## UT Austin Junior Numerical Analysis and Applied Math Group

### Welcome to the website for Jr. NAP!

 The Jr. Numerical Analysis and Applied Math (NAP) seminar is on Thursdays from 2:00-3:00 in RLM 10.176.If you are interested in giving a talk, please email svallelian_at_math

#### 2/20: Charlie

posted Feb 20, 2014, 2:29 PM by Sarah Vallelian

 Title: Mullins-Sekerka Problem and LaplaceEquation on Multiply Connected RegionsAbstract: Mullins-Sekerka problem is a free boundary problem commonly used inmodeling crystal growth or solidification and liquidation where material movement is governed by diffusion and no surface tension. The talk will focus on why Laplace Equation in non-conventional regions is of interest, and the peculiarity of existence, uniqueness, and numerical methods under this circumstance.

#### 2/13: Christina

posted Feb 15, 2014, 10:27 AM by Sarah Vallelian

 Title: Numerical methods for multiscale inverse problemsAbstract: We will consider inverse problems for multiscale partial differentialequations of the form $-\div \left(\aeps\nabla u^\epsilon\right)+b^{\epsilon}u^{\epsilon} = f$ in which solution data is used to determine coefficients in the equation. Such problems contain both the general difficulty of finding an inverse and the challenge of multiscale modeling, which is hard even for forward computations. The problem in its full generality is typically ill-posed and one approach is to reduce the dimensionality of the original problem by just considering the inverse of an effective equation without microscale $\epsilon$. We will here include microscale features directly in the inverse problem. In order to reduce the dimension of the unknowns and avoid ill-posedness, we will assume that the microscale can be accurately parametrized by piecewise smooth coefficients. We indicate in numerical examples how the technique can be applied to medical imaging and exploration seismology.

#### 11/14: Rohit

posted Nov 14, 2013, 12:36 PM by Sarah Vallelian

 Title: Regularity of the solution for Obstacles exhibiting non-local behavior Abstract: Starting from an optimal cash management problem and its formulation as a stochastic impulse control problem we will derive an obstacle problem where the obstacle exhibits non-local behavior. Generalizing the 1-d situation we will discuss some properties of the solution as well as introduce the notion of a quasi-variational inequality. The objective of the talk will be to relate the probabilistic interpretation of the problem to its analytic reformulation and discuss some applications.

#### 10/31: Sara

posted Oct 29, 2013, 4:08 PM by Sarah Vallelian

 Title: MRI Made Easy Abstract: Magnetic resonance imaging (MRI) is a test that uses a magnetic field and pulses of radio wave energy to make pictures of organs and structures inside the body. This talk will provide a very brief introduction to the physics and techniques of MRI.

#### 10/17: Charlie

posted Oct 21, 2013, 9:44 AM by Sarah Vallelian

 Title: A Level Set Approach to Mullins-Sekerka problem and the Regularization of Layer PotentialsAbstract: We will look over the Mullins-Sekerka problems and introduce the level set approach, some difficulties and possible solutions to the local expansion of layer potentials.

#### 10/10: Jamie

posted Oct 9, 2013, 3:47 PM by Sarah Vallelian

 Title: An Overview of Numerical Methods for Conservation Laws Abstract: We will first look at the mathematical properties of conservation laws. Then look at a variety of numerical methods that attempt to solve them. Hopefully in the process shedding some light on the difficulties that arise while attempting to model the general problem.

#### 10/3: Jane

posted Oct 2, 2013, 12:47 PM by Sarah Vallelian

 Title: Can a finite element method perform arbitrarily badly?Abstract: The talk will simply go through the paper of the same title given by Ivo Babushka and John Osborn, 1999. In that paper, they construct a toy elliptic boundary value problem which converges arbitrary slowly in almost all reasonable norms. Moreover, adaptive procedures cannot save the convergence rate. The problem is 1D with piecewise polynomial elements, and the rest is just elementary arithmetic. But revisit such an easy case may suggest a different view of classical methods.

#### 9/26: Chris

posted Sep 27, 2013, 12:13 PM by Sarah Vallelian

 Title: An eigenvalue optimization problem for graph partitioningAbstract: We begin by reviewing the graph partitioning problem and its applications to data clustering. We then proceed to introduce a new non-convex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified and a novel rearrangement algorithm is proposed, which we show is strictly decreasing and converges in a finite number of iterations to a local minimum of the relaxed objective function. We end by discussing some applications, as well as connections to other problems such as Nonnegative Matrix Factorization and Reaction Diffusion equations. This is joint work with Braxton Osting and \'Edouard Oudet.

#### 9/19: Christina

posted Sep 18, 2013, 8:47 PM by Sarah Vallelian

 Title: Nonuniform sampling theory and multiscale computationAbstract: We consider multiscale functions of the type that are studied in averaging and homogenization theory and in multiscale modeling. Typical examples are two-scale functions f(x, x/), (0 <  1) that are periodic in the second variable. We prove that under certain band limiting conditions these multiscale functions can be uniquely and stably recovered from non-uniform samples of optimal rate. The goal of this study is to establish the close relation between computational grids in multiscale modeling and sampling strategies developed in information theory.

#### 9/12: Jamie

posted Sep 15, 2013, 12:45 PM by Sarah Vallelian   [ updated Sep 15, 2013, 12:46 PM ]

 Title: An Integration Based Interpolation Technique for Cell Averaged DataAbstract: Numerical solutions to conservation laws give the evolution of cell averaged data over time. The techniques used for solving this class of problems require the knowledge of the data at precise points. In this talk I will give an overview of the traditional WENO, weighted essentially non-oscillatory, interpolation technique and explain its limitations. Then present a different integration based I-WENO scheme which overcomes many of the difficulties of the traditional scheme.

1-10 of 27