The local minimax method(LMM)proposed by Li and Zhou(2001,2002)is an efficient method to solve nonlinear elliptic partial differential equations(PDEs)with certain variational structures for multiple solutions.The stee...The local minimax method(LMM)proposed by Li and Zhou(2001,2002)is an efficient method to solve nonlinear elliptic partial differential equations(PDEs)with certain variational structures for multiple solutions.The steepest descent direction and the Armijo-type step-size search rules are adopted in Li and Zhou(2002)and play a significant role in the performance and convergence analysis of traditional LMMs.In this paper,a new algorithm framework of the LMMs is established based on general descent directions and two normalized(strong)Wolfe-Powell-type step-size search rules.The corresponding algorithm framework,named the normalized Wolfe-Powell-type LMM(NWP-LMM),is introduced with its feasibility and global convergence rigorously justified for general descent directions.As a special case,the global convergence of the NWP-LMM combined with the preconditioned steepest descent(PSD)directions is also verified.Consequently,it extends the framework of traditional LMMs.In addition,conjugate-gradient-type(CG-type)descent directions are utilized to speed up the NWP-LMM.Finally,extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared with different algorithms in the LMM’s family to indicate the effectiveness and robustness of our algorithms.In practice,the NWP-LMM combined with the CG-type direction performs much better than its known LMM companions.展开更多
First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking...First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking purposes, inexact proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes applied to infinite-dimensional elliptic optimal control problems and to validate the theoretical estimates.展开更多
This paper considers the existence problem of an elliptic equation, which is equivalent to solving the so called prescribing conformal Gaussian curvature problem on the hyperbolic disc H^2. An existence result is prov...This paper considers the existence problem of an elliptic equation, which is equivalent to solving the so called prescribing conformal Gaussian curvature problem on the hyperbolic disc H^2. An existence result is proved. In particular, K（x） is allowed to be unbounded above.展开更多
This paper considers a semilinear elliptic equation on a n-dimensional complete noncompact R.iemannian manifold, which is a generalization of the well known Yamabe equation. An existence result is proved.
Following work carried out earlier on linear-quadratic optimal control for linear finitedimensional stationary systems we report,in this article,on extension of some of those results to certaininfinite dimensional sys...Following work carried out earlier on linear-quadratic optimal control for linear finitedimensional stationary systems we report,in this article,on extension of some of those results to certaininfinite dimensional systems;in particular a class of PDE systems of elliptic type.These systems arestudied in the now familiar framework developed by J.L.Lions and E.Magenes,specialized to asubclass of such systems important in a variety of applications.As an extended example this paperstudies an optimal redistribution problem in a groundwater flow system governed by Darcy's equation,presenting both analytic and computational work related to such problems.展开更多
The authors study the continuity estimate of the solutions of almost harmonic maps with the perturbation term f in a critical integrability class(Zygmund class)L^(n/2)log^(q) L,n is the dimension with n≥3.They prove ...The authors study the continuity estimate of the solutions of almost harmonic maps with the perturbation term f in a critical integrability class(Zygmund class)L^(n/2)log^(q) L,n is the dimension with n≥3.They prove that when q>n/2 the solution must be continuous and they can get continuity modulus estimates.As a byproduct of their method,they also study boundary continuity for the almost harmonic maps in high dimension.展开更多
In this paper we study the L_(p) dual Minkowski problem for the case p<0<q.We prove for any positive smooth function f on S^(1),there exists an F:R^(+)→R^(-),such that if F(q)<p<0 or 0<q<-F(-p)then ...In this paper we study the L_(p) dual Minkowski problem for the case p<0<q.We prove for any positive smooth function f on S^(1),there exists an F:R^(+)→R^(-),such that if F(q)<p<0 or 0<q<-F(-p)then there is a smooth and strictly convex body solving the planar L_(p) dual Minkowski problem.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.12171148 and 11771138)the Construct Program of the Key Discipline in Hunan Province.Wei Liu was supported by National Natural Science Foundation of China(Grant Nos.12101252 and 11971007)+2 种基金supported by National Natural Science Foundation of China(Grant No.11901185)National Key Research and Development Program of China(Grant No.2021YFA1001300)the Fundamental Research Funds for the Central Universities(Grant No.531118010207).
文摘The local minimax method(LMM)proposed by Li and Zhou(2001,2002)is an efficient method to solve nonlinear elliptic partial differential equations(PDEs)with certain variational structures for multiple solutions.The steepest descent direction and the Armijo-type step-size search rules are adopted in Li and Zhou(2002)and play a significant role in the performance and convergence analysis of traditional LMMs.In this paper,a new algorithm framework of the LMMs is established based on general descent directions and two normalized(strong)Wolfe-Powell-type step-size search rules.The corresponding algorithm framework,named the normalized Wolfe-Powell-type LMM(NWP-LMM),is introduced with its feasibility and global convergence rigorously justified for general descent directions.As a special case,the global convergence of the NWP-LMM combined with the preconditioned steepest descent(PSD)directions is also verified.Consequently,it extends the framework of traditional LMMs.In addition,conjugate-gradient-type(CG-type)descent directions are utilized to speed up the NWP-LMM.Finally,extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared with different algorithms in the LMM’s family to indicate the effectiveness and robustness of our algorithms.In practice,the NWP-LMM combined with the CG-type direction performs much better than its known LMM companions.
文摘First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking purposes, inexact proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes applied to infinite-dimensional elliptic optimal control problems and to validate the theoretical estimates.
基金Supported by the China National Education Committee Science Foundation
文摘This paper considers the existence problem of an elliptic equation, which is equivalent to solving the so called prescribing conformal Gaussian curvature problem on the hyperbolic disc H^2. An existence result is proved. In particular, K（x） is allowed to be unbounded above.
文摘This paper considers a semilinear elliptic equation on a n-dimensional complete noncompact R.iemannian manifold, which is a generalization of the well known Yamabe equation. An existence result is proved.
文摘Following work carried out earlier on linear-quadratic optimal control for linear finitedimensional stationary systems we report,in this article,on extension of some of those results to certaininfinite dimensional systems;in particular a class of PDE systems of elliptic type.These systems arestudied in the now familiar framework developed by J.L.Lions and E.Magenes,specialized to asubclass of such systems important in a variety of applications.As an extended example this paperstudies an optimal redistribution problem in a groundwater flow system governed by Darcy's equation,presenting both analytic and computational work related to such problems.
文摘The authors study the continuity estimate of the solutions of almost harmonic maps with the perturbation term f in a critical integrability class(Zygmund class)L^(n/2)log^(q) L,n is the dimension with n≥3.They prove that when q>n/2 the solution must be continuous and they can get continuity modulus estimates.As a byproduct of their method,they also study boundary continuity for the almost harmonic maps in high dimension.
基金supported by National Natural Science Foundation of China(Grant Nos.11971424 and 11571304)。
文摘In this paper we study the L_(p) dual Minkowski problem for the case p<0<q.We prove for any positive smooth function f on S^(1),there exists an F:R^(+)→R^(-),such that if F(q)<p<0 or 0<q<-F(-p)then there is a smooth and strictly convex body solving the planar L_(p) dual Minkowski problem.
