10:00 - 12:00 30/06/2020
Existence and regularity for time fractional diffusion equations and systems
Trần Bảo Ngọc
Abstract: In this seminar, we talk about time fractional diffusion equations containing Caputo’s fractional derivatives. We focus on
- Existence and regularity of final value problems for time diffusion equations
- Existence and regularity of final value problems for time diffusion systems
We discuss both linear and nonlinear cases. Here, spectral theory and fixed point theorems are mainly employed to establish existence and uniqueness of solutions. Then, by making uses of Sobolev embeddings on the Hilbert scales and fractional Sobolev spaces, we obtain some use regularities for the solution.
Speaker: Lý Kim Hà
A brief tour of Cauchy - Riemann equations
Abstract: The main purpose of this talk is to provide a (very) short history of Cauchy-Riemann equations from 1960's. Some related problems are also mentioned.
10:00 9/1/2020, Room F207.
Seminar on PDE
Room E 202B
Calderon-Zygmund theory for nonlinear PDEs and applications
14:00 – 15:00
Speaker: Prof. Tuoc Phan (University of Tennessee – Knoxville, US)
Well-posedness of a fractional degenerate forward-backward problem
15:00 – 16:00
Speaker: Prof. Tan Do (Vietnamese-German University, Vietnam)
Mitigating the Cost of data-driven PDE-constrained Inverse Problems Using Dimensionality Reduction and Deep Learning
Bùi Thanh Tân, Univ. of Texas Austin
Given a hierarchy of reduced-order models to solve the inverse problems for quantities of interest, each model with varying levels of fidelity and computational cost, a deep learning framework is proposed to improve the models by learning the errors between each successive levels.
By statistically modeling errors of reduced order models and using training data involving forward solves of the reduced order models and the higher fidelity model, we train deep neural networks to learn the error between successive levels of the hierarchy of reduced order models thereby improving their error bounds. The training of the deep neural networks occurs during the offline phase and the error bounds can be improved online as new training data is observed.
To mitigate the big-data aspect in inverse problems, we have developed a randomized misfit approach that blends random projection theory in high dimensions and inverse problem theory to effectively reduce high-dimensional data while preserving the accuracy of inverse solution.