We introduce a fast solver for the phase field crystal(PFC)and functionalized Cahn-Hilliard(FCH)equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov’s accelera...We introduce a fast solver for the phase field crystal(PFC)and functionalized Cahn-Hilliard(FCH)equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov’s accelerated gradient descent(PAGD)method.We discretize these problems with a Fourier collocation method in space,and employ various second-order schemes in time.We observe a significant speedup with this solver when compared to the preconditioned gradient descent(PGD)method.With the PAGD solver,fully implicit,second-order-in-time schemes are not only feasible to solve the PFC and FCH equations,but also do so more efficiently than some semi-implicit schemes in some cases where accuracy issues are taken into account.Benchmark computations of four different schemes for the PFC and FCH equations are conducted and the results indicate that,for the FCH experiments,the fully implicit schemes(midpoint rule and BDF2 equipped with the PAGD as a nonlinear time marching solver)perform better than their IMEX versions in terms of computational cost needed to achieve a certain precision.For the PFC,the results are not as conclusive as in the FCH experiments,which,we believe,is due to the fact that the nonlinearity in the PFC is milder nature compared to the FCH equation.We also discuss some practical matters in applying the PAGD.We introduce an averaged Newton preconditioner and a sweeping-friction strategy as heuristic ways to choose good preconditioner parameters.The sweeping-friction strategy exhibits almost as good a performance as the case of the best manually tuned parameters.展开更多
The image reconstruction of electrical impedance tomography(EIT)is a nonlinear and ill-posed inverse problem and the imaging results are easily affected by measurement noise,which needs to be solved by using regulariz...The image reconstruction of electrical impedance tomography(EIT)is a nonlinear and ill-posed inverse problem and the imaging results are easily affected by measurement noise,which needs to be solved by using regularization methods.The iterative regularization method has become a focus of the research due to its ease of implementation.To deal with the ill-posed and ill-conditional problems in image reconstruction,the inexact Newton-Landweber iterative method is considered and the Nesterov’s acceleration strategy is introduced.One Nesterov-type accelerated version of the inexact Newton-Landweber iteration is presented to determine the conductivity distributions inside an object from electrical measurements made on the surface.In order to further optimize the acceleration,both the steepest descent step-length and the minimal error step-length are exploited during the iterative image reconstruction process.Landweber iteration and its accelerated version are also implemented for comparison.All algorithms are terminated by the discrepancy principle.Finally,the performance is tested by reporting numerical simulations to verify the remarkable acceleration efficiency of the proposed method.展开更多
In this paper,the author is concerned with the problem of achieving Nash equilibrium in noncooperative games over networks.The author proposes two types of distributed projected gradient dynamics with accelerated conv...In this paper,the author is concerned with the problem of achieving Nash equilibrium in noncooperative games over networks.The author proposes two types of distributed projected gradient dynamics with accelerated convergence rates.The first type is a variant of the commonly-known consensus-based gradient dynamics,where the consensual terms for determining the actions of each player are discarded to accelerate the learning process.The second type is formulated by introducing the Nesterov's accelerated method into the distributed projected gradient dynamics.The author proves convergence of both algorithms with at least linear rates under the common assumption of Lipschitz continuity and strongly monotonicity.Simulation examples are presented to validate the outperformance of the proposed algorithms over the well-known consensus-based approach and augmented game based approach.It is shown that the required number of iterations to reach the Nash equilibrium is greatly reduced in the proposed algorithms.These results could be helpful to address the issue of long convergence time in partial-information Nash equilibrium seeking algorithms.展开更多
基金NSF grants DMS-1720213,DMS-1719854,and DMS-2012634NSF grants DMS-1720213 and DMS-2111228.The work of S.M.Wise was partially supported by DMS-1719854 and DMS-2012634.
文摘We introduce a fast solver for the phase field crystal(PFC)and functionalized Cahn-Hilliard(FCH)equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov’s accelerated gradient descent(PAGD)method.We discretize these problems with a Fourier collocation method in space,and employ various second-order schemes in time.We observe a significant speedup with this solver when compared to the preconditioned gradient descent(PGD)method.With the PAGD solver,fully implicit,second-order-in-time schemes are not only feasible to solve the PFC and FCH equations,but also do so more efficiently than some semi-implicit schemes in some cases where accuracy issues are taken into account.Benchmark computations of four different schemes for the PFC and FCH equations are conducted and the results indicate that,for the FCH experiments,the fully implicit schemes(midpoint rule and BDF2 equipped with the PAGD as a nonlinear time marching solver)perform better than their IMEX versions in terms of computational cost needed to achieve a certain precision.For the PFC,the results are not as conclusive as in the FCH experiments,which,we believe,is due to the fact that the nonlinearity in the PFC is milder nature compared to the FCH equation.We also discuss some practical matters in applying the PAGD.We introduce an averaged Newton preconditioner and a sweeping-friction strategy as heuristic ways to choose good preconditioner parameters.The sweeping-friction strategy exhibits almost as good a performance as the case of the best manually tuned parameters.
基金National Natural Science Foundation of China(12101204,12261021)Heilongjiang Provincial Natural Science Foundation of China(LH2023A018)Modern Numerical Method Course for Research Program on Teaching Reform of Degree and Postgraduate Education of Heilongjiang University(2024)。
文摘The image reconstruction of electrical impedance tomography(EIT)is a nonlinear and ill-posed inverse problem and the imaging results are easily affected by measurement noise,which needs to be solved by using regularization methods.The iterative regularization method has become a focus of the research due to its ease of implementation.To deal with the ill-posed and ill-conditional problems in image reconstruction,the inexact Newton-Landweber iterative method is considered and the Nesterov’s acceleration strategy is introduced.One Nesterov-type accelerated version of the inexact Newton-Landweber iteration is presented to determine the conductivity distributions inside an object from electrical measurements made on the surface.In order to further optimize the acceleration,both the steepest descent step-length and the minimal error step-length are exploited during the iterative image reconstruction process.Landweber iteration and its accelerated version are also implemented for comparison.All algorithms are terminated by the discrepancy principle.Finally,the performance is tested by reporting numerical simulations to verify the remarkable acceleration efficiency of the proposed method.
基金supported by the National Natural Science Foundation of China under Grant No.T2322023Hunan Provincial Natural Science Foundation of China under Grant No.2022JJ20018。
文摘In this paper,the author is concerned with the problem of achieving Nash equilibrium in noncooperative games over networks.The author proposes two types of distributed projected gradient dynamics with accelerated convergence rates.The first type is a variant of the commonly-known consensus-based gradient dynamics,where the consensual terms for determining the actions of each player are discarded to accelerate the learning process.The second type is formulated by introducing the Nesterov's accelerated method into the distributed projected gradient dynamics.The author proves convergence of both algorithms with at least linear rates under the common assumption of Lipschitz continuity and strongly monotonicity.Simulation examples are presented to validate the outperformance of the proposed algorithms over the well-known consensus-based approach and augmented game based approach.It is shown that the required number of iterations to reach the Nash equilibrium is greatly reduced in the proposed algorithms.These results could be helpful to address the issue of long convergence time in partial-information Nash equilibrium seeking algorithms.
