Discovering underlying partial differential equations(PDEs)from observational data has important impllications across fields.It bridges the gap between theory and observation,enhancing our understandingof complex syst...Discovering underlying partial differential equations(PDEs)from observational data has important impllications across fields.It bridges the gap between theory and observation,enhancing our understandingof complex systems in applications.In this paper,we propose a novel approach,termed physics-informed sparse optimization(PIS),for learning surface PDEs.Our approach incorporates both L_(2) physicsinformed model loss and L1 regularization penalty terms in the loss function,enabling the identification of specific physical terms within the surface PDEs.The unknown function and the differential operators on surfaces are approximated by some extrinsic meshless methods.We provide practical demonstrations of the algorithms including linear and nonlinear systems.The numerical experiments on spheres and various other surfaces demonstrate the effectiveness of the proposed approach in simultaneously achieving precise solution prediction and identification of unknown PDEs.展开更多
This paper deals with the solvability and the convergence of a class of unsymmetric Meshless Local Petrov-Galerkin(MLPG)method with radial basis function(RBF)kernels generated trial spaces.Local weak-form testings are...This paper deals with the solvability and the convergence of a class of unsymmetric Meshless Local Petrov-Galerkin(MLPG)method with radial basis function(RBF)kernels generated trial spaces.Local weak-form testings are done with stepfunctions.It is proved that subject to sufficiently many appropriate testings,solvability of the unsymmetric RBF-MLPG resultant systems can be guaranteed.Moreover,an error analysis shows that this numerical approximation converges at the same rate as found in RBF interpolation.Numerical results(in double precision)give good agreement with the provided theory.展开更多
We analyze a least-squares asymmetric radial basis function collocation method for solving the modified Helmholtz equations.In the theoretical part,we proved the convergence of the proposed method providing that the c...We analyze a least-squares asymmetric radial basis function collocation method for solving the modified Helmholtz equations.In the theoretical part,we proved the convergence of the proposed method providing that the collocation points are sufficiently dense.For numerical verification,direct solver and a subspace selection process for the trial space(the so-called adaptive greedy algorithm)is employed,respectively,for small and large scale problems.展开更多
Pattern formations by Gierer-Meinhardt(GM)activator-inhibitor model are considered in this paper.By linear analysis,critical value of bifurcation parameter can be evaluated to ensure Turing instability.Numerical simul...Pattern formations by Gierer-Meinhardt(GM)activator-inhibitor model are considered in this paper.By linear analysis,critical value of bifurcation parameter can be evaluated to ensure Turing instability.Numerical simulations are tested by using second order semi-implicit backward difference methods for time discretization and the meshless Kansa method for spatially discretization.We numerically show the convergence of our algorithm.Pattern transitions in irregular domains are shown.We also provide various parameter settings on some irregular domains for different patterns appeared in nature.To further simulate patterns in reality,we construct different kinds of animal type domains and obtain desired patterns by applying proposed parameter settings.展开更多
Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the ...Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the pipe.The interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer layer.The Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary data.In this work,we assume that the measurements are available on the whole outer boundary on an annulus domain.By imposing reasonable assumptions,the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization method.Relationship between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given.A novel numerical algorithm is proposed and numerical examples are given.展开更多
We considered the point source identification problems for heat equations from noisy observation data taken at the minimum number of spatially fixed measurement points.We aim to identify the unknown number of sources ...We considered the point source identification problems for heat equations from noisy observation data taken at the minimum number of spatially fixed measurement points.We aim to identify the unknown number of sources and their locations along with their strengths.In our previous work,we proved that minimum measurement points needed under the noise-free setting.In this paper,we extend the proof to cover the noisy cases over a border class of source functions.We show that if the regularization parameter is chosen properly,the problem can be transformed into a poles identification problem.A reconstruction scheme is proposed on the basis of the developed theoretical results.Numerical demonstrations in 2D and 3D conclude the paper.展开更多
The Fourth International Conference on Scientific Computing and Partial Differential Equations(SCPDE)was held at Hong Kong Baptist University from December 5 to December 9,2011(http://www.math.hkbu.edu.hk/SCPDE11/).It...The Fourth International Conference on Scientific Computing and Partial Differential Equations(SCPDE)was held at Hong Kong Baptist University from December 5 to December 9,2011(http://www.math.hkbu.edu.hk/SCPDE11/).It was a sequel to similar conferences held in Hong Kong(2002,2005 and 2008).The conference aims to promote research interests in scientific computation.In SCPDE 2011,there were over 100 participants from several countries and regions participated in the conference.The Programme included sixteen plenary speakers,thirty invited talks,thirty five contributed talks.Four main themes were organized to review recent scientific developments and explore exciting new directions in mathematical modeling and computational methods;in particular,they are Numerical Analysis and Applications of Volterra Functional Equations,Stochastic Computations,Electromagnetic and Acoustic Scattering,and Numerical Partial Differential Equations.展开更多
The Third International Conference on Scientific Computing and Partial Differential Equations(SCPDE)was held from December 8 to December 12,2008 at China Hong Kong Baptist University.It was a sequel to similar confere...The Third International Conference on Scientific Computing and Partial Differential Equations(SCPDE)was held from December 8 to December 12,2008 at China Hong Kong Baptist University.It was a sequel to similar conferences held in Hong Kong region(2002 and 2005).The conference aims to promote research interests in scientific computation.In SCPDE 2008,there were 118 participants from seventeen countries and regions participated in the conference.The Programme included seventeen plenary addresses,thirty invited talks,twenty five contributed talks and seven poster presentations.展开更多
基金supported by NSFC(No.12101310)NSF of Jiangsu Province(No.BK20210315)+2 种基金the Fundamental Research Funds for the Central Universities(No.30923010912)the work of the second author was supported by the General Research Fund(GRF No.12301520,12301021,12300922)of Hong Kong Research Grant Councilthe work of the corresponding author was supported by NSFC(No.11901377)and NSFC(No.12171093).
文摘Discovering underlying partial differential equations(PDEs)from observational data has important impllications across fields.It bridges the gap between theory and observation,enhancing our understandingof complex systems in applications.In this paper,we propose a novel approach,termed physics-informed sparse optimization(PIS),for learning surface PDEs.Our approach incorporates both L_(2) physicsinformed model loss and L1 regularization penalty terms in the loss function,enabling the identification of specific physical terms within the surface PDEs.The unknown function and the differential operators on surfaces are approximated by some extrinsic meshless methods.We provide practical demonstrations of the algorithms including linear and nonlinear systems.The numerical experiments on spheres and various other surfaces demonstrate the effectiveness of the proposed approach in simultaneously achieving precise solution prediction and identification of unknown PDEs.
基金supported by the CERG Grant of the Hong Kong Research Grant Council and the FRG Grant of the Hong Kong Baptist University.
文摘This paper deals with the solvability and the convergence of a class of unsymmetric Meshless Local Petrov-Galerkin(MLPG)method with radial basis function(RBF)kernels generated trial spaces.Local weak-form testings are done with stepfunctions.It is proved that subject to sufficiently many appropriate testings,solvability of the unsymmetric RBF-MLPG resultant systems can be guaranteed.Moreover,an error analysis shows that this numerical approximation converges at the same rate as found in RBF interpolation.Numerical results(in double precision)give good agreement with the provided theory.
基金supported by CERG Grants of Hong Kong Research Grant CouncilFRG grants of Hong Kong Baptist University.
文摘We analyze a least-squares asymmetric radial basis function collocation method for solving the modified Helmholtz equations.In the theoretical part,we proved the convergence of the proposed method providing that the collocation points are sufficiently dense.For numerical verification,direct solver and a subspace selection process for the trial space(the so-called adaptive greedy algorithm)is employed,respectively,for small and large scale problems.
基金supported by a Hong Kong Research Grant Council GRF Grant,and a Hong Kong Baptist University FRG Grant.
文摘Pattern formations by Gierer-Meinhardt(GM)activator-inhibitor model are considered in this paper.By linear analysis,critical value of bifurcation parameter can be evaluated to ensure Turing instability.Numerical simulations are tested by using second order semi-implicit backward difference methods for time discretization and the meshless Kansa method for spatially discretization.We numerically show the convergence of our algorithm.Pattern transitions in irregular domains are shown.We also provide various parameter settings on some irregular domains for different patterns appeared in nature.To further simulate patterns in reality,we construct different kinds of animal type domains and obtain desired patterns by applying proposed parameter settings.
基金supported by a CERG Grant of Hong Kong Research Grant Council,a FRG grant of Hong Kong Baptist University,and was partially supported by the NSFC Project No.19971116.
文摘Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the pipe.The interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer layer.The Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary data.In this work,we assume that the measurements are available on the whole outer boundary on an annulus domain.By imposing reasonable assumptions,the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization method.Relationship between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given.A novel numerical algorithm is proposed and numerical examples are given.
文摘We considered the point source identification problems for heat equations from noisy observation data taken at the minimum number of spatially fixed measurement points.We aim to identify the unknown number of sources and their locations along with their strengths.In our previous work,we proved that minimum measurement points needed under the noise-free setting.In this paper,we extend the proof to cover the noisy cases over a border class of source functions.We show that if the regularization parameter is chosen properly,the problem can be transformed into a poles identification problem.A reconstruction scheme is proposed on the basis of the developed theoretical results.Numerical demonstrations in 2D and 3D conclude the paper.
文摘The Fourth International Conference on Scientific Computing and Partial Differential Equations(SCPDE)was held at Hong Kong Baptist University from December 5 to December 9,2011(http://www.math.hkbu.edu.hk/SCPDE11/).It was a sequel to similar conferences held in Hong Kong(2002,2005 and 2008).The conference aims to promote research interests in scientific computation.In SCPDE 2011,there were over 100 participants from several countries and regions participated in the conference.The Programme included sixteen plenary speakers,thirty invited talks,thirty five contributed talks.Four main themes were organized to review recent scientific developments and explore exciting new directions in mathematical modeling and computational methods;in particular,they are Numerical Analysis and Applications of Volterra Functional Equations,Stochastic Computations,Electromagnetic and Acoustic Scattering,and Numerical Partial Differential Equations.
文摘The Third International Conference on Scientific Computing and Partial Differential Equations(SCPDE)was held from December 8 to December 12,2008 at China Hong Kong Baptist University.It was a sequel to similar conferences held in Hong Kong region(2002 and 2005).The conference aims to promote research interests in scientific computation.In SCPDE 2008,there were 118 participants from seventeen countries and regions participated in the conference.The Programme included seventeen plenary addresses,thirty invited talks,twenty five contributed talks and seven poster presentations.
