Applied Mathematics and Machine Learning
Department of Mathematics and Statistics
Texas Tech University
Test abstract
In this talk, I will introduce several low-rank tensor approaches for solving high-dimensional PDEs.In this talk, I will introduce several low-rank tensor approaches for solving high-dimensional PDEs.The subject of the talk are biological and biomedical problems that we model using systems of chemotactic equations with cross-diffusion terms. I will present recent results related to quasi-periodic behavior of biological system arising from chemotactic framework of Keller-Segel.
In the first part of the talk, model representing traveling wave phenomena of bacteria dynamics in the presence of the diffusion in the media source of attraction (e.g. food, drug, etc.) will be presented. We use celebrated Keller-Segel framework using system of partial differential equation with cross-diffusion and reactive terms in case of traveling wave pattern stability. We use closed form type solution of Keller-Segel system when diffusion coefficient for source of the attractor is equal zero. Stability of the traveling band type solution for complete dynamical system with respect to parameters of equation initial and boundary data is investigated in detail. For this purpose Lyapunov type functional is constructed and non-linear Gronwall differential inequality for Energy functional is derived. Obtained result provide explicit estimate for differences between base-line solution with no diffusion in source term and solution of complete realistic system of equation.
Second one is joint research with Angie Piec and Bobby Nettle and dedicated to light driven spatial Algae-Daphinia dynamics as primitive evolutionary model with chemotaxis. First constructed analytically time and spatially dependent solution of system of two equation, which model Algae-Daphinia dynamics and proved using maximum principle machinery that this solution is unique.
Then I proved that this solution is stable depending on the relations between chemotactic and diffusion coefficients, and reactive terms, using Sobolev embedding theorems, and Energy functional.
It is well-known that the three-dimensional (3D) Navier-Stokes equations have found much applications in our real world, e.g., aerospace, fluid mechanics, and even medical sciences. In PDE & Analysis community, it is also famous for being selected as one of the seven Millennium Prize Problems declared by the Clay Research Institute. Indeed, while the 2D Navier-Stokes equations is L2-norm critical (and hence it was solved almost a century ago by Leray), the 3D Navier-Stokes equations is L2-norm supercritical, and similarly to many other supercritical nonlinear PDEs (e.g., wave equation), the global regularity of its solution has remained extremely difficult. In 2009, Terence Tao proved the global regularity of logarithmically supercritical wave equation and this new direction has spread to many other PDEs (Boussinesq, Euler, magnetohydrodynamics, Navier-Stokes, quasi-geostrophic). I will describe the difficulty of the Navier-Stokes equations using fractional Laplacians, and then use harmonic analysis tools such as the Littlewood-Paley decomposition and Bernstein's inequality to describe this new phenomenon. The talk will have some Analysis content, but I will do my best to make it accessible to graduate students. Deep neural networks and other deep learning methods have very successfully been applied to the numerical approximation of high-dimensional nonlinear parabolic partial differential equations (PDEs). In particular, simulations indicate that algorithms based on deep learning overcome the curse of dimensionality in the numerical approximation of solutions of semilinear PDEs. For certain linear PDEs this has also been proved mathematically. However, proofs in the more general case have been difficult to rigorously formulate. To that end, we present a novel framework to alleviate this issue. In this talk, we will briefly introduce the concept of deep learning and then define an abstract framework which is amenable to proving theoretical results. We then prove some auxiliary results regarding the representation of particular approximations of a class of PDEs. In particular, we prove in the case of semilinear heat equations with gradient-independent nonlinearities that the numbers of parameters of the employed deep neural networks grow at most polynomially in both the PDE dimension and the reciprocal of the prescribed approximation accuracy. Our proof relies on recently introduced full history recursive multilevel Picard approximations of semilinear PDEs.