The Jr. Numerical Analysis and Applied Math (NAP) seminar is on Thursdays from 2:003:00 in RLM 10.176.
If you are interested in giving a talk, please email svallelian_at_math 
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 lowrank matrices. 
posted Mar 18, 2014, 9:22 AM by Sarah Vallelian
Title: Algorithms and Computational Complexity
Abstract: An algorithm is any welldefined 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, welldefined 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. 
posted Feb 20, 2014, 2:29 PM by Sarah Vallelian
Title: MullinsSekerka Problem and LaplaceEquation on Multiply Connected Regions
Abstract: MullinsSekerka 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 nonconventional regions is of interest,
and the peculiarity of existence, uniqueness, and numerical methods under
this circumstance.

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 illposed 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 illposedness, 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.

posted Nov 14, 2013, 12:36 PM by Sarah Vallelian
Title: Regularity of the solution for Obstacles exhibiting nonlocal 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 nonlocal behavior. Generalizing the 1d situation we will discuss some properties of the solution as well as introduce the notion of a quasivariational inequality. The objective of the talk will be to relate the probabilistic interpretation of the problem to its analytic reformulation and discuss some applications. 
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. 
posted Oct 21, 2013, 9:44 AM by Sarah Vallelian
Title: A Level Set Approach to MullinsSekerka problem and the Regularization of Layer Potentials
Abstract: We will look over the MullinsSekerka problems and introduce the level set approach, some difficulties and possible solutions to the local expansion of layer potentials. 
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. 
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.

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 nonconvex 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. 
