# Mathematics Department Seminar

## Computational and Applied Mathematics Seminar

Spring 2018

Mondays at 4:10 p.m. in 401 Carver

The CAM Seminar is organized in the ISU Mathematics Department. It brings speakers from inside and outside of ISU, raising issues and exchanging ideas on topics of current interest in the are of computational and applied mathematics.

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4/27 Friday 3:10pm--4:00pm, in 401 Carver

The characterization of the scattering data for the matrix Schr\"odinger operator on the half line

Tuncay Aktosun
Department of Mathematics
University of Texas at Arlington
Arlington, TX 76019-0408, USA

ABSTRACT: The matrix Schr\"odinger equation is considered on the half line with the general selfadjoint boundary condition described by two boundary matrices satisfying certain appropriate restrictions. It is assumed that the matrix potential is integrable, is selfadjoint, and has a finite first moment. The corresponding scattering data set is constructed, and such scattering data sets are characterized by providing a set of necessary and sufficient conditions assuring the existence and uniqueness of the one-to-one correspondence between the scattering data set and the input data set containing the potential and boundary matrices. The work presented here provides a generalization of the classical result by Agranovich and Marchenko from the Dirichlet boundary condition to the general selfadjoint boundary condition.

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02/26

Inverse resonance problems for the Schrödinger operator on the real line with mixed given data

Xiao-Chuan Xu

Nanjing University of Science and Technology

Abstract: In this work, we study inverse resonance problems for the Schrödinger
operator on the real line with the potential supported in [0, 1]. In general, all eigenvalues
and resonances cannot uniquely determine the potential. (i) It is shown that if the
potential is known a priori on [0, 1/2], then the unique recovery of the potential on
the whole interval from all eigenvalues and resonances is valid. (ii) If the potential
is known a priori on [0, a], then for the case a > 1/2, infinitely many eigenvalues
and resonances can be missing for the unique determination of the potential, and for
the case a < 1/2, all eigenvalues and resonances plus a part of so-called sign-set
can uniquely determine the potential. (iii) It is also shown that all eigenvalues and
resonances, together with a set of logarithmic derivative values of eigenfunctions and
wave-functions at 1/2, can uniquely determine the potential.

03/05

Positive and free energy satisfying schemes for diffusion with interaction potentials

Wumaier Maimaitiyiming

Department of Mathematics, ISU

Abstract: Aggregation and chemotaxis are important phenomena in biology, and they can be modeled by a class of diffusion equations with interaction potentials. In this work, we design and analyze a second order accurate free energy satisfying finite volume method for solving such equations. The schemes (both semi-discrete and fully discrete) are shown to satisfy free energy dissipation law, preserve non-negativity for the solution and conserve total mass at discrete level. These properties guarantee that computed numerical solutions are probability density, and schemes are energy stable. One and two-dimensional numerical examples are given to demonstrate the effectiveness of the scheme.

03/19

LOCAL HOLDER REGULARITY FOR QUASILINEAR
PARABOLIC EQUATISON

Sukung Huang

Department of Mathematics
Yonsei University, South Korea

Abstract: For parabolic p-Laplacian type of equations, where the structure is generalized in Orlicz space, we deliver H\"{o}lder regularity for degenerate and singular type of equations using a unified method of proof relying on geometric characters. Also I will explain recent results of H\"{o}lder regularity for porous medium type of equations. This result also explains local regularity of a
coupled system consisting of a degenerate porous medium type of Keller-Segel system and Stokes system modeling the motion of swimming bacteria living in a fluid and consuming oxygen.

3/26

Vector and Matrix Optimal Mass Transport: Theory and Algorithm

Hongxin Chen

Department of ECE, Iowa State Univ.

Abstract: In many applications such as color image processing, data has more than one piece of information associated with each spatial coordinate, and in such cases the classical optimal mass transport (OMT) must be generalized to handle vector-valued or matrix-valued densities. In this talk, I will discuss the vector and matrix optimal mass transport. Both the theory and algorithm will be covered. On the theory side, we have a rigorous mathematical formulation for these setups and provide analytical results including existence of solutions and strong duality. On the algorithm side, we have a simple, scalable, and parallelizable methods to solve the vector and matrix-OMT problems. This algorithm is in fact closely related to compress sensing.

04/02

A min-max representation of elliptic operators, and applications

Russell Schwabb
Michigan State University

ABSTRACT:  We call operators that enjoy the global comparison property
elliptic'' operators.  This means that the operator preserves
ordering between any two functions in its domain, whose graphs are
ordered and that agree at a point-- i.e. the operator evaluated at
this location will have the same ordering.  This is a generalization
of the fact that we teach to calculus students that at the point of a
local maximum, any $C^2$ function must satisfy $f''(x_0)\leq 0$.  It
turns out that not only does this property serve as a defining feature
for many nonlinear partial differential and integro-differential
equations, but furthermore, we will present a recent result that shows
the global comparison property implies such an operator must have a
familiar form that is common to nonlinear elliptic equations.  Time
permitting, we will elaborate on what this characterization may mean
for the interplay between integro-differential equations and
(nonlinear) Dirichlet-to-Neumann mappings and free boundary problems
like the Hele-Shaw flow.

4/9

A Grazing Gaussian Beam

James Ralston

UCLA

ABSTRACT: Gaussian beams are asymptotic solutions concentrated on a single ray path in the high frequency limit. The interaction of Gaussian beams with boundaries has been studied in several settings. A case which is not covered arises when the ray path is tangent to a concave boundary. This is usually called a “grazing” ray. The question that we investigate is how a beam following a grazing ray will interact with the boundary. In the example discussed here we are able to compute the amplitude of the asymptotic solution on the central ray and see the influence of grazing the boundary.

4/16

Theory and modeling of island formation during deposition:
On-top of surfaces vs intercalated just under surfaces

James Evans

Department of Physics, Mathematics, and Ames Lab, Iowa State Univ.

4/23

Self-organized dynamics: aggregation and flocking

Changhui Tan

Rice University

Abstract: Self-organized behaviors are commonly observed in nature and human societies, such as bird flocks, fish swarms and human crowds. In this talk, I will present some celebrated mathematical models, with simple small-scale interactions which lead to the emergence of global behaviors: aggregation and flocking. I will discuss the models in different scales: from microscopic agent-based dynamics, through kinetic mean-field descriptions, to macroscopic fluid systems.  In particular, the macroscopic models can be viewed as compressible Euler equations with nonlocal interactions. I will show some recent results on the global wellposedness theory of the systems, large time behaviors, and interesting connections to some classical equations in fluid mechanics.

4/27

The characterization of the scattering data for the matrix Schr\"odinger operator on the half line

Tuncay Aktosun
Department of Mathematics
University of Texas at Arlington
Arlington, TX 76019-0408, USA

ABSTRACT: The matrix Schr\"odinger equation is considered on the half line with the general selfadjoint boundary condition described by two boundary matrices satisfying certain appropriate restrictions. It is assumed that the matrix potential is integrable, is selfadjoint, and has a finite first moment. The corresponding scattering data set is constructed, and such scattering data sets are characterized by providing a set of necessary and sufficient conditions assuring the existence and uniqueness of the one-to-one correspondence between the scattering data set and the input data set containing the potential and boundary matrices. The work presented here provides a generalization of the classical result by Agranovich and Marchenko from the Dirichlet boundary condition to the general selfadjoint boundary condition.

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