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Petr N. Vabishchevich - One of the best experts on this subject based on the ideXlab platform.
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approximation of a fractional power of an Elliptic Operator
Numerical Linear Algebra With Applications, 2020Co-Authors: Petr N. VabishchevichAbstract:Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an Elliptic Operator. In a number of works, approximations of such a nonlocal Operator are constructed on the basis of an integral representation with a singular integrand. In the present paper, new and more convenient integral representations are proposed for Operators with fractional powers. Approximations are based on the classical quadrature formulas. The results of numerical experiments on the accuracy of quadrature formulas are presented. The proposed approximations are used for numerical solving a model two-dimensional boundary value problem for fractional diffusion.
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identification of a time dependent right hand side of an unsteady equation with a fractional power of an Elliptic Operator
International Conference on Large-Scale Scientific Computing, 2019Co-Authors: Petr N. VabishchevichAbstract:An inverse problem of identifying the right-hand side is considered for an unsteady equation with a fractional power of the Elliptic Operator. We consider the case when the time-dependent right-hand side is unknown. The redefinition (additional information) is associated with the known solution at an internal point (points) of the computational domain. The computational algorithm is based on a special decomposition of the solution of the unsteady problem during a transition from the previous time level to the next one. The related auxiliary problems are direct boundary value problems for stationary equations with fractional powers of Elliptic Operators. Some features of the proposed computational algorithm are demonstrated by the results of numerical experiments for a model 2D inverse problem.
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identification of the right hand side of an equation with a fractional power of an Elliptic Operator
International Conference on Numerical Methods and Applications, 2018Co-Authors: Petr N. VabishchevichAbstract:An inverse problem of identifying the right-hand side of an equation with a fractional power of an Elliptic Operator by the solution is considered. The direct problem is solved via solving a Cauchy problem for a pseudo-parabolic equation. The problem of identifying the right-hand side is reduced to a retrospective problem for this pseudo-parabolic equation. An iterative method is employed to adjust the initial condition. The results of numerical experiments for a 2D inverse problem are presented.
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numerical solution of time dependent problems with a fractional power Elliptic Operator
Computational Mathematics and Mathematical Physics, 2018Co-Authors: Petr N. VabishchevichAbstract:A time-dependent problem in a bounded domain for a fractional diffusion equation is considered. The first-order evolution equation involves a fractional-power second-order Elliptic Operator with Robin boundary conditions. A finite-element spatial approximation with an additive approximation of the Operator of the problem is used. The time approximation is based on a vector scheme. The transition to a new time level is ensured by solving a sequence of standard Elliptic boundary value problems. Numerical results obtained for a two-dimensional model problem are presented.
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Numerical Solution of Time-Dependent Problems with Fractional Power Elliptic Operator
Computational Methods in Applied Mathematics, 2017Co-Authors: Petr N. VabishchevichAbstract:Abstract An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves a fractional power of an Elliptic Operator of second order. Finite element approximation in space is employed. To construct approximation in time, standard two-level schemes are used. The approximate solution at a new time-level is obtained as a solution of a discrete problem with the fractional power of the Elliptic Operator. A Padé-type approximation is constructed on the basis of special quadrature formulas for an integral representation of the fractional power Elliptic Operator using explicit schemes. A similar approach is applied in the numerical implementation of implicit schemes. The results of numerical experiments are presented for a test two-dimensional problem.
Boris N. Khoromskij - One of the best experts on this subject based on the ideXlab platform.
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range separated tensor decomposition of the discretized dirac delta and Elliptic Operator inverse
Journal of Computational Physics, 2020Co-Authors: Boris N. KhoromskijAbstract:Abstract In this paper, we introduce the Operator dependent range-separated (RS) tensor approximation of the discretized Dirac delta function (distribution) in R d . It is constructed by application of the Elliptic Operator to the RS tensor representation of the associated Green kernel discretized on the d-dimensional Cartesian grid. The proposed local-global decomposition of the Dirac delta can be applied for solving the potential equations in a non-homogeneous medium when the density in the right-hand side is given by a large sum of pointwise singular charges. As an example of applications, we describe the regularization scheme for solving the Poisson-Boltzmann equation that models the electrostatics in bio-molecules. We show how the idea of the Operator dependent RS tensor decomposition of the Dirac delta can be generalized to the closely related problem on range-separated tensor representation of the Elliptic resolvent. This approach paves the way for application of tensor numerical methods to Elliptic problems with non-regular data. Numerical tests confirm the expected localization properties of the RS tensor approximation of the Dirac delta represented on a tensor grid in 3D.
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range separated tensor representation of the discretized multidimensional dirac delta and Elliptic Operator inverse
arXiv: Numerical Analysis, 2018Co-Authors: Boris N. KhoromskijAbstract:In this paper, we introduce the Operator dependent range-separated tensor approximation of the discretized Dirac delta in $\mathbb{R}^d$. It is constructed by application of the discrete Elliptic Operator to the range-separated decomposition of the associated Green kernel discretized on the Cartesian grid in $\mathbb{R}^d$. The presented Operator dependent local-global splitting of the Dirac delta can be applied for solving the potential equations in non-homogeneous media when the density in the right-hand side is given by the large sum of pointwise singular charges. We show how the idea of the Operator dependent RS splitting of the Dirac delta can be extended to the closely related problem on the range separated tensor representation of the Elliptic resolvent. The numerical tests confirm the expected localization properties of the obtained Operator dependent approximation of the Dirac delta represented on a tensor grid. As an example of application, we consider the regularization scheme for solving the Poisson-Boltzmann equation for modeling the electrostatics in bio-molecules.
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tensor approach to optimal control problems with fractional d dimensional Elliptic Operator in constraints
arXiv: Numerical Analysis, 2018Co-Authors: Gennadij Heidel, Boris N. Khoromskij, Venera Khoromskaia, Volker SchulzAbstract:We introduce the tensor numerical method for solution of the $d$-dimensional optimal control problems with fractional Laplacian type Operators in constraints discretized on large spacial grids. It is based on the rank-structured approximation of the matrix valued functions of the corresponding fractional Elliptic Operator. The functions of finite element (finite difference) Laplacian on a tensor grid are diagonalized by using the fast Fourier transform (FFT) matrix and then the low rank tensor approximation to the multi-dimensional core diagonal tensor is computed. The existence of low rank canonical approximation to the class of matrix valued functions of the fractional Laplacian is proved based on the sinc quadrature approximation method applied to the integral transform of the generating function. The equation for the control function is solved by the PCG method with the rank truncation at each iteration step where the low Kronecker rank preconditioner is precomputed by using the canonical decomposition of the core tensor for the inverse of system matrix. The right-hand side, the solution, and the governing Operator are maintained in the rank-structured tensor format. Numerical tests for the 2D and 3D control problems confirm the linear complexity scaling of the method in the univariate grid size.
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hierarchical tensor product approximation to the inverse and related Operators for high dimensional Elliptic problems
Computing, 2005Co-Authors: Ivan P. Gavrilyuk, Wolfgang Hackbusch, Boris N. KhoromskijAbstract:The class of H -matrices allows an approximate matrix arithmetic with almost linear complexity. In the present paper, we apply the H-matrix technique combined with the Kronecker tensor-product approximation (ef. [2. 20]) to represent the inverse of a discrete Elliptic Operator in a hypercube 0.1 d = Kd on the ease of a high spatial dimension d. In this data-sparse format, we also represent the Operator exponential, the fractional power of an Elliptic Operator as well as the solution Operator of the matrix Lyapunov-Sylvester equation. The complexity of our approximations can be estimated by C(dn log2 n) where N - n2 is the discrete problem size.
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Hierarchical Tensor-Product Approximation to the Inverse and Related Operators for High-Dimensional Elliptic Problems
Computing, 2005Co-Authors: Ivan P. Gavrilyuk, Wolfgang Hackbusch, Boris N. KhoromskijAbstract:The class of -matrices allows an approximate matrix arithmetic with almost linear complexity. In the present paper, we apply the -matrix technique combined with the Kronecker tensor-product approximation (cf. [2, 20]) to represent the inverse of a discrete Elliptic Operator in a hypercube (0, 1)^ d ∈ℝ^ d in the case of a high spatial dimension d . In this data-sparse format, we also represent the Operator exponential, the fractional power of an Elliptic Operator as well as the solution Operator of the matrix Lyapunov-Sylvester equation. The complexity of our approximations can be estimated by ( d n log ^ q n ), where N = n ^ d is the discrete problem size.
Hugo Ferreira - One of the best experts on this subject based on the ideXlab platform.
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Fundamental solutions for the wave Operator on static Lorentzian manifolds with timelike boundary
Letters in Mathematical Physics, 2019Co-Authors: Claudio Dappiaggi, Nicoló Drago, Hugo FerreiraAbstract:We consider the wave Operator on static, Lorentzian manifolds with timelike boundary, and we discuss the existence of advanced and retarded fundamental solutions in terms of boundary conditions. By means of spectral calculus, we prove that answering this question is equivalent to studying the self-adjoint extensions of an associated Elliptic Operator on a Riemannian manifold with boundary ( M , g ). The latter is diffeomorphic to any constant time hypersurface of the underlying background. In turn, assuming that ( M , g ) is of bounded geometry, this problem can be tackled within the framework of boundary triples. These consist of the assignment of two surjective, trace Operators from the domain of the adjoint of the Elliptic Operator onto an auxiliary Hilbert space $${\mathsf {h}}$$ h , which is the third datum of the triple. Self-adjoint extensions of the underlying Elliptic Operator are in one-to-one correspondence with self-adjoint Operators $$\Theta $$ Θ on $${\mathsf {h}}$$ h . On the one hand, we show that, for a natural choice of boundary triple, each $$\Theta $$ Θ can be interpreted as the assignment of a boundary condition for the original wave Operator. On the other hand, we prove that, for each such $$\Theta $$ Θ , there exists a unique advanced and retarded fundamental solution. In addition, we prove that these share the same structural property of the counterparts associated with the wave Operator on a globally hyperbolic spacetime.
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Fundamental solutions for the wave Operator on static Lorentzian manifolds with timelike boundary
2019Co-Authors: Dappiaggi Claudio, Drago Nicolo', Hugo FerreiraAbstract:We consider the wave Operator on static, Lorentzian manifolds with timelike boundary, and we discuss the existence of advanced and retarded fundamental solutions in terms of boundary conditions. By means of spectral calculus, we prove that answering this question is equivalent to studying the self-adjoint extensions of an associated Elliptic Operator on a Riemannian manifold with boundary (M, g). The latter is diffeomorphic to any constant time hypersurface of the underlying background. In turn, assuming that (M, g) is of bounded geometry, this problem can be tackled within the framework of boundary triples. These consist of the assignment of two surjective, trace Operators from the domain of the adjoint of the Elliptic Operator onto an auxiliary Hilbert space h , which is the third datum of the triple. Self-adjoint extensions of the underlying Elliptic Operator are in one-to-one correspondence with self-adjoint Operators Θ on h . On the one hand, we show that, for a natural choice of boundary triple, each Θ can be interpreted as the assignment of a boundary condition for the original wave Operator. On the other hand, we prove that, for each such Θ , there exists a unique advanced and retarded fundamental solution. In addition, we prove that these share the same structural property of the counterparts associated with the wave Operator on a globally hyperbolic spacetime
Philippe Tchamitchian - One of the best experts on this subject based on the ideXlab platform.
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The solution of the Kato square root problem for second order Elliptic Operators on Rn
The Annals of Mathematics, 2002Co-Authors: Pascal Auscher, Steve Hofmann, Michael T. Lacey, Alan Mcintosh, Philippe TchamitchianAbstract:We prove the Kato conjecture for Elliptic Operators on Jfin. More precisely, we establish that the domain of the square root of a uniformly complex Elliptic Operator L =-div (AV) with bounded measurable coefficients in IEtn iS the
Claudio Dappiaggi - One of the best experts on this subject based on the ideXlab platform.
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Fundamental solutions for the wave Operator on static Lorentzian manifolds with timelike boundary
Letters in Mathematical Physics, 2019Co-Authors: Claudio Dappiaggi, Nicoló Drago, Hugo FerreiraAbstract:We consider the wave Operator on static, Lorentzian manifolds with timelike boundary, and we discuss the existence of advanced and retarded fundamental solutions in terms of boundary conditions. By means of spectral calculus, we prove that answering this question is equivalent to studying the self-adjoint extensions of an associated Elliptic Operator on a Riemannian manifold with boundary ( M , g ). The latter is diffeomorphic to any constant time hypersurface of the underlying background. In turn, assuming that ( M , g ) is of bounded geometry, this problem can be tackled within the framework of boundary triples. These consist of the assignment of two surjective, trace Operators from the domain of the adjoint of the Elliptic Operator onto an auxiliary Hilbert space $${\mathsf {h}}$$ h , which is the third datum of the triple. Self-adjoint extensions of the underlying Elliptic Operator are in one-to-one correspondence with self-adjoint Operators $$\Theta $$ Θ on $${\mathsf {h}}$$ h . On the one hand, we show that, for a natural choice of boundary triple, each $$\Theta $$ Θ can be interpreted as the assignment of a boundary condition for the original wave Operator. On the other hand, we prove that, for each such $$\Theta $$ Θ , there exists a unique advanced and retarded fundamental solution. In addition, we prove that these share the same structural property of the counterparts associated with the wave Operator on a globally hyperbolic spacetime.