A Continuation Multiple Shooting Method for Wasserstein Geodesic Equation

On Homotopy Continuation Method for Computing Multiple Solutions to the Henon Equation

Motivated by numerical examples in solving semilinear elliptic PDEs for multiple solutions, some properties of Newton homotopy continuation method, such as its continuation on symmetries, the Morse index and certain functional structures, are established. Those results provide useful information on selecting initial points for the method to find desired solutions. As an application, a bifurcati...

Complementary Condensing for the Direct Multiple Shooting Method

In this contribution we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems typically arise from the convexification of integer control decisions. We treat this problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved using an SQP method. Concerning...

Direct multiple shooting method for solving approximate shortest path problems

Nonlinear diffusion: Geodesic Convexity is equivalent to Wasserstein Contraction

It is well known that nonlinear diffusion equations can be interpreted as a gradient flow in the space of probability measures equipped with the Euclidean Wasserstein distance. Under suitable convexity conditions on the nonlinearity, due to R. J. McCann [10], the associated entropy is geodesically convex, which implies a contraction type property between all solutions with respect to this dista...

Periodic Solutions of the Hamilton - Jacobi Equation by the Shooting Method : a Technique for Beam Dynamics

Periodic solutions of the Hamilton-Jacobi equation determine invariant tori in phase space. The Fourier spectrum of a torus with respect to angular coordinates gives useful information about nonlinear resonances and their potential for causing instabilities. We describe a method to solve the Hamilton-Jacobi equation for an arbitrary accelerator lattice. The method works with Fourier modes of th...