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[UC] AMS Conference, Applied PDEs and Opt

Nov 9, 2019, at UC Riverside

1. Additional theoretical works  on Sphere covering inequality

2. Weak Solutions of Mean Field Game Master Equations

Background: Mean field game equation is to find Nash equilibrium for an N-player differential game (Game theory) when N is big enough.

Monotonicity is required for unique sol. Otherwise counterexamples exist, multiple sols appear.

3. Asymptotic Analysis of a Simple Thermostat Problem

check the spectrum, pairs of eigenvalues merge and approach to imaginary axis.

4. Super-resolution in imaging high contrast target from the perspective of scattering coefficients (Yat Tin Chow, UCR)

inverse problem: see objects in measuring physical entities, and reconstructing the data. Abbe-Rayleigh resolution limit: minimal distance we can resolve. some people try to break the limit (super resolution). Breaking diffraction barrier near field imaging, etc. Our contribution: aim at mathematical understanding of naturally observe super resolution.

Forward problem: have a equation, find the measurement. Inverse problem: inverse way, given measurement.

scattering coefficients. For incident field of mode, decay property and transformation rules.

5.  Computer-assisted proofs for optimization methods and fixed-point iterations

convergence analysis is equivalent to semidefinite optimization program. algorithm: Douglas-Rachford splitting.

optimization problem: DRS iteration, find min contraction factor rho. assume operator A is mu-strongly monotone. Used Tr and other tools to transform the problem into a convex optimization problem which is desired. Use Mathematica to get the convergence rate by computing the equivalent optimization problem.

Fast prototyping. Numerically test whether the target algorithm convergent, if so, then try to prove it manually. If the computer tells you that it’s impossible, then try to add more assumptions until the algorithm convergent.

6. Particle Swarm Optimization-Based Source Seeking with Obstacle Avoidance.

partical swam optimization can be used to direct robot movement.