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In the present work, we develop a novel particle method for a general class of mean field control problems, with source and terminal constraints. Specific examples of the problems we consider include the dynamic formulation of thep-Wasserstein metric, optimal transport around an obstacle, and measure transport subject to acceleration controls. Unlike existing numerical approaches, our particle method is meshfree and does not require global knowledge of an underlying cost function or of the terminal constraint. A key feature of our approach is a novel way of enforcing the terminal constraint via a soft, nonlocal approximation, inspired by recent work on blob methods for diffusion equations.We prove convergence of our particle approximation to solutions of the continuum mean-field control problem in the sense of Γ-convergence. A byproduct of our result is an extension of existing discrete-to-continuum convergence results for mean field control problems to more general state and measure costs, as arise when modeling transport around obstacles, and more general constraint sets, including controllable linear time invariant systems. Finally, we conclude by implementing our method numerically and using it to compute solutions the example problems discussed above. We conduct a detailed numerical investigation of the convergence properties of our method, as well as its behavior in sampling applications and for approximation of optimal transport maps. 

As a counterpoint to classical stochastic particle methods for linear diffusion equations, such as Langevin dynamics for the Fokker-Planck equation, we develop a deterministic particle method for the weighted porous medium equation and prove its convergence on bounded time intervals. This generalizes related work on blob methods for unweighted porous medium equations. From a numerical analysis perspective, our method has several advantages: it is meshfree, preserves the gradient flow structure of the underlying PDE, converges in arbitrary dimension, and captures the correct asymptotic behavior in simulations. 

In this work, we build a unifying framework to interpolate between density-driven and geometry-based algorithms for data clustering and, specifically, to connect the mean shift algorithm with spectral clustering at discrete and continuum levels. We seek this connection through the introduction of Fokker–Planck equations on data graphs. Besides introducing new forms of mean shift algorithms on graphs, we provide new theoretical insights on the behavior of the family of diffusion maps in the large sample limit as well as provide new connections between diffusion maps and mean shift dynamics on a fixed graph. Several numerical examples illustrate our theoretical findings and highlight the benefits of interpolating density-driven and geometry-based clustering algorithms. 

Abstract Combining the classical theory of optimal transport with modern operator splitting techniques, we develop a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media, materials science, and biological swarming. Our method proceeds as follows: first, we discretize in time, either via the classical JKO scheme or via a novel Crank–Nicolson-type method we introduce. Next, we use the Benamou–Brenier dynamical characterization of the Wasserstein distance to reduce computing the solution of the discrete time equations to solving fully discrete minimization problems, with strictly convex objective functions and linear constraints. Third, we compute the minimizers by applying a recently introduced, provably convergent primal dual splitting scheme for three operators (Yan in J Sci Comput 1–20, 2018). By leveraging the PDEs’ underlying variational structure, our method overcomes stability issues present in previous numerical work built on explicit time discretizations, which suffer due to the equations’ strong nonlinearities and degeneracies. Our method is also naturally positivity and mass preserving and, in the case of the JKO scheme, energy decreasing. We prove that minimizers of the fully discrete problem converge to minimizers of the spatially continuous, discrete time problem as the spatial discretization is refined. We conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our approach, including higher-order convergence our novel Crank–Nicolson-type method, when compared to the classical JKO method.