This paper develops and analyzes a stochastic derivative-free optimization strategy.A key feature is the state-dependent adaptive variance.We prove global convergence in probability with algebraic rate and give the qu...This paper develops and analyzes a stochastic derivative-free optimization strategy.A key feature is the state-dependent adaptive variance.We prove global convergence in probability with algebraic rate and give the quantitative results in numerical examples.A striking fact is that convergence is achieved without explicit information of the gradient and even without comparing different objective function values as in established methods such as the simplex method and simulated annealing.It can otherwise be compared to annealing with state-dependent temperature.展开更多
This paper gives a systematic introduction to HMM,the heterogeneous multiscale methods,including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using H...This paper gives a systematic introduction to HMM,the heterogeneous multiscale methods,including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular problem.This is illustrated by examples from several application areas,including complex fluids,micro-fluidics,solids,interface problems,stochastic problems,and statistically self-similar problems.Emphasis is given to the technical tools,such as the various constrained molecular dynamics,that have been developed,in order to apply HMM to these problems.Examples of mathematical results on the error analysis of HMM are presented.The review ends with a discussion on some of the problems that have to be solved in order to make HMM a more powerful tool.展开更多
基金partially supported by the National Science Foundation through grants DMS-2208504(BE),DMS-1913309(KR),DMS-1937254(KR),and DMS-1913129(YY)support from Dr.Max Rossler,the Walter Haefner Foundation,and the ETH Zurich Foundation.
文摘This paper develops and analyzes a stochastic derivative-free optimization strategy.A key feature is the state-dependent adaptive variance.We prove global convergence in probability with algebraic rate and give the quantitative results in numerical examples.A striking fact is that convergence is achieved without explicit information of the gradient and even without comparing different objective function values as in established methods such as the simplex method and simulated annealing.It can otherwise be compared to annealing with state-dependent temperature.
基金supported in part by NSF grant DMS99-73341The work of Xiantao Li is supported in part by ONR grant N00014-01-1-0674 and DOE grant DE-FG02-03ER25587The work of Vanden-Eijnden is supported in part by NSF grants DMS02-09959 and DMS02-39625.
文摘This paper gives a systematic introduction to HMM,the heterogeneous multiscale methods,including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular problem.This is illustrated by examples from several application areas,including complex fluids,micro-fluidics,solids,interface problems,stochastic problems,and statistically self-similar problems.Emphasis is given to the technical tools,such as the various constrained molecular dynamics,that have been developed,in order to apply HMM to these problems.Examples of mathematical results on the error analysis of HMM are presented.The review ends with a discussion on some of the problems that have to be solved in order to make HMM a more powerful tool.
