We introduce a new tensor integration method for time-dependent partial differential equations(PDEs)that controls the tensor rank of the PDE solution via time-dependent smooth coordinate transformations.Such coordinat...We introduce a new tensor integration method for time-dependent partial differential equations(PDEs)that controls the tensor rank of the PDE solution via time-dependent smooth coordinate transformations.Such coordinate transformations are obtained by solving a sequence of convex optimization problems that minimize the component of the PDE operator responsible for increasing the tensor rank of the PDE solution.The new algorithm improves upon the non-convex algorithm we recently proposed in Dektor and Venturi(2023)which has no guarantee of producing globally optimal rank-reducing coordinate transformations.Numerical applications demonstrating the effectiveness of the new coordinate-adaptive tensor integration method are presented and discussed for prototype Liouville and Fokker-Planck equations.展开更多
基金supported by the U.S.Air Force Office of Scientific Research(AFOSR)grant FA9550-20-1-0174the U.S.Army Research Office(ARO)grant W911NF-18-1-0309.
文摘We introduce a new tensor integration method for time-dependent partial differential equations(PDEs)that controls the tensor rank of the PDE solution via time-dependent smooth coordinate transformations.Such coordinate transformations are obtained by solving a sequence of convex optimization problems that minimize the component of the PDE operator responsible for increasing the tensor rank of the PDE solution.The new algorithm improves upon the non-convex algorithm we recently proposed in Dektor and Venturi(2023)which has no guarantee of producing globally optimal rank-reducing coordinate transformations.Numerical applications demonstrating the effectiveness of the new coordinate-adaptive tensor integration method are presented and discussed for prototype Liouville and Fokker-Planck equations.
