Numerical Methods for Partial Differential Equations

Numerical methods for partial differential equation have experienced steady developments in the past several decades.Most of the remaining challenges fall into two categories,both of which are associated with the curse of the dimensionality.In the first category,the func-tion of the partial differential equation is high dimensional,with examples from many-body quantum mechanics,molecular dynamics,to kinetic equations and control problems.In the second category,the objective of the interest is the high dimensional map from the parameters of the equation to the solution or vice versa,with examples such as inverse problems and un-certainty quantifications.The curse of the dimensionality has been the bottleneck for further breakthroughs in the numerical solution of partial differential equation problems.