scholarly journals Numerical algorithms without saturation for the Schroedinger equation of hydrogen atom

2018 ◽  
pp. 215-218
Author(s):  
С.Д. Алгазин
Keyword(s):  
Eigenvalue Problem ◽  
Unit Sphere ◽  
Separation Of Variables ◽  
Numerical Algorithms ◽  
Numerical Results ◽  
Mathematical Apparatus ◽  
Entire Space ◽  
Lyman Alpha ◽  
The Laplace Operator ◽  
Reduced Problem

Математически проблема сводится к задаче на собственные значения для оператора Лапласа во всем пространстве с кулоновским потенциалом. Для численного решения этой задачи применяется новый математический аппарат, разработанный автором. Инверсией относительно единичной сферы задача сводится к проблеме собственных значений в проколотом в центре единичном шаре. Граничное условие в бесконечности (нулевое) переходит в центр шара. В шаре можно исключить периодическую переменную $\varphi$ и построить дискретизацию, наследующую свойство разделения переменных дифференциального оператора ($h$-матрица). По $\varphi$ выбиралось 11 точек. Клетки $\Lambda_0$, $\Lambda_1$, $\Lambda_2$, $\Lambda_3$, $\Lambda_4$ и $\Lambda_5$ в $h$-матрице соответствуют линиям Lyman, Balmer, Paschen, Brackett, Pfund и Humphreys. Из рассмотрения, представленных расчетов видим, что $\alpha$-линия Lyman определена с точностью $5.43\%$. Таким образом, совпадение результатов расчетов с теоретическими значениями удовлетворительное. Mathematically, the problem under consideration is reduced to the eigenvalue problem for the Laplace operator in the entire space with the Coulomb potential. The new mathematical apparatus developed by the author is applied to the numerical solution of the reduced problem. This problem is reduced to the eigenvalue problem in the unit ball punctured at the center after inversion with respect to the unit sphere. The null boundary condition at infinity is transformed to the condition at the center of the unit sphere. In the sphere it is possible to split off the periodic variable $\varphi$ and to construct the discretization inheriting the property of the separation of variables of the differential operator (the $h$-matrix). Eleven points is chosen based on the values of $\varphi$. The blocks $\Lambda_0$, $\Lambda_1$, $\Lambda_2$, $\Lambda_3$, $\Lambda_4$, and $\Lambda_5$ of the $h$-matrix correspond to the Lyman, Balmer, Paschen, Brackett, Pfund, and Humphreys lines. From the obtained numerical results, it follows that the Lyman-alpha line is determined with the accuracy equal to 5.43\%. Thus, the coincidence of the numerical results with the theoretical values is satisfactory.

Preface to the special issue, CMAM 2011, no. 3.

2011 ◽  
Vol 11 (3) ◽  
pp. 272
Author(s):  
Ivan Gavrilyuk ◽  
Boris Khoromskij ◽  
Eugene Tyrtyshnikov
Keyword(s):  
Separation Of Variables ◽  
Numerical Algorithms ◽  
Multilevel Methods ◽  
High Dimensional ◽  
Special Issue ◽  
High Dimensions ◽  
Guiding Principle ◽  
Tensor Methods ◽  
Closed World ◽  
The One

Abstract In the recent years, multidimensional numerical simulations with tensor-structured data formats have been recognized as the basic concept for breaking the "curse of dimensionality". Modern applications of tensor methods include the challenging high-dimensional problems of material sciences, bio-science, stochastic modeling, signal processing, machine learning, and data mining, financial mathematics, etc. The guiding principle of the tensor methods is an approximation of multivariate functions and operators with some separation of variables to keep the computational process in a low parametric tensor-structured manifold. Tensors structures had been wildly used as models of data and discussed in the contexts of differential geometry, mechanics, algebraic geometry, data analysis etc. before tensor methods recently have penetrated into numerical computations. On the one hand, the existing tensor representation formats remained to be of a limited use in many high-dimensional problems because of lack of sufficiently reliable and fast software. On the other hand, for moderate dimensional problems (e.g. in "ab-initio" quantum chemistry) as well as for selected model problems of very high dimensions, the application of traditional canonical and Tucker formats in combination with the ideas of multilevel methods has led to the new efficient algorithms. The recent progress in tensor numerical methods is achieved with new representation formats now known as "tensor-train representations" and "hierarchical Tucker representations". Note that the formats themselves could have been picked up earlier in the literature on the modeling of quantum systems. Until 2009 they lived in a closed world of those quantum theory publications and never trespassed the territory of numerical analysis. The tremendous progress during the very recent years shows the new tensor tools in various applications and in the development of these tools and study of their approximation and algebraic properties. This special issue treats tensors as a base for efficient numerical algorithms in various modern applications and with special emphases on the new representation formats.

Improving the Performance of Numerical Algorithms for the Bethe‐Salpeter Eigenvalue Problem

PAMM ◽  
2018 ◽  
Vol 18 (1) ◽  
Author(s):  
Peter Benner ◽  
Andreas Marek ◽  
Carolin Penke
Keyword(s):  
Eigenvalue Problem ◽  
Numerical Algorithms

scholarly journals Reasoning about Measures of Unmeasurable Sets

2020 ◽  
Author(s):  
Marco Console ◽  
Matthias Hofer ◽  
Leonid Libkin
Keyword(s):  
Asymptotic Behavior ◽  
Unit Sphere ◽  
Ad Hoc ◽  
Discrete Analog ◽  
Uniform Measure ◽  
Entire Space ◽  
Vc Dimension ◽  
Know How ◽  
First Order ◽  
Specific Subset

In a variety of reasoning tasks, one estimates the likelihood of events by means of volumes of sets they define. Such sets need to be measurable, which is usually achieved by putting bounds, sometimes ad hoc, on them. We address the question how unbounded or unmeasurable sets can be measured nonetheless. Intuitively, we want to know how likely a randomly chosen point is to be in a given set, even in the absence of a uniform distribution over the entire space. To address this, we follow a recently proposed approach of taking intersection of a set with balls of increasing radius, and defining the measure by means of the asymptotic behavior of the proportion of such balls taken by the set. We show that this approach works for every set definable in first-order logic with the usual arithmetic over the reals (addition, multiplication, exponentiation, etc.), and every uniform measure over the space, of which the usual Lebesgue measure (area, volume, etc.) is an example. In fact we establish a correspondence between the good asymptotic behavior and the finiteness of the VC dimension of definable families of sets. Towards computing the measure thus defined, we show how to avoid the asymptotics and characterize it via a specific subset of the unit sphere. Using definability of this set, and known techniques for sampling from the unit sphere, we give two algorithms for estimating our measure of unbounded unmeasurable sets, with deterministic and probabilistic guarantees, the latter being more efficient. Finally we show that a discrete analog of this measure exists and is similarly well-behaved.

K-L Expansion of a Random Process in Wavelet Bases

1998 ◽  
Author(s):  
Om P. Agrawal ◽  
Shantaram S. Pai
Keyword(s):  
Eigenvalue Problem ◽  
Wavelet Transforms ◽  
Series Representation ◽  
Numerical Algorithms ◽  
Analytical Techniques ◽  
Random Processes ◽  
Random Coefficients ◽  
Engineering Systems ◽  
Wavelet Bases ◽  
Orthogonal Wavelets

Abstract Random processes play a significant role in stochastic analysis of mechanical systems, structures, fluid mechanics, and other engineering systems. In this paper, a numerical method for series representation of random processes, with specified mean and correlation functions, in wavelet bases is presented. In this method, the Karhunen-Loeve expansion approach is used to represent a process as a linear sum of orthonormal eigenfunctions with uncorrelated random coefficients. The correlation and the eigenfunctions are approximated as truncated linear sums of compactly supported orthogonal wavelets. The eigenfunctions satisfy an integral eigenvalue problem. Using the above approximations, the integral eigenvalue problem is converted to a matrix (finite dimensional) eigenvalue problem. Numerical algorithms are discussed to compute one- and two-dimensional wavelet transforms of certain functions, and the resulting equations are solved to obtain the eigenvalues and the eigenfunctions. The scheme provides an improvement over other existing schemes. Two examples are considered to show the feasibility and effectiveness of this method. Numerical studies show that the results obtained using this method compare well with analytical techniques.

scholarly journals Characterization of Clifford Torus in Three-Spheres

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 718
Author(s):  
Dong-Soo Kim ◽  
Young Ho Kim ◽  
Jinhua Qian
Keyword(s):  
Laplace Operator ◽  
Unit Sphere ◽  
Three Dimensional ◽  
Gauss Map ◽  
Dimensional Sphere ◽  
Clifford Torus ◽  
Closed Surfaces ◽  
Dimensional Unit ◽  
The Laplace Operator

We characterize spheres and the tori, the product of the two plane circles immersed in the three-dimensional unit sphere, which are associated with the Laplace operator and the Gauss map defined by the elliptic linear Weingarten metric defined on closed surfaces in the three-dimensional sphere.

Elastic Layer Pressed Against a Half Space

1974 ◽  
Vol 41 (3) ◽  
pp. 703-707 ◽  
Author(s):  
K. C. Tsai ◽  
J. Dundurs ◽  
L. M. Keer
Keyword(s):  
Eigenvalue Problem ◽  
Integral Equations ◽  
Half Space ◽  
Elastic Layer ◽  
Numerical Results ◽  
Fredholm Integral Equations ◽  
Auxiliary Functions ◽  
Receding Contact ◽  
Two Bodies

The paper considers the elastic layer which is pressed against a half space by loads that are not necessarily symmetric about the center of the loaded region. It is shown that the receding contact between the two bodies can be treated by means of superposition, leading to two homogeneous Fredholm integral equations for auxiliary functions that are directly related to the contact tractions. The determination of the extent of contact and the shift between the load and contact intervals can be viewed as an eigenvalue problem of the homogeneous integral equations. Specific numerical results are given for two types of triangular loads, and a comparison is made with certain symmetric loads.

Development of Differential Quadrature Related Generalized Methods, Discrete Element Analysis Methods and EDQ Based Time Integration Methods

2005 ◽  
Author(s):  
Chang-New Chen
Keyword(s):  
Time Integration ◽  
Numerical Algorithms ◽  
Discrete Element ◽  
Numerical Results ◽  
Differential Quadrature ◽  
Element Analysis ◽  
Integration Methods ◽  
Analysis Methods ◽  
Time Integration Methods

Development of differential quadrature related generalized methods, discrete element analysis methods and EDQ based time integration methods has been carried out the last few years. The related numerical algorithms are summarized and presented. Numerical results are also presented.

scholarly journals Numerical Algorithms for Computing an Arbitrary Singular Value of a Tensor Sum

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 211
Author(s):  
Asuka Ohashi ◽  
Tomohiro Sogabe
Keyword(s):  
Eigenvalue Problem ◽  
Large Scale ◽  
Numerical Algorithms ◽  
Direct Methods ◽  
Coefficient Matrix ◽  
Lanczos Method ◽  
Singular Value ◽  
Preconditioned Conjugate Gradient ◽  
Schur Decomposition ◽  
Pcg Method

We consider computing an arbitrary singular value of a tensor sum: T:=In⊗Im⊗A+In⊗B⊗Iℓ+C⊗Im⊗Iℓ∈Rℓmn×ℓmn, where A∈Rℓ×ℓ, B∈Rm×m, C∈Rn×n. We focus on the shift-and-invert Lanczos method, which solves a shift-and-invert eigenvalue problem of (TTT−σ˜2Iℓmn)−1, where σ˜ is set to a scalar value close to the desired singular value. The desired singular value is computed by the maximum eigenvalue of the eigenvalue problem. This shift-and-invert Lanczos method needs to solve large-scale linear systems with the coefficient matrix TTT−σ˜2Iℓmn. The preconditioned conjugate gradient (PCG) method is applied since the direct methods cannot be applied due to the nonzero structure of the coefficient matrix. However, it is difficult in terms of memory requirements to simply implement the shift-and-invert Lanczos and the PCG methods since the size of T grows rapidly by the sizes of A, B, and C. In this paper, we present the following two techniques: (1) efficient implementations of the shift-and-invert Lanczos method for the eigenvalue problem of TTT and the PCG method for TTT−σ˜2Iℓmn using three-dimensional arrays (third-order tensors) and the n-mode products, and (2) preconditioning matrices of the PCG method based on the eigenvalue and the Schur decomposition of T. Finally, we show the effectiveness of the proposed methods through numerical experiments.

Sign in / Sign up

Export Citation Format

Share Document