Computing tasks may often be posed as optimization problems.The objective functions for real-world scenarios are often nonconvex and/or nondifferentiable.State-of-the-art methods for solving these problems typically o...Computing tasks may often be posed as optimization problems.The objective functions for real-world scenarios are often nonconvex and/or nondifferentiable.State-of-the-art methods for solving these problems typically only guarantee convergence to local minima.This work presents Hamilton-Jacobi-based Moreau adaptive descent(HJ-MAD),a zero-order algorithm with guaranteed convergence to global minima,assuming continuity of the objective function.The core idea is to compute gradients of the Moreau envelope of the objective(which is"piece-wise convex")with adaptive smoothing parameters.Gradients of the Moreau envelope(i.e.,proximal operators)are approximated via the Hopf-Lax formula for the viscous Hamilton-Jacobi equation.Our numerical examples illustrate global convergence.展开更多
We review the level set methods for computing multi-valued solutions to a class of nonlinear first order partial differential equations,including Hamilton-Jacobi equations,quasi-linear hyperbolic equations,and conserv...We review the level set methods for computing multi-valued solutions to a class of nonlinear first order partial differential equations,including Hamilton-Jacobi equations,quasi-linear hyperbolic equations,and conservative transport equations with multi-valued transport speeds.The multivalued solutions are embedded as the zeros of a set of scalar functions that solve the initial value problems of a time dependent partial differential equation in an augmented space.We discuss the essential ideas behind the techniques,the coupling of these techniques to the projection of the interaction of zero level sets and a collection of applications including the computation of the semiclassical limit for Schr¨odinger equations and the high frequency geometrical optics limits of linear wave equations.展开更多
基金partially funded by AFOSR MURI FA9550-18-502,ONR N00014-18-1-2527,N00014-18-20-1-2093,N00014-20-1-2787supported by the NSF Graduate Research Fellowship under Grant No.DGE-1650604.
文摘Computing tasks may often be posed as optimization problems.The objective functions for real-world scenarios are often nonconvex and/or nondifferentiable.State-of-the-art methods for solving these problems typically only guarantee convergence to local minima.This work presents Hamilton-Jacobi-based Moreau adaptive descent(HJ-MAD),a zero-order algorithm with guaranteed convergence to global minima,assuming continuity of the objective function.The core idea is to compute gradients of the Moreau envelope of the objective(which is"piece-wise convex")with adaptive smoothing parameters.Gradients of the Moreau envelope(i.e.,proximal operators)are approximated via the Hopf-Lax formula for the viscous Hamilton-Jacobi equation.Our numerical examples illustrate global convergence.
基金the National Science Foundation under Grant DMS05-05975Osher’s research was supported by AFOSR Grant FA9550-04-0143NSF DMS-0513394 and the Sloan Foundation。
文摘We review the level set methods for computing multi-valued solutions to a class of nonlinear first order partial differential equations,including Hamilton-Jacobi equations,quasi-linear hyperbolic equations,and conservative transport equations with multi-valued transport speeds.The multivalued solutions are embedded as the zeros of a set of scalar functions that solve the initial value problems of a time dependent partial differential equation in an augmented space.We discuss the essential ideas behind the techniques,the coupling of these techniques to the projection of the interaction of zero level sets and a collection of applications including the computation of the semiclassical limit for Schr¨odinger equations and the high frequency geometrical optics limits of linear wave equations.
