Theoretical argumentation for so-called suitable spatial condition is conducted by the aid of homotopy framework to demonstrate that the proposed boundary condition does guarantee that the over-specification boundary ...Theoretical argumentation for so-called suitable spatial condition is conducted by the aid of homotopy framework to demonstrate that the proposed boundary condition does guarantee that the over-specification boundary condition resulting from an adjoint model on a limited-area is no longer an issue, and yet preserve its well-poseness and optimal character in the boundary setting. The ill-poseness of over-specified spatial boundary condition is in a sense, inevitable from an adjoint model since data assimilation processes have to adapt prescribed observations that used to be over-specified at the spatial boundaries of the modeling domain. In the view of pragmatic implement, the theoretical framework of our proposed condition for spatial boundaries indeed can be reduced to the hybrid formulation of nudging filter, radiation condition taking account of ambient forcing, together with Dirichlet kind of compatible boundary condition to the observations prescribed in data assimilation procedure. All of these treatments, no doubt, are very familiar to mesoscale modelers. Key words Variational data assimilation - Adjoint model - Over-specified partial boundary condition This research work is sponsored by the National Key Programme for Developing Basic Sciences (G1998040907), the Project of Natural Science Foundation of Jiangsu Province (BK99020), the President Foundation of Nanjing University (985) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.展开更多
Recent years have witnessed growing interests in solving partial differential equations by deep neural networks,especially in the high-dimensional case.Unlike classical numerical methods,such as finite difference meth...Recent years have witnessed growing interests in solving partial differential equations by deep neural networks,especially in the high-dimensional case.Unlike classical numerical methods,such as finite difference method and finite element method,the enforcement of boundary conditions in deep neural networks is highly nontrivial.One general strategy is to use the penalty method.In the work,we conduct a comparison study for elliptic problems with four different boundary conditions,i.e.,Dirichlet,Neumann,Robin,and periodic boundary conditions,using two representative methods:deep Galerkin method and deep Ritz method.In the former,the PDE residual is minimized in the least-squares sense while the corresponding variational problem is minimized in the latter.Therefore,it is reasonably expected that deep Galerkin method works better for smooth solutions while deep Ritz method works better for low-regularity solutions.However,by a number of examples,we observe that deep Ritz method can outperform deep Galerkin method with a clear dependence of dimensionality even for smooth solutions and deep Galerkin method can also outperform deep Ritz method for low-regularity solutions.Besides,in some cases,when the boundary condition can be implemented in an exact manner,we find that such a strategy not only provides a better approximate solution but also facilitates the training process.展开更多
In this paper, we extend the reliable modification of the Adomian Decom-position Method coupled to the Lesnic’s approach to solve boundary value problems and initial boundary value problems with mixed boundary condit...In this paper, we extend the reliable modification of the Adomian Decom-position Method coupled to the Lesnic’s approach to solve boundary value problems and initial boundary value problems with mixed boundary conditions for linear and nonlinear partial differential equations. The method is applied to different forms of heat and wave equations as illustrative examples to exhibit the effectiveness of the method. The method provides the solution in a rapidly convergent series with components that can be computed iteratively. The numerical results for the illustrative examples obtained show remarkable agreement with the exact solutions. We also provide some graphical representations for clear-cut comparisons between the solutions using Maple software.展开更多
We establish an extension result of existence and partial regularity for the nonzero Neumann initial-boundary value problem of the Landau-Lifshitz equation with nonpositive anisotropy constants in three or four dimens...We establish an extension result of existence and partial regularity for the nonzero Neumann initial-boundary value problem of the Landau-Lifshitz equation with nonpositive anisotropy constants in three or four dimension space. The partial regularity is proved up to the boundary and this result is an important supplement to those for the Dirichlet problem or the homogeneous Neumann problem.展开更多
When an incoming water wave is parallel to a porous breakwater, a paradoxical phenomenon exists in that by strictly following the potential flow boundary condition of normal flux continuity on the interfaces, the wate...When an incoming water wave is parallel to a porous breakwater, a paradoxical phenomenon exists in that by strictly following the potential flow boundary condition of normal flux continuity on the interfaces, the water wave permeates the wall completely, regardless of breakwater porosity. To account for this paradoxical phenomenon when solving the problem of water waves obliquely impacting on a thin porous wall, a new partial-slipping boundary condition on the thin porous wall for potential flow is proposed. Analytical results show that when the water wave is parallel to a thin porous wall （i.e., the incident angle equals to 90~）, the transmitted wave side remains quiescent, i.e., the transmitted wave side does not capture any wave energy when no viscous effect exists. This reveals that the above-mentioned paradoxical investigated in this study, which provides proper boundary information. phenomenon disappears. The viscous boundary layer effect is also conditions on a thin porous wall for viscous flows and detailed flow展开更多
We consider the questions of boundary regularity for weak solutions of second-order nonlinear elliptic systems under the natural growth condition. We obtain a general criterion for a weak solution to be regular in the...We consider the questions of boundary regularity for weak solutions of second-order nonlinear elliptic systems under the natural growth condition. We obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. The proof yields directly the optimal regularity for the solution in this neighborhood. This result is new for the situation under the natural growth conditions.展开更多
The purpose of this paper is to investigate the existence, uniqueness and dynamics of a nonlinear system of partial differential equations with nonlocal and coupled boundary conditions which is motivated by a model pr...The purpose of this paper is to investigate the existence, uniqueness and dynamics of a nonlinear system of partial differential equations with nonlocal and coupled boundary conditions which is motivated by a model problem arising from quasi-state thermoelasticity. A sufficient condition for the uniqueness of a steady-state solution is obtained. The behavior of solutions to the evolution problem and the relation between the solutions to the evolution problem and its corresponding steady-state problem are also discussed.展开更多
基金the National Key Programme for Developing Basic Sciences(G1998040907)the Project of Natural Science Foundation of Jiangsu Pr
文摘Theoretical argumentation for so-called suitable spatial condition is conducted by the aid of homotopy framework to demonstrate that the proposed boundary condition does guarantee that the over-specification boundary condition resulting from an adjoint model on a limited-area is no longer an issue, and yet preserve its well-poseness and optimal character in the boundary setting. The ill-poseness of over-specified spatial boundary condition is in a sense, inevitable from an adjoint model since data assimilation processes have to adapt prescribed observations that used to be over-specified at the spatial boundaries of the modeling domain. In the view of pragmatic implement, the theoretical framework of our proposed condition for spatial boundaries indeed can be reduced to the hybrid formulation of nudging filter, radiation condition taking account of ambient forcing, together with Dirichlet kind of compatible boundary condition to the observations prescribed in data assimilation procedure. All of these treatments, no doubt, are very familiar to mesoscale modelers. Key words Variational data assimilation - Adjoint model - Over-specified partial boundary condition This research work is sponsored by the National Key Programme for Developing Basic Sciences (G1998040907), the Project of Natural Science Foundation of Jiangsu Province (BK99020), the President Foundation of Nanjing University (985) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
基金the grants NSFC 11971021National Key R&D Program of China(No.2018YF645B0204404)NSFC 11501399(R.Du)。
文摘Recent years have witnessed growing interests in solving partial differential equations by deep neural networks,especially in the high-dimensional case.Unlike classical numerical methods,such as finite difference method and finite element method,the enforcement of boundary conditions in deep neural networks is highly nontrivial.One general strategy is to use the penalty method.In the work,we conduct a comparison study for elliptic problems with four different boundary conditions,i.e.,Dirichlet,Neumann,Robin,and periodic boundary conditions,using two representative methods:deep Galerkin method and deep Ritz method.In the former,the PDE residual is minimized in the least-squares sense while the corresponding variational problem is minimized in the latter.Therefore,it is reasonably expected that deep Galerkin method works better for smooth solutions while deep Ritz method works better for low-regularity solutions.However,by a number of examples,we observe that deep Ritz method can outperform deep Galerkin method with a clear dependence of dimensionality even for smooth solutions and deep Galerkin method can also outperform deep Ritz method for low-regularity solutions.Besides,in some cases,when the boundary condition can be implemented in an exact manner,we find that such a strategy not only provides a better approximate solution but also facilitates the training process.
文摘In this paper, we extend the reliable modification of the Adomian Decom-position Method coupled to the Lesnic’s approach to solve boundary value problems and initial boundary value problems with mixed boundary conditions for linear and nonlinear partial differential equations. The method is applied to different forms of heat and wave equations as illustrative examples to exhibit the effectiveness of the method. The method provides the solution in a rapidly convergent series with components that can be computed iteratively. The numerical results for the illustrative examples obtained show remarkable agreement with the exact solutions. We also provide some graphical representations for clear-cut comparisons between the solutions using Maple software.
基金Supported by the Science Foundation of Zhejiang Sci-Tech University(No.0905828-Y)
文摘We establish an extension result of existence and partial regularity for the nonzero Neumann initial-boundary value problem of the Landau-Lifshitz equation with nonpositive anisotropy constants in three or four dimension space. The partial regularity is proved up to the boundary and this result is an important supplement to those for the Dirichlet problem or the homogeneous Neumann problem.
基金supported by the National Science Council,(Grant No.NSC92-2611-E002-029)
文摘When an incoming water wave is parallel to a porous breakwater, a paradoxical phenomenon exists in that by strictly following the potential flow boundary condition of normal flux continuity on the interfaces, the water wave permeates the wall completely, regardless of breakwater porosity. To account for this paradoxical phenomenon when solving the problem of water waves obliquely impacting on a thin porous wall, a new partial-slipping boundary condition on the thin porous wall for potential flow is proposed. Analytical results show that when the water wave is parallel to a thin porous wall （i.e., the incident angle equals to 90~）, the transmitted wave side remains quiescent, i.e., the transmitted wave side does not capture any wave energy when no viscous effect exists. This reveals that the above-mentioned paradoxical investigated in this study, which provides proper boundary information. phenomenon disappears. The viscous boundary layer effect is also conditions on a thin porous wall for viscous flows and detailed flow
基金Supported by NSF(No. 10531020) of Chinathe Program of 985 Innovation Engineering on Information in Xiamen University (2004-2007) and NCETXMU
文摘We consider the questions of boundary regularity for weak solutions of second-order nonlinear elliptic systems under the natural growth condition. We obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. The proof yields directly the optimal regularity for the solution in this neighborhood. This result is new for the situation under the natural growth conditions.
文摘The purpose of this paper is to investigate the existence, uniqueness and dynamics of a nonlinear system of partial differential equations with nonlocal and coupled boundary conditions which is motivated by a model problem arising from quasi-state thermoelasticity. A sufficient condition for the uniqueness of a steady-state solution is obtained. The behavior of solutions to the evolution problem and the relation between the solutions to the evolution problem and its corresponding steady-state problem are also discussed.
