Monday, November 2, 2015 - 14:05 , Location: Skiles 005 , Professor James von Brecht , Cal State University, Long Beach , Organizer: Martin Short
In this talk, I will discuss mathematical models and tools for analyzing physical and biological processes that exhibit co-dimension one characteristics. Examples include the assembly of inorganic polyoxometalate (POM) macroions into hollow spherical structures and the assembly of surfactant molecules into micelles and vesicles. I will characterize when such structures can arise in the context of isotropic and anisotropic models, as well as applications of these insights to physical models of these behaviors.
Thursday, October 29, 2015 - 11:00 , Location: Skiles 006 , Philippe Chartier , INRIA Rennes, Université de Rennes I, ENS Rennes , Philippe.Chartier@inria.fr , Organizer: Molei Tao
Joint with School of Math Colloquium. Special time (colloquium time).
In this talk, I will introduce B-series, which are formal series indexed by trees, and briefly expose the two laws operating on them. The presentation of algebraic aspects will here be focused on applications to numerical analysis. I will then show how B-series can be used on two examples: modified vector fields techniques, which allow for the construction of arbitrarly high-order schemes, and averaging methods, which lie at the core of many numerical schemes highly-oscillatory evolution equations. Ultimately and if time permits, I will illustrate how these concepts lead to the accelerated simulation of the rigid body and the (nonlinear) Schrödinger equations. A significant part of the talk will remain expository and aimed at a general mathematical audience.
Tuesday, October 27, 2015 - 12:30 , Location: Skiles 005 , Venkat Chandrasekaran , Cal Tech , Organizer: Greg Blekherman
Due to its favorable analytical properties, the relative entropy function plays a prominent role in a variety of contexts in information theory and in statistics. In this talk, I'll discuss some of the beneficial computational properties of this function by describing a class of relative-entropy-based convex relaxations for obtaining bounds on signomials programs (SPs), which arise commonly in many problems domains. SPs are non-convex in general, and families of NP-hard problems can be reduced to SPs. By appealing to representation theorems from real algebraic geometry, we show that sequences of bounds obtained by solving increasingly larger relative entropy programs converge to the global optima for broad classes of SPs. The central idea underlying our approach is a connection between the relative entropy function and efficient proofs of nonnegativity via the arithmetic-geometric-mean inequality. (Joint work with Parikshit Shah.)
Monday, October 26, 2015 - 14:00 , Location: Skiles 005 , Professor Maarten de Hoop , Rice University , firstname.lastname@example.org , Organizer:
We consider an inverse problem for an inhomogeneous wave equation with discrete-in-time sources, modeling a seismic rupture. We assume that the sources occur along an unknown path with subsonic velocity, and that data is collected over time on some detection surface. We explore the question of uniqueness for these problems, and show how to recover the times and locations of sources microlocally first, and then the smooth part of the source assuming that it is the same at each source location. In case the sources (now all different) are (roughly speaking) non-negative and of limited oscillation in space, and sufficiently separated in space-time, which is a model for microseismicity, we present an explicit reconstruction, requiring sufficient local energy decay. (Joint research with L. Oksanen and J. Tittelfitz)
Simultaneous Random and Optimized Sources and Detectors for Efficient Optimization in Inverse ProblemsMonday, October 19, 2015 - 14:00 , Location: Skiles 005 , Eric de Sturler , Department of Mathematics, Virginia Tech , email@example.com , Organizer: Sung Ha Kang
In nonlinear inverse problems, we often optimize an objective function involving many sources, where each source requires the solution of a PDE. This leads to the solution of a very large number of large linear systems for each nonlinear function evaluation, and potentially additional systems (for detectors) to evaluate or approximate a Jacobian. We propose a combination of simultaneous random sources and detectors and optimized (for the problem) sources and detectors to drastically reduce the number of systems to be solved. We apply our approach to problems in diffuse optical tomography.This is joint work with Misha Kilmer and Selin Sariaydin.
Wednesday, October 14, 2015 - 14:00 , Location: Skiles 270 , Vira Babenko , The University of Utah , firstname.lastname@example.org , Organizer: Sung Ha Kang
A wide variety of questions which range from social and economic sciences to physical and biological sciences lead to functions with values that are sets in finite or infinite dimensional spaces, or that are fuzzy sets. Set-valued and fuzzy-valued functions attract attention of a lot of researchers and allow them to look at numerous problems from a new point of view and provide them with new tools, ideas and results. In this talk we consider a generalized concept of such functions, that of functions with values in so-called L-space, that encompasses set-valued and fuzzy functions as special cases and allow to investigate them from the common point of view. We will discus several problems of Approximation Theory and Numerical Analysis for functions with values in L-spaces. In particular numerical methods of solution of Fredholm and Volterra integral equations for such functions will be presented.
Monday, October 5, 2015 - 14:00 , Location: Skiles 005 , Felix Lieder , Mathematisches Institut Lehrstuhl für Mathematische Optimierung , email@example.com , Organizer:
Survival can be tough: Exposing a bacterial strain to new environments will typically lead to one of two possible outcomes. First, not surprisingly, the strain simply dies; second the strain adapts in order to survive. In this talk we are concerned with the hardness of survival, i.e. what is the most eﬃcient (smartest) way to adapt to new environments? How many new abilities does a bacterium need in order to survive? Here we restrict our focus on two speciﬁc bacteria, namely E.coli and Buchnera. In order to answer the questions raised, we ﬁrst model the underlying problem as an NP-hard decision problem. Using a re-weighted l1-regularization approach, well known from image reconstruction, we then approximate ”good” solutions. A numerical comparison between these ”good” solutions and the ”exact” solutions concludes the talk.
Monday, September 28, 2015 - 14:05 , Location: Skiles 005 , Dr. Christina Frederick , GA Tech , Organizer: Martin Short
I will discuss inverse problems involving elliptic partial differential equations with highly oscillating coefficients. The multiscale nature of such problems poses a challenge in both the mathematical formulation and the numerical modeling, which is hard even for forward computations. I will discuss uniqueness of the inverse in certain problem classes and give numerical methods for inversion that can be applied to problems in medical imaging and exploration seismology.
Monday, September 14, 2015 - 14:00 , Location: Skiles 005 , Associate Professor Hongchao Zhang , Department of Mathematics and Center for Computational & Technology (CCT) at Louisiana State University , firstname.lastname@example.org , Organizer:
In this talk, we discuss a very efficient algorithm for projecting a point onto a polyhedron. This algorithm solves the projeciton problem through its dual and fully exploits the sparsity. The SpaRSA (Sparse Reconstruction by Separable Approximation) is used to approximately identify active constraints in the polyhedron, and the Dual Active Set Algorithm (DASA) is used to compute a high precision solution. Some interesting convergence properties and very promising numerical results compared with the state-of-the-art software IPOPT and CPLEX will be discussed in this talk.
Monday, April 20, 2015 - 15:05 , Location: Skiles 005 , Dr. Antonio Cicone , L'Aquila, Italy , Organizer: Haomin Zhou
Given a finite set of matrices F, the Markovian Joint Spectral Radius represents the maximal rate of growth of products of matrices in F when the matrices are multiplied each other following some Markovian law. This quantity is important, for instance, in the study of the so called zero stability of variable stepsize BDF methods for the numerical integration of ordinary differential equations. Recently Kozyakin, based on a work by Dai, showed that, given a set F of N matrices of dimension d and a graph G, which represents the admissible products, it is possibile to compute the Markovian Joint Spectral Radius of the couple (F,G) as the classical Joint Spectral Radius of a new set of N matrices of dimension N*d, which are produced as a particular lifting of the matrices in F. Clearly by this approach the exact evaluation or the simple approximation of the Markovian Joint Spectral Radius becomes a challenge even for reasonably small values of N and d. In this talk we briefly review the theory of the Joint Spectral Radius, and we introduce the Markovian Joint Spectral Radius. Furthermore we address the question whether it is possible to reduce the exact calculation computational complexity of the Markovian Joint Spectral Radius. We show that the problem can be recast as the computation of N polytope norms in dimension d. We conclude the presentation with some numerical examples. This talk is based on a joint work with Nicola Guglielmi from the University of L'Aquila, Italy, and Vladimir Yu. Protasov from the Moscow State University, Russia.