Speaker: Dr. Joshua Lee Padgett, Department of Mathematical Sciences, University of Arkansas
Date: Wednesday, September 23, 2020, 4:00 PM – 5:00 PM
Title: Beating the curse of dimensionality in high-dimensional partial differential equations
Abstract: In recent years, high dimensional partial differential equations (PDEs) have become a topic of extreme interest due to their occurrence in numerous scientific fields. Examples of such equations include the Schr{\”o}dinger equation in quantum many-body problems, the nonlinear Black-Scholes equation for pricing financial derivatives, and the Hamilton-Jacobi-Bellman equation in dynamic programming. In each of these cases, standard numerical techniques suffer from the so-called curse of dimensionality, which refers to the computational complexity of an employed approximation method growing exponentially as a function of the dimension of the underlying problem. This phenomenon is what prevents traditional numerical algorithms, such as finite differences and finite element methods, from being efficiently employed in problems with more than, say, ten dimensions. The purpose of this talk is to introduce a novel approximation algorithm known as the multilevel Picard (MLP) approximation method for beating the curse of dimensionality in the case of semilinear PDEs. This talk will focus on developing an understanding of the methods used to develop and analyze MLP approximations, as one must have a strong understanding of concepts from real and stochastic analysis, probability theory, and stochastic fixed point equations. As such, we will also discuss the need for a deeper understanding of pure mathematics to help further develop the field of applied mathematics. Numerical examples will be provided in order to provide experimental verification of the obtained results. This talk should be accessible to all students with a basic understanding of analysis (with intuition for more advanced topics provided).
Bio: Josh Padgett is an Assistant Professor in the Department of Mathematical Sciences at the University of Arkansas. Prior to joining the University of Arkansas, Josh was a postdoc in the Department of Mathematics and Statistics at Texas Tech University. Josh earned his Ph.D. in mathematics at Baylor University under the supervision of Qin Sheng. His undergraduate research focused on the metabolic features of tumor cells and the occurrence of the so-called Warburg effect. Josh is also an affiliated faculty member at the Center for Astrophysics, Space Physics, and Engineering Research and is a honorary adjunct faculty at Texas Tech University. His research lies at the intersection of pure, applied, and computational mathematics. Current research interests include applied mathematics, numerical analysis, geometric and Lie group integration methods, mathematics of deep learning, operator splitting methods, algebraic structures of numerical methods, fractional differential equations, stochastic differential equations, and the use of spectral and operator theory in theoretical and experimental physics.
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