DEPARTMENT OF MATHEMATICAL SCIENCES

APPLIED AND COMPUTATIONAL MATHEMATICS SEMINAR

**Speaker:** Ratna Khatri, George Mason University

**Title: ***
Role of Nonlocal operators in Inverse Problems and Deep Learning
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**Abstract:**
In this talk, we will discuss the applications of nonlocal operators in three kinds of inverse problems. At first we introduce an external source identification problem with fractional partial differential equation (PDE) as constraints. Our motivation to introduce this new class of inverse problems stems from the fact that the classical PDE models only allow the source/control to be placed on the boundary or inside the observation domain where the PDE is fulfilled. Our new approach allows us to place the source/control outside and away from the observation domain. The second problem is motivated by imaging science where we propose to use the fractional Laplacian as a regularizer. In addition, we create a bilevel optimization neural network (BONNet) to learn the optimal regularization parameters, like the strength of regularization and the fractional exponent. As our model problem, we consider tomographic reconstruction and show an improvement in the reconstruction quality, especially for limited data. Lastly, we introduce a novel deep neural network with a rigorous mathematical framework. Our network allows information passage across all the network layers.

**Time:** Friday, February 14, 2020, 1:30-2:30pm

**Place:** Exploratory Hall, Room 4106

Department of Mathematical Sciences

George Mason University

4400 University Drive, MS 3F2

Fairfax, VA 22030-4444

http://math.gmu.edu/

Tel. 703-993-1460, Fax. 703-993-1491