Solution of quantum eigenvalue problems by means of algebraic consistency conditions

The coarse-mesh finite difference (CMFD) scheme is a very effective nonlinear diffusion acceleration method for neutron transport calculations. CMFD can become unstable and fail to converge when the computational cell optical thickness is relatively large in k-eigenvalue problems or diffusive fixed-source problems. Some variants and fixups have been developed to enhance the stability of CMFD, including the partial current-based CMFD (pCMFD), optimally diffusive CMFD (odCMFD), and linear prolongation-based CMFD (lpCMFD). Linearized Fourier analysis has proven to be a very reliable and accurate tool to investigate the convergence rate and stability of such coupled high-order transport/low-order diffusion iterative schemes. It is shown in this paper that the use of different transport solvers in Fourier analysis may have some potential implications on the development of stabilizing techniques, which is exemplified by the odCMFD scheme. A modification to the artificial diffusion coefficients of odCMFD is proposed to improve its stability. In addition, two explicit expressions are presented to calculate local optimal successive overrelaxation (SOR) factors for lpCMFD to further enhance its acceleration performance for fixed-source problems and k-eigenvalue problems, respectively.

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A reduction method for a class of elliptic eigenvalue problems

Beiträge zur Numerischen Mathematik Band 4 ◽

10.1515/9783486992793-004 ◽

1975 ◽

pp. 33-42

Keyword(s):

Reduction Method ◽

Eigenvalue Problems ◽

Elliptic Eigenvalue Problems

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Bootstrap bounds on closed Einstein manifolds

Journal of High Energy Physics ◽

10.1007/jhep10(2020)069 ◽

2020 ◽

Vol 2020 (10) ◽

Author(s):

James Bonifacio ◽

Kurt Hinterbichler

Keyword(s):

Compact Riemannian Manifold ◽

Tree Level ◽

Conformal Field Theories ◽

Einstein Manifolds ◽

Consistency Conditions ◽

Conformal Field ◽

Geometric Data ◽

Conformal Bootstrap ◽

Laplacian Operators ◽

Kaluza Klein

Abstract A compact Riemannian manifold is associated with geometric data given by the eigenvalues of various Laplacian operators on the manifold and the triple overlap integrals of the corresponding eigenmodes. This geometric data must satisfy certain consistency conditions that follow from associativity and the completeness of eigenmodes. We show that it is possible to obtain nontrivial bounds on the geometric data of closed Einstein manifolds by using semidefinite programming to study these consistency conditions, in analogy to the conformal bootstrap bounds on conformal field theories. These bootstrap bounds translate to constraints on the tree-level masses and cubic couplings of Kaluza-Klein modes in theories with compact extra dimensions. We show that in some cases the bounds are saturated by known manifolds.

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Entropy numbers of embedding maps between Besov spaces with an application to eigenvalue problems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500015341 ◽

1981 ◽

Vol 90 (1-2) ◽

pp. 63-70 ◽

Cited By ~ 5

Author(s):

Bernd Carl

Keyword(s):

Banach Space ◽

Asymptotic Behaviour ◽

Besov Spaces ◽

Eigenvalue Problems ◽

Function Spaces ◽

Sequence Spaces ◽

Entropy Numbers ◽

Diagonal Operators

SynopsisIn this paper we determine the asymptotic behaviour of entropy numbers of embedding maps between Besov sequence spaces and Besov function spaces. The results extend those of M. Š. Birman, M. Z. Solomjak and H. Triebel originally formulated in the language of ε-entropy. It turns out that the characterization of embedding maps between Besov spaces by entropy numbers can be reduced to the characterization of certain diagonal operators by their entropy numbers.Finally, the entropy numbers are applied to the study of eigenvalues of operators acting on a Banach space which admit a factorization through embedding maps between Besov spaces.The statements of this paper are obtained by results recently proved elsewhere by the author.

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