Scientific Computing Seminar
Universitat Automa de Barcelona, Spain | |
PLEASE NOTE CHANGE IN DAY AND TIME FOR THIS SEMINAR ONLY |
Abstract: We investigate different models that are intended to describe the small mean free path regime of a kinetic equation, with a particular attention paid to the moment closure by entropy minimization. We design a specific asymptotic-induced numerical strategy which is able to treat the stiff terms of the asymptotic regime. We evaluate on numerics the performance of the method and the ability of the reduced models to capture the main features of the full kinetic equation. This is a work in collaboration with Th. Goudon, P. Lafitte, F. Vecil.
Division of Applied Mathematics -
Probability Seminar
Abstract:
The subject of the talk is the distribution law of
$\int^{\infty}_{0} V (W_{s} + \mu s)ds$ and the joint law of
$\int^{t}_{0} V (W_{s}ds,W_{t}$, with $(W_{t})_{t \geq 0}$ taken
to be some standard
Brownian motion, for certain choices for the parameter
$\mu \in \Re$ and for the function $V : \Re \mapsto \Re$.
These laws play an important role in many areas of analysis, physics,
probability theory, statistics and, in the special
case $V (x) := e^{\sigma x} \in \Re,$ mathematical finance and actuarial
science. Somewhat unexpectedly, integrals of exponential Brownian
motion feature also in the analysis of hyperbolic spaces { see [3].
The most intriguing aspect of the ``theory'' of such functionals is
that it connects some seemingly unrelated domains - for example,
it allows one to develop some useful exponential counterparts
of Levy's and Pitman's theorems (see [4]). Furthermore,
this ``theory'' leads to some striking identities about exponential
functionals of Brownian bridge and Brownian motion processes (see [1]).
While the distribution laws of integral functionals of Brownian
motion have been studied for quite some time (see [5] and [2]),
tractable expressions for the associated densities are still
difficult to obtain (usually the densities are characterized in
terms of the inverse Laplace transform in the time domain, which,
by way of Lamperti's representation, leads to the renowned
integral formula obtained by M. Yor [7]). The main objective of the
talk is to present a new approach to the study of the distribution
law of the integral of geometric Brownian motion. In particular,
we will show that M. Yor's formula [7] is the least tractable member
of a cluster of integral formulas. In addition, we will give an
independent (and considerably simpler) proof of some striking
identities about exponential functionals of Brownian motion and
Brownian bridge and will obtain even more identities of the
same ``striking'' type.
References:
[1] C. Donati-Martin, H. Matsumoto and M. Yor (2000). On Striking
Identities About the Exponential
Functionals of the Brownian Bridge and Brownian Motion.
Endre Cski 65, Period. Math. Hungar., 41 no. 1-2, pp: 103-119.
[2] Daniel Dufresne (1990). The Distribution of a Perpetuity
with Application to Risk Theory and Pension Funding.
Scand. Act. Journal, 90, pp: 39-79.
[3] N. Ikeda and H. Matsumoto (1999). Brownian Motion on the
Hyperbolic Plane and Selberg Trace Formula.
J. Funct. Anal., 163, pp: 63-110.
[4] H. Matsumoto and M. Yor(2001). An Analog of Pitman's 2M - X
Theorem, II: The Role of the Generalized Inverse Gaussian Laws.
Nagoya Math J., 162, pp: 65-86.
[5] Mark Kac (1949). On the Distributions of Certain Wiener
Functionals. Trans. Amer. Math. Soc., 65, pp: 63-110.
[6] Marc Yor (1980). Loi de L'indice du Lacet Brownien, et
Distribution de Hartman-Watson. Zeit. fr Wahr. und Verw.
Gebiete, 53 no. 1, pp: 71-95.
[7] Marc Yor (1992). On Some Exponential Functionals of Brownian
Motion. Adv. Appl. Prob., 24 pp: 509-531.
Boston University, Dept. of Mathematics and Statistics, 143 Bay State Road, Boston, MA, 02215
Brown University Computer Science Theory Lunch Seminar
Abstract: Solving a linear system is one of the most fundamental computational problems. Unfortunately, the basic algorithm that most of us learn (Gaussian Elimination) is often useless in practice due to slow running time or stability issues. Instead, it is more common to use iterative solvers, the simplest ones being steepest descent and conjugate gradient.
The snag with iterative solvers is that their performance often depends on the "condition number" of the given system, so it is common to modify the system by applying a "preconditioner" matrix which reduces the condition number. This raises a key question: given a linear system, how can we find a good preconditioner?
In this work, we develop a variant of conjugate gradient method which *automatically* constructs good preconditioners. The general idea is very simple. We run the conjugate gradient method until it "gets stuck". The fact that it is stuck then implies a way to modify the preconditioner so that the conjugate gradient steps will be "less stuck" in the future.
This talk will be self-contained -- the audience only needs to know basic linear algebra, and how to interpret pictures of algorithms that are stuck.
Joint work with John Dunagan, Microsoft Research.
Brown Analysis Seminar
Transatlantic Seminar
Abstract: I will discuss a 2D nonlinear dispersive wave equation which arises in an elastic plate. In certain cases this equation possesses solitary wave solutions as ground states to a minimization problem. I will show that all ground states are nonlinearly stable.
PDE Seminar
Department of Mathematics Colloquium
Abstract: It is a classical and extremely difficult problem to prove theorems about prime values of irreducible polynomials over the integers. For example, it is still not known if there are infinitely primes of the form $n^{2} + 1$. There is a long history of analogies between the integers and polynomials (in one variable) over a finite field, so one can formulate an analogous problem in this other setting. It was discovered several years ago (joint work with K. Conrad and R. Gross) that there are some surprises upon making the translation. After illustrating these surprises via some explicit examples, and discussing theorems that heuristically predict these phenomena, the main goal of the talk is to motivate (by examples) and discuss more recent asymptotic results as the finite field and the polynomial being specialized are allowed to vary. These asymptotic results make precise the sense in which the surprising behavior in the function field case should be rather typical.
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