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3. Sampling and optimization

    1. Asymptotic behavior of nonlinear kinetic Fokker-Planck equations

          Consider the following nonlinear modifications of Fokker-Planck equations $$\partial_t f = Div(A[f]\nabla_xf) + B[f]\nabla_x V$$ where $A[f]$ is a matrix that depends nonlinearly on $f$ and $V$ is a potential. Current results are mostly for sampling from a Gaussian where the inverse Hession of the covariance matrix can be computed explicitly.

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      Problem 3.1.

      Prove convergence... Determine convergence rates...

        • Sampling high dimension distributions


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          Problem 3.2.

          Find a measure $\mu$ supported on a hypersurface $S \subset \mathbb{R}^n$ and a nonlinear, degenerate, parabolic equation for which solutions converge to $\mu$.

          Remark: This should require a diffusion that is degenerate or becomes degenerate in the limit as $t \to \infty$.

            • Consensus based optimization with time dependent temperature

                  Consensus based optimization has been proposed to do global optimization of a non-convex objective function without gradient descent.

              To do this, one works with the thermalization of the measure of the solution of the system at time $t$ and as $T^{-1} \to \infty$, this weighted mean converges to a global minimum.

              In particular, one fixes $T^{-1}$ and takes the limit $t \to \infty$ and the solutions collapses to a dirac delta. Then, one proves that this point is arbitrarily close to the global minimum if $T$ is taken to be small enough.

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              Problem 3.3.

              [Franca Hoffman] Can we design a time-dependent global optimization such that you can still prove convergence to the global minimizer? Can we find an optimal cooling schedule?

                  Cite this as: AimPL: Integro-differential equations in many-particle interacting systems, available at http://aimpl.org/manyparticle.