An efficient SQP algorithm for solving nonlinear degenerate problems is proposed in the paper. At each iteration of the algorithm, a quadratic programming subproblem, which is always feasible by introducing a slack va...An efficient SQP algorithm for solving nonlinear degenerate problems is proposed in the paper. At each iteration of the algorithm, a quadratic programming subproblem, which is always feasible by introducing a slack variable, is solved to obtain a search direction. The steplength along this direction is computed by employing the 1∞ exact penalty function through Armijo-type line search scheme. The algorithm is proved to be convergent globally under mild conditions.展开更多
In this paper, the evaluation of discretely sampled Asian options was considered by numerically solving the associated partial differential equations with the Legendre spectral method. Double average options were disc...In this paper, the evaluation of discretely sampled Asian options was considered by numerically solving the associated partial differential equations with the Legendre spectral method. Double average options were discussed as examples. The problem is a parabolic one on a finite domain whose equation degenerates into ordinary differential equations on the boundaries. A fully discrete scheme was established by using the Legendre spectral method in space and the Crank-Nicolson finite difference scheme in time. The stability and convergence of the scheme were analyzed. Numerical results show that the method can keep the spectral accuracy in space for such degenerate problems.展开更多
The authors consider the existence and regularity of the oblique derivative problem:where P is a second order elliptic differential operator on Rn,Ωis a bounded domain in Rn and is a unit vector field on the boundary...The authors consider the existence and regularity of the oblique derivative problem:where P is a second order elliptic differential operator on Rn,Ωis a bounded domain in Rn and is a unit vector field on the boundary of Ω(which may be tangential to the boundary).All above are assumed with limited smoothness. The authors show that solution u has an elliptic gain from f in Holder spaces(Theorem 1.1). The authors obtain LP regualrity of solution in Theorem 1.3, which generalizes the results in [7] to the limited smooth case. Some of the application nonlinear problems are also discussed.展开更多
This is a continuation of the previous paper [6]. The authors prove Holder and Lp regulaxity of operators collstructed from the oblique derivaive problem in [6] by establishing estimates of pseudodifferential operator...This is a continuation of the previous paper [6]. The authors prove Holder and Lp regulaxity of operators collstructed from the oblique derivaive problem in [6] by establishing estimates of pseudodifferential operators with parameters.展开更多
In this work we present a family of relaxation schemes for nonlinear convection diffusion problems,which can tackle also the cases of degenerate diffusion and of convection dominated regimes.The schemes proposed can a...In this work we present a family of relaxation schemes for nonlinear convection diffusion problems,which can tackle also the cases of degenerate diffusion and of convection dominated regimes.The schemes proposed can achieve any order of accuracy,give non-oscillatory solutions even in the presence of singularities and their structure depends only weakly on the particular PDE being integrated.One and two dimensional results are shown,and a nonlinear stability estimate is given.展开更多
We investigate the impact of a high-degree vertex in Turán problems for degenerate hypergraphs(including graphs).We say an r-graph F is bounded if there exist constantsα,β>0 such that for large n,every n-ver...We investigate the impact of a high-degree vertex in Turán problems for degenerate hypergraphs(including graphs).We say an r-graph F is bounded if there exist constantsα,β>0 such that for large n,every n-vertex F-free r-graph with a vertex of degree at least\(\alpha\left({\matrix{{n-1}\cr{r-1}}}\right)\)has fewer than(1−β)·ex(n,F)edges.The boundedness property is crucial for recent works that aim to extend the classical Hajnal–Szemerédi Theorem(Toward a density Corrádi–Hajnal theorem for degenerate hypergraphs.J.Combin.Theory Ser.B,172,221–262(2025))and the anti-Ramsey theorems of Erdős–Simonovits–Sós(Tight bounds for rainbow partial F-tiling in edge-colored complete hypergraphs.J.Graph Theory,110(4),457–467(2025)).We show that many well-studied degenerate hypergraphs,such as all even cycles,most complete bipartite graphs,and the expansion of most complete bipartite graphs,are bounded.In addition,to prove the boundedness of the expansion of complete bipartite graphs,we introduce and solve a Zarankiewicz-type problem for 3-graphs,strengthening a theorem by Kostochka–Mubayi–Verstraëte(Turán problems and shadows III:expansions of graphs.SIAM J.Discrete Math.,29(2),868–876(2015)).展开更多
In this paper, we review some results on the spectral methods. We first consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems, including degenerated and singular different...In this paper, we review some results on the spectral methods. We first consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems, including degenerated and singular differential equations. Then we present the generalized Jacobi quasi-orthogonal approximation and its applica- tions to the spectral element methods for high order problems with mixed inhomogeneous boundary conditions. We also discuss the related spectral methods for non-rectangular domains and the irrational spectral methods for unbounded domains. Next, we consider the Hermite spectral method and the generalized Hermite spec- tral method with their applications. Finally, we consider the Laguerre spectral method and the generalized Laguerre spectral method for many problems defined on unbounded domains. We also present the generalized Laguerre quasi-orthogonal approximation and its applications to certain problems of non-standard type and exterior problems.展开更多
Partition property plays a central role in domain decomposition methods. Existing theory essentially assumes certain ellipticity. We prove the partition property for prod lems without ellipticity which are of practica...Partition property plays a central role in domain decomposition methods. Existing theory essentially assumes certain ellipticity. We prove the partition property for prod lems without ellipticity which are of practical importance. Example applications include implicit schemes applied to degenerate parabolic partial differential equations arising from superconductors, superfluids and liquid crystals. With this partition property, Schwarz algorithms can be applied to general non-elliptic problems with an h-independent optimal convergence rate. Application to the time-dependent Ginzburg-Landau model of superconductivity is illustrated and numerical results are presented.展开更多
基金Supported by the National Natural Science Foundation of China(No.10671060)the Specialized Research Found for the Doctoral Program of Higher Education(No.20030532006)
文摘An efficient SQP algorithm for solving nonlinear degenerate problems is proposed in the paper. At each iteration of the algorithm, a quadratic programming subproblem, which is always feasible by introducing a slack variable, is solved to obtain a search direction. The steplength along this direction is computed by employing the 1∞ exact penalty function through Armijo-type line search scheme. The algorithm is proved to be convergent globally under mild conditions.
文摘In this paper, the evaluation of discretely sampled Asian options was considered by numerically solving the associated partial differential equations with the Legendre spectral method. Double average options were discussed as examples. The problem is a parabolic one on a finite domain whose equation degenerates into ordinary differential equations on the boundaries. A fully discrete scheme was established by using the Legendre spectral method in space and the Crank-Nicolson finite difference scheme in time. The stability and convergence of the scheme were analyzed. Numerical results show that the method can keep the spectral accuracy in space for such degenerate problems.
文摘The authors consider the existence and regularity of the oblique derivative problem:where P is a second order elliptic differential operator on Rn,Ωis a bounded domain in Rn and is a unit vector field on the boundary of Ω(which may be tangential to the boundary).All above are assumed with limited smoothness. The authors show that solution u has an elliptic gain from f in Holder spaces(Theorem 1.1). The authors obtain LP regualrity of solution in Theorem 1.3, which generalizes the results in [7] to the limited smooth case. Some of the application nonlinear problems are also discussed.
文摘This is a continuation of the previous paper [6]. The authors prove Holder and Lp regulaxity of operators collstructed from the oblique derivaive problem in [6] by establishing estimates of pseudodifferential operators with parameters.
基金This work was supported by MIUR/PRIN2005 project“Modellistica numerica per il cal-colo scientifico ed applicazioni avanzate”.
文摘In this work we present a family of relaxation schemes for nonlinear convection diffusion problems,which can tackle also the cases of degenerate diffusion and of convection dominated regimes.The schemes proposed can achieve any order of accuracy,give non-oscillatory solutions even in the presence of singularities and their structure depends only weakly on the particular PDE being integrated.One and two dimensional results are shown,and a nonlinear stability estimate is given.
基金Supported by the National Key R&D Program of China(Grant No.2023YFA1010202)supported by the Central Guidance on Local Science and Technology Development Fund of Fujian Province(Grant No.2023L3003)+1 种基金supported by the National Natural Science Foundation of China(Grant Nos.12501487,12571367)supported by the Excellent Young Talents Program(Overseas)of the National Natural Science Foundation of China and by the ERC Advanced Grant(Grant No.101020255)。
文摘We investigate the impact of a high-degree vertex in Turán problems for degenerate hypergraphs(including graphs).We say an r-graph F is bounded if there exist constantsα,β>0 such that for large n,every n-vertex F-free r-graph with a vertex of degree at least\(\alpha\left({\matrix{{n-1}\cr{r-1}}}\right)\)has fewer than(1−β)·ex(n,F)edges.The boundedness property is crucial for recent works that aim to extend the classical Hajnal–Szemerédi Theorem(Toward a density Corrádi–Hajnal theorem for degenerate hypergraphs.J.Combin.Theory Ser.B,172,221–262(2025))and the anti-Ramsey theorems of Erdős–Simonovits–Sós(Tight bounds for rainbow partial F-tiling in edge-colored complete hypergraphs.J.Graph Theory,110(4),457–467(2025)).We show that many well-studied degenerate hypergraphs,such as all even cycles,most complete bipartite graphs,and the expansion of most complete bipartite graphs,are bounded.In addition,to prove the boundedness of the expansion of complete bipartite graphs,we introduce and solve a Zarankiewicz-type problem for 3-graphs,strengthening a theorem by Kostochka–Mubayi–Verstraëte(Turán problems and shadows III:expansions of graphs.SIAM J.Discrete Math.,29(2),868–876(2015)).
基金supported by National Natural Science Foundation of China(Grant No.11171227)Fund for Doctoral Authority of China(Grant No.20123127110001)+1 种基金Fund for E-institute of Shanghai Universities(Grant No.E03004)Leading Academic Discipline Project of Shanghai Municipal Education Commission(Grant No.J50101)
文摘In this paper, we review some results on the spectral methods. We first consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems, including degenerated and singular differential equations. Then we present the generalized Jacobi quasi-orthogonal approximation and its applica- tions to the spectral element methods for high order problems with mixed inhomogeneous boundary conditions. We also discuss the related spectral methods for non-rectangular domains and the irrational spectral methods for unbounded domains. Next, we consider the Hermite spectral method and the generalized Hermite spec- tral method with their applications. Finally, we consider the Laguerre spectral method and the generalized Laguerre spectral method for many problems defined on unbounded domains. We also present the generalized Laguerre quasi-orthogonal approximation and its applications to certain problems of non-standard type and exterior problems.
基金This work was supported in part by Hong Kong RGC DAG93/94 SC10, Competitive Earmarked ResearchGrant HKUST593/94E and the speci
文摘Partition property plays a central role in domain decomposition methods. Existing theory essentially assumes certain ellipticity. We prove the partition property for prod lems without ellipticity which are of practical importance. Example applications include implicit schemes applied to degenerate parabolic partial differential equations arising from superconductors, superfluids and liquid crystals. With this partition property, Schwarz algorithms can be applied to general non-elliptic problems with an h-independent optimal convergence rate. Application to the time-dependent Ginzburg-Landau model of superconductivity is illustrated and numerical results are presented.
