic environment on MATLAB. On the digital computers, real numbers are approximated with finite precision. Thus the rounding errors are inevitable, and in particular, their rapid growth due to numerical instability are serious from the viewpoint of the reliability. Mltiple-precision arithmetic has a possibility to overcome the difficulty, since it enables numerical computation without rounding errors virtually by the use of enough precision. In the presentation, we discuss the influence of rounding errors in simple examples. Some effective use of multiple-precision arithmetic is demonstrated in advanced sciences. Finally, we introduce its design and implementation on MATLAB, and show performance comparisons with other environments.

## September 27

## SIAM Northern Section Conference

University of WyomingThe First Annual SIAM Northern Section Conference will be held at the University of Wyoming on September 27-29, 2019. Details can be found at http://www.uwyo.edu/mathstats/siam/.

## October 4

## Towards Intelligent Dynamic Universal Simulation Methods for Multi-Scale Problems

Prof. Stefan Heinz, Department of Mathematics and Statistcs, University of WyomingAccurate and feasible simulations of turbulent flow around aircraft and wind turbines suffer from two major problems. On the one hand, computationally very efficient pure Reynolds-averaged Navier-Stokes (RANS) methods often fail to describe the main features of such flows because of their lacking ability to resolve turbulent flows. On the other hand, pure Large Eddy Simulation (LES) methods, which are capable of simulating resolved flow, are computationally way too expensive for real wall-bounded turbulent flows. The development of solutions to these problems via the design of hybrid methods involving both RANS and LES elements takes place now over decades. The need to address some essential problems that were basically unaddressed so far will be explained first. Then, a theoretical solution to the problem considered is presented in the frame of well-known turbulence models. Applications to periodic hill flows including flow separation illustrate the potential of these new simulation methods. From a mathematical viewpoint, the new equations represent Navier-Stokes type equations that are intelligent, expected to be universal, and they can be made dynamic. The simulation concept can be also used to address other multi-scale problems than considered here.

## October 11

## On the Mean Speed of Bistable Transition Fronts in Unbounded Domains

Dr. Hongjun Guo, Department of Mathematics and Statistcs, University of WyomingIn this talk, I will present the existence and further properties of propagation speeds of transition fronts for bistable reaction-diffusion equations in exterior domains and in some domains with multiple cylindrical branches. In exterior domains, we show that all transition fronts with complete propagation propagate with the same global mean speed, which turns out to be equal to the uniquely defined planar speed. In domains with multiple cylindrical branches, I will show that the solutions emanating from some branches and propagating completely are transition fronts propagating with the unique planar speed. I will also give some geometrical and scaling conditions on the domain, either exterior or with multiple cylindrical branches, which guarantee that any transition front has a global mean speed.

## October 18

## Behavior of Solutions in Stochastic Critical and Supercritical Focusing nonlinear Schrödinger Equation

Prof. Annie Millet, University of Paris 1We study nonlinear Schrö̈dinger (NLS) equation with focusing nonlinearity, subject to additive or multiplicative stochastic perturbations driven by an infinite dimensional Brownian motion. Under the appropriate assumptions on the space covariance of the driving noise, previously A. de Bouard and A. Debussche established the

Hlocal well-posedness for energy sub-critical nonlinearity, and global well-posedness in the mass-subcritical case. In our work we study the^{1}L-critical, intercritical and energy (^{2}H)-critical cases of stochastic NLS, and obtain quantitative estimates on the blow-up time when the mass, energy and^{1}L-norm of the gradient of the initial condition are controlled by similar quantities of the ground state. This completes blow-up results proved by A. de Bouard and A. Debussche for energy sub-critical nonlinearities.^{2}This is joint work with Svetlana Roudenko (Florida International University).

## October 25

## TBD

Liqiang Li, Department of Mathematics and Statistcs, University of WyomingTBD

## November 1

## Is Clustering Advantageous in Statistical Ill-Posed Linear Inverse Problems

Dr. Rasika Rajapakshage, Department of Mathematics and Statistcs, University of WyomingIn many statistical linear inverse problems, one needs to recover classes of similar curves from their noisy images under an operator that does not have a bounded inverse. Problems of this kind appear in many areas of application. Routinely, in such problems clustering is carried out at the pre-processing step and then the inverse problem is solved for each of the cluster averages separately. As a result, the errors of the procedures are usually examined for the estimation step only. The objective of this paper is to examine, both theoretically and via simulations, the effect of clustering on the accuracy of the solutions of general ill-posed linear inverse problems. In particular, we assume that one observes

X,_{m}= A f_{m}+ ∇ ε_{m}m=1, ..., M, where functionsfcan be grouped into_{m}Kclasses and one needs to recover a vector functionf= (f. We construct an estimators for_{1}, ..., f_{M})^{T}as a solution of a penalized optimization problem which corresponds to the clustering before estimation setting. We derive an oracle inequality for its precision and confirm that the estimator is minimax optimal or nearly minimax optimal up to a logarithmic factor of the number of observations. One of the advantages of our estimation procedure is that we do not assume that the number of clusters is known in advance. Subsequently, we compare the accuracy of the above procedure with the precision of estimation without clustering and clustering following the recovery of each of the unknown functions separately. We complement our study with the real data example.fWe conclude that clustering at pre-processing step is beneficial when the problem is moderately ill-posed. It should be applied with extreme care when the problem is severely ill-posed.

## November 8

## Fisk Lecture

Prof. Hari Bercovici, Department of Mathematics, University of IndianaSee the Fisk Colloquium announcment.

## November 15

## TBD

Liqiang Li, Department of Mathematics and Statistcs, University of WyomingTBD

## November 22

## Rubik's Cubes

Cedar Wiseman, Department of Mathematics and Statistcs, University of WyomingTBD

## December 6

## Quantun Computing

Prof. Dan Stanescu, Department of Mathematics and Statistcs, University of WyomingTBD

## December 13

## Rationally Predicting I-80 Closures

Clay Carper, Department of Mathematics and Statistcs, University of WyomingTBD