## Seminars and Colloquia Schedule

### Thesis defense: Invariance of random matrix

Series
Time
Thursday, May 12, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
JunTao DuanGeorgia institute of technology

Random matrix has been found useful in many real world applications. The celebrated Johnson-Lindenstrauss lemma states that certain geometric structure of deterministic vectors is preserved when projecting high dimensional space $R^n$ to a lower dimensional space $R^m$. However, when random vectors are concerned, it is still unclear how the distribution of the geometry is affected by random matrices. Since random projection or embedding introduces dependence to independent random vectors, does it imply random matrices are inferior for transforming random vectors?

We will start with establishing a new  central limit theorem  for random variables with certain product dependence structure. At the same time, we obtain its Berry-Esseen type rate of convergence. Then we apply this general central limit theorem to random projections and embeddings of two independent random vectors $X, Z$. In particular, we show the distribution of inner product structure is preserved by random matrices. Roughly speaking, two independent random vectors remain "independent" in the randomly projected lower dimensional space or randomly embedded high dimensional space. More importantly, we also quantitatively characterize the distortion of distribution introduced by random matrices. The error term has a bound at most $O(\frac{1}{\sqrt{m}} + \frac{1}{\sqrt{n}})$.

Then we also establish the fact that random matrices have low distortion on the norm of a random vector. It is first justified by establishing concentration of the projected or embedded norm under sub-Gaussian assumptions. A central limit theorem for the randomly projected norm is established as well similar to the CLT for inner product.

### Approximation of invariant manifolds for Parabolic PDEs over irregular domains

Series
CDSNS Colloquium
Time
Friday, May 13, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Jorge GonzalezGeorgia Tech

The computation of invariant manifolds for parabolic PDE is an important problem due to its many applications. One of the main difficulties is dealing with irregular high dimensional domains when the classical Fourier methods are not applicable, and it is necessary to employ more sophisticated numerical methods. This work combines the parameterization method based on an invariance equation for the invariant manifold, with the finite element method. We implement the method for a variety of example problems having both polynomial and non-polynomial nonlinearities, on non-convex two-dimensional polygonal domains (not necessary simply connected), for equilibrium solutions. We implement a-posteriori error indicators which provide numerical evidence of the accuracy of the computations. This is a joint work with J.D Mireles-James, and Necibe Tuncer.