# Analysis Seminar

## Analysis Seminar - Spring 2018

The seminar meets every other Monday at 3:30pm in Rolla G5.

Speaker: John Singler (Missouri S&T)

Title: Global attractors in a low dimensional model for transition to turbulence

Speaker: Jason Murphy (Missouri S&T)

Title: Scattering below the ground state for nonlinear Schrödinger equations

Abstract: The ground state solution to the nonlinear Schrödinger equation (NLS) is a global, non-scattering solution that often provides a threshold between scattering and blowup. In this talk, we will discuss new, simplified proofs of scattering below the ground state threshold (joint with B. Dodson), as well as some extensions to other models of NLS (joint with R. Killip, M. Visan, J. Zheng, as well as with C. Miao and J. Lu).

Speaker: Wenqing Hu (Missouri S&T)

Title: A random perturbation approach to some stochastic approximation algorithms in optimization

Abstract: Many large-scale learning problems in modern statistics and machine learning can be reduced to solving stochastic optimization problems, i.e., the search for (local) minimum points of the expectation of an objective random function (loss function). These optimization problems are usually solved by certain stochastic approximation algorithms, which are recursive update rules with random inputs in each iteration. In this talk, we will be considering various types of such stochastic approximation algorithms, including the stochastic gradient descent, the stochastic composite gradient descent, as well as the stochastic heavy-ball method. By introducing approximating diffusion processes to the discrete recursive schemes, we will analyze the convergence of the diffusion limits to these algorithms via delicate techniques in stochastic analysis and asymptotic methods, in particular via random perturbations of dynamical systems. This talk is based on a series of joint works with Chris Junchi Li (Princeton), Weijie Su (UPenn) and Haoyi Xiong (Missouri S&T).

## Fall 2017

Title: On the fast convergence of random perturbations of the gradient flow

Abstract: We consider in this work small random perturbations (of multiplicative noise type) of the gradient flow. We rigorously prove that under mild conditions, when the potential function is a Morse function with additional strong saddle condition, the perturbed gradient flow converges to the neighborhood of local minimizers in O(ln(ε^{-1})) time on the average, where ε>0 is the scale of the random perturbation. Under a change of time scale, this indicates that for the diffusion process that approximates the stochastic gradient method, it takes (up to logarithmic factor) only a linear time of inverse stepsize to evade from all saddle points and hence it implies a fast convergence of its discrete--time counterpart.

Title: Almost sure scattering for the energy-critical NLS

Abstract: We consider the defocusing energy-critical nonlinear Schrödinger equation in four space dimensions with radial (i.e. spherically-symmetric) initial data below the energy space. In this setting, the problem is known to be ill-posed. Nonetheless, we can show that for suitably randomized radial initial data, one obtains global well-posedness and scattering almost surely. This is joint work with R. Killip and M. Visan.

Title: Boundary layers for viscous incompressible flow (part I)

Abstract: Prandtl boundary layer theory is a conundrum in mathematical fluid dynamics. In this talk, I shall give an overview of the current status on the resolution of the theory, including a positive result for a family of parallel pipe flow as well as a counter example.

Title: Boundary layers for viscous incompressible flow (part II)

Abstract: Prandtl boundary layer theory is a conundrum in mathematical fluid dynamics. In this talk, I shall give an overview of the current status on the resolution of the theory, including a positive result for a family of parallel pipe flow as well as a counter example.

Title: Large deviations and averaging for systems of slow-fast stochastic reaction-diffusion equations

Abstract: We study a large deviation principle for a system of stochastic reaction-diffusion equations (SRDEs) with a separation of fast and slow components and small noise in the slow component. The derivation of the large deviation principle is based on the weak convergence method in infinite dimensions, which results in studying averaging for controlled SRDEs. By appropriate choice of the parameters, the fast process and the associated control that arises from the weak convergence method decouple from each other. We show that in this decoupling case one can use the weak convergence method to characterize the limiting process via a "viable pair" that captures the limiting controlled dynamics and the effective invariant measure simultaneously. The characterization of the limit of the controlled slow-fast processes in terms of viable pair enables us to obtain a variational representation of the large deviation action functional. Due to the infinite-dimensional nature of our set-up, the proof of tightness as well as the analysis of the limit process and in particular the proof of the large deviations lower bound is considerably more delicate here than in the finite-dimensional situation. Smoothness properties of optimal controls in infinite dimensions (a necessary step for the large deviations lower bound) need to be established. We emphasize that many issues that are present in the infinite dimensional case, are completely absent in finite dimensions.

For more information about the analysis seminar, contact Dr. David Grow.