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

3/20: Chris

posted Mar 18, 2014, 9:23 AM by Sarah Vallelian

Title: Randomized Methods in Numerical Linear Algebra 

Abstract: In this talk we will introduce the basics of random matrix theory, and use these tools to develop randomized techniques for computing the eigenvalues and eigenvectors of large matrices. In particular, we will sketch the proofs of some error bounds for the Nystr\"om method and for some simple SVD algorithms applied to low-rank matrices.

2/27: Tian

posted Mar 18, 2014, 9:22 AM by Sarah Vallelian

Title: Algorithms and Computational Complexity 

Abstract: An algorithm is any well-defined computational procedure that takes some value, or set of values, as input and produces some value, or set of values as output. In other words, algorithms are like road maps for accomplishing a given, well-defined task. One of the most important aspects of an algorithm is how fast it is. It is often easy to come up with an algorithm to solve a problem, but if the algorithm is too slow, it may be not worth trying at all. Since the exact speed of an algorithm depends on where the algorithm is run, as well as the exact details of its implementation, typically runtime relative to the size of the input is talked. We will introduce the notations for computational complexity and also how to perform an algorithm analysis.

2/20: Charlie

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

Title: Mullins-Sekerka Problem and LaplaceEquation on Multiply Connected Regions

Abstract: Mullins-Sekerka problem is a free boundary problem commonly used in
modeling 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 problems

Abstract: We will consider inverse problems for multiscale partial differential
equations 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 Potentials

Abstract: 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 partitioning

Abstract: 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.

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