This paper introduces a domain decomposition preconditioner for elliptic equations with rough coefficients. The coarse space of the domain decomposition method is constructed via the so-called rough polyharmonic splin...This paper introduces a domain decomposition preconditioner for elliptic equations with rough coefficients. The coarse space of the domain decomposition method is constructed via the so-called rough polyharmonic splines （RPS for short）. As an approximation space of the eUiptic problem, RPS is known to recover the quasi-optimal convergence rate and attain the quasi-optimal localization property. The authors lay out the formulation of the RPS based domain decomposition preconditioner, and numerically verify the performance boost of this method through several examples.展开更多
The remainders and the convergence of cardinal polyharmonic spline interpolation are studied, and the asymptotic behavior of the best approximation by polyharmonic spline and the average K-width of some class of smoot...The remainders and the convergence of cardinal polyharmonic spline interpolation are studied, and the asymptotic behavior of the best approximation by polyharmonic spline and the average K-width of some class of smooth functions defined on the Euclidean space Rn are determined.展开更多
This paper concerns the convex optimal control problem governed by multiscale elliptic equations with arbitrarily rough L∞coefficients,which has not only complex coupling between nonseparable scales and nonlinearity,...This paper concerns the convex optimal control problem governed by multiscale elliptic equations with arbitrarily rough L∞coefficients,which has not only complex coupling between nonseparable scales and nonlinearity,but also important applications in composite materials and geophysics.We use one of the recently developed numerical homogenization techniques,the so-called Rough Polyharmonic Splines(RPS)and its generalization(GRPS)for the efficient resolution of the elliptic operator on the coarse scale.Those methods have optimal convergence rate which do not rely on the regularity of the coefficients nor the concepts of scale-separation or periodicity.As the iterative solution of the nonlinearly coupled OCP-OPT formulation for the optimal control problem requires solving the corresponding(state and co-state)multiscale elliptic equations many times with different right hand sides,numerical homogenization approach only requires one-time pre-computation on the fine scale and the following iterations can be done with computational cost proportional to coarse degrees of freedom.Numerical experiments are presented to validate the theoretical analysis.展开更多
In this paper,we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator ∆^(2) ± λ^(2).Similar to the derivation of fundamental solutions,it is non-tr...In this paper,we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator ∆^(2) ± λ^(2).Similar to the derivation of fundamental solutions,it is non-trivial to derive particular solutions for higher order differential operators.In this paper,we provide a simple algebraic factorization approach to derive particular solutions for these types of differential operators in 2D and 3D.The main focus of this paper is its simplicity in the sense that minimal mathematical background is required for numerically solving higher order partial differential equations such as thin plate vibration.Three numerical examples in both 2D and 3D are given to validate particular solutions we derived.展开更多
基金supported by the National Natural Science Foundation of China(No.11471214)the One Thousand Plan of China for young scientists
文摘This paper introduces a domain decomposition preconditioner for elliptic equations with rough coefficients. The coarse space of the domain decomposition method is constructed via the so-called rough polyharmonic splines （RPS for short）. As an approximation space of the eUiptic problem, RPS is known to recover the quasi-optimal convergence rate and attain the quasi-optimal localization property. The authors lay out the formulation of the RPS based domain decomposition preconditioner, and numerically verify the performance boost of this method through several examples.
文摘The remainders and the convergence of cardinal polyharmonic spline interpolation are studied, and the asymptotic behavior of the best approximation by polyharmonic spline and the average K-width of some class of smooth functions defined on the Euclidean space Rn are determined.
基金supported by the State Key Program of National Natural Science Foundation of China(No.11931003)National Natural Science Foundation of China(Nos.41974133,11871339,11861131004).
文摘This paper concerns the convex optimal control problem governed by multiscale elliptic equations with arbitrarily rough L∞coefficients,which has not only complex coupling between nonseparable scales and nonlinearity,but also important applications in composite materials and geophysics.We use one of the recently developed numerical homogenization techniques,the so-called Rough Polyharmonic Splines(RPS)and its generalization(GRPS)for the efficient resolution of the elliptic operator on the coarse scale.Those methods have optimal convergence rate which do not rely on the regularity of the coefficients nor the concepts of scale-separation or periodicity.As the iterative solution of the nonlinearly coupled OCP-OPT formulation for the optimal control problem requires solving the corresponding(state and co-state)multiscale elliptic equations many times with different right hand sides,numerical homogenization approach only requires one-time pre-computation on the fine scale and the following iterations can be done with computational cost proportional to coarse degrees of freedom.Numerical experiments are presented to validate the theoretical analysis.
文摘In this paper,we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator ∆^(2) ± λ^(2).Similar to the derivation of fundamental solutions,it is non-trivial to derive particular solutions for higher order differential operators.In this paper,we provide a simple algebraic factorization approach to derive particular solutions for these types of differential operators in 2D and 3D.The main focus of this paper is its simplicity in the sense that minimal mathematical background is required for numerically solving higher order partial differential equations such as thin plate vibration.Three numerical examples in both 2D and 3D are given to validate particular solutions we derived.
