Method Of Analytic Continuation Research Articles

Analytic continuation of harmonic sums near the integer values

We present a simple method for analytic continuation of harmonic sums near negative and positive integer numbers. We provide a precomputed database for the exact expansion of harmonic sums over a small parameter near these integer numbers, along with MATHEMATICA code, which shows the application of the database for actual problems. We also provide the FORM code that was used to obtain the database mentioned above. The applications of the obtained database for the study of evolution equations in the quantum field theory are discussed.

A fractional Landweber iterative regularization method for stable analytic continuation

In this paper, we consider the problem of analytic continuation of the analytic function $g(z) = g(x+iy)$ on a strip domain Ω = $\{z = x+iy\in \mathbb{C}|\, x\in\mathbb{R}, 0 < y < y_0\}$, where the data is given only on the line $y = 0$. This problem is a severely ill-posed problem. We propose the fraction Landweber iterative regularization method to deal with this problem. Under the a priori and a posteriori regularization parameter choice rule, we all obtain the error estimates between the regularization solution and the exact solution. Some numerical examples are given to verify the efficiency and accuracy of the proposed methods.

On the solution of one integro-differential equation with singular and hypersingular integrals

A linear integro-differential equation of the first order given on a closed curve located on the complex plane is studied. The coefficients of the equation have a special structure. The equation contains a singular integral, which can be understood as the main value by Cauchy, and a hypersingular integral which can be understood as the end part by Hadamard. The analytical continuation method is applied. The equation is reduced to a sequential solution of the Riemann boundary value problem and two linear differential equations. The Riemann problem is solved in the class of analytic functions with special points. Differential equations are solved in the class of analytical functions on the complex plane. The conditions for the solvability of the original equation are explicitly given. The solution of the equation when these conditions are fulfilled is also given explicitly. Examples are considered. A non-obvious special case is analyzed.

Simultaneous inversion of two initial values for a time‐fractional diffusion‐wave equation

AbstractThis study is devoted to recovering two initial values for a time‐fractional diffusion‐wave equation from boundary Cauchy data. We provide the uniqueness result for recovering two initial values simultaneously by the method of Laplace transformation and analytic continuation. And then we use a nonstationary iterative Tikhonov regularization method to solve the inverse problem and propose a finite dimensional approximation algorithm to find good approximations to the initial values. Numerical examples in one‐ and two‐dimensional cases are provided to show the effectiveness of the proposed method.

A backward problem for distributed order diffusion equation: uniqueness and numerical solution

In this paper we consider the identification of the initial condition for a distributed order diffusion equation. We first prove the unique existence and the regularity properties of the strong solution on the bounded temporal-spacial domain based on the eigenfunction expansions. The ill-posedness of the backward problem is interpreted by the compactness of the observation operator. Next the Laplace transformation technique and analytic continuation method are adopted to prove the uniqueness of the backward problem. Then for stabilizing the ill-posed problem, the backward problem is formulated as a Tikhonov type optimization problems, and the conjugate gradient method is adopted to solve the optimization problem with the help of the variational adjoint technique. Finally four numerical examples are given to show the efficiency and stability of the proposed method.

SIF determination in finite double-edge cracked orthotropic composite using J-integral and digital image correlation

This paper demonstrates ability to evaluate the stress intensity factor in a double-edge cracked finite [013/905/013] graphite-epoxy orthotropic tensile composite from J-integral and a single digital image correlation recorded displacement field. Relevant stress/strain/displacement information is obtained by processing the measured displacement data with stress functions, conformal mapping and analytical continuation method. A normalized stress intensity factor of 2.05 is determined experimentally for the present geometry, loading and material. Plane-stress is assumed. The technique is not restricted to a laboratory environment and concepts are applicable to other structural shapes, loads and crack configurations. Unlike purely numerical or theoretical approaches, the present method does not require knowing the external loading. This is significant as loads are often unknown in practice. Results are verified independently.

A hypersingular integro-differential equation of the Euler type

In this paper, we study an integro-differential equation on a closed curve located on the complex plane. The integrals included in the equation are understood as a finite part by Hadamard. The coefficients of the equation have a particular structure. The analytical continuation method is applied. The equation is reduced to a boundary value linear conjugation problem for analytic functions and linear Euler differential equations in the domains of the complex plane. Solutions of the Euler equations, which are unambiguous analytical functions, are sought. The conditions of solvability of the initial equation are given explicitly. The solution of the initial equation obtained under these conditions is also given explicitly. Examples are considered.

Analytic Continuation Methods for Multivalued Functions of One Variable and Their Application to the Solution of Algebraic Equations

The paper discusses several methods of analytic continuation of a multivalued function of one variable given on a part of its Riemann surface in the form of a Puiseux series generated by the power function z = w1/ρ, where ρ 1/2 and ρ ≠ 1. We present a many-sheeted variant of G.Polya’s theorem describing the relation between the indicator and conjugate diagrams for entire functions of exponential type. The description is based on V. Bernstein’s construction for the many-sheeted indicator diagram of an entire function of order ρ ≠ 1 and normal type. The summation domain of the “proper” Puiseux series (the many-sheeted “Borel polygon”) is found with the use of a generalization of the Borel method. This result seems to be new even in the case of a power series. The theory is applied to describe the domains of analytic continuation of Puiseux series representing the inverses of rational functions. As a consequence, a new approach to the solution of algebraic equations is found.

Calculation of resonances by analytical continuation: role of asymptotic behavior of coupling function

The regularized analytical continuation method (RAC) counts among the most powerful modern method for determination of resonance energies and resonance widths of low energy atomic and molecular systems including large systems of biological importance. By this method the resonance state is transformed into bound state by addition of an attractive potential. In this paper the choice of the perturbation potential is discussed and the performance of the method studied using model potential. It is shown that for correct applications of the RAC method the knowledge of the asymptotic behavior of the basic ingredient of the method, the coupling function λ(κ), is important. In this paper we find the correct asymptotic form of λ(κ) for broad class of realistic potentials and for nonzero values of angular momentum. New RAC formula convenient for singular perturbation potentials is proposed.

Average spectrum method for analytic continuation: Efficient blocked-mode sampling and dependence on the discretization grid

The average spectrum method is a promising approach for the analytic continuation of imaginary time or frequency data to the real axis. It determines the analytic continuation of noisy data from a functional average over all admissible spectral functions, weighted by how well they fit the data. Its main advantage is the apparent lack of adjustable parameters and smoothness constraints, using instead the information on the statistical noise in the data. Its main disadvantage is the enormous computational cost of performing the functional integral. Here we introduce an efficient implementation, based on the singular value decomposition of the integral kernel, eliminating this problem. It allows us to analyze the behavior of the average spectrum method in detail. We find that the discretization of the real-frequency grid, on which the spectral function is represented, biases the results. The distribution of the grid points plays the role of a default model while the number of grid points acts as a regularization parameter. We give a quantitative explanation for this behavior, point out the crucial role of the default model and provide a practical method for choosing it, making the average spectrum method a reliable and efficient technique for analytic continuation.

An inverse source problem for parabolic equations with local measurements

Inverse source problems for partial differential equations are of interest both in mathematics and in engineering. In this paper, we consider the problem of recovering the spatial contamination source in the diffusion model with the local measurements at fixed times. One example is constructed to show that the local measurements at one fixed time are not enough to determine the source. We propose the problem of recovering the source with the local measurements at two different fixed times. The uniqueness and conditional stability estimate are proved by the analytic continuation method.

The backward problem of parabolic equations with the measurements on a discrete set

Abstract The backward problems of parabolic equations are of interest in the study of both mathematics and engineering. In this paper, we consider a backward problem for the one-dimensional heat conduction equation with the measurements on a discrete set. The uniqueness for recovering the initial value is proved by the analytic continuation method. We discretize this inverse problem by a finite element method to deduce a severely ill-conditioned linear system of algebra equations. In order to overcome the ill-posedness, we apply the discrete Tikhonov regularization with the generalized cross validation rule to obtain a stable numerical approximation to the initial value. Numerical results for three examples are provided to show the effect of the measurement data.

Stress Field for a Coated Triangle-Like Hole Problem in Plane Elasticity

ABSTRACTThe stress field induced by an edge dislocation or a point force located near a coated triangle-like hole in an infinite plate is provided in this paper. Based on the method of analytical continuation and the technique of conformal mapping in conjunction with the alternation technique, a series solution for the displacement and stresses in the coating layer and the matrix is obtained analytically. Examples for the interaction between an edge dislocation and a coated triangle-like hole for various material constant combinations, coating thicknesses and shape factors are discussed. The analysis discovers that the so-called trapping mechanism of dislocations is more likely to exist near a coated triangle-like hole. The result shows that the dislocation will first be repelled by the coating layer and then attracted by a hole when the coating layer is slightly stiffer than the matrix. However, when the coating layer is sufficiently thin, the dislocation will always be attracted by a hole even the coating layer is stiffer than the matrix.

Scattering amplitudes and contour deformations

We employ a scalar model to exemplify the use of contour deformations when solving Lorentz-invariant integral equations for scattering amplitudes. In particular, we calculate the onshell 2 -> 2 scattering amplitude for the scalar system. The integrals produce branch cuts in the complex plane of the integrand which prohibit a naive Euclidean integration path. By employing contour deformations, we can also access the kinematical regions associated with the scattering amplitude in Minkowski space. We show that in principle a homogeneous Bethe-Salpeter equation, together with analytic continuation methods such as the Resonances-via-Pad\'e method, is sufficient to determine the resonance pole locations on the second Riemann sheet. However, the scalar model investigated here does not produce resonance poles above threshold but instead virtual states on the real axis of the second sheet, which pose difficulties for analytic continuation methods. To address this, we calculate the scattering amplitude on the second sheet directly using the two-body unitarity relation which follows from the scattering equation.

Resonance cross-section formula for low-energy elastic scattering

A simple analytical formula for partial wave cross section describing low-energy elastic scattering in the presence of a shape resonance located near the elastic threshold is proposed. The formula is based on the extension of the regularized method of analytical continuation in the coupling constant (RAC) recently proposed by Hor\'a\ifmmode \check{c}\else \v{c}\fi{}ek and co-workers. The RAC method provides the complex energies of resonances and virtual states requiring as input information a set of real bound state energies of a perturbed system for which standard highly sophisticated numerical codes may be employed. No scattering calculations are needed. These parameters define the low-energy partial wave resonance scattering cross section. Cross sections are calculated for the $2{s}^{2}\ensuremath{\varepsilon}p\phantom{\rule{0.16em}{0ex}}^{2}P^{0}$ state of beryllium and $3{s}^{2}\ensuremath{\varepsilon}p\phantom{\rule{0.16em}{0ex}}^{2}P^{0}$ of magnesium. The results compare well with recently published data.

On the use of natural electromagnetic fields to search for deep polarized objects

In article the possibility of detection of the deep polarized objects in natural electromagnetic fields is experimentally proved on the basis of synchronous measurements of a vertical component of long-period variations of magnetic field of Earth. Interpretation of data of the induced polarization is executed by method of analytical continuation of geophysical fields down (towards sources). Experimental works are completed near Lekhta (Karelia) in 2017.

Bose–Einstein condensation in two-dimensional traps

In two-dimensional traps, since the theoretical study of Bose–Einstein condensation will encounter the problem of divergence, the actual contribution of the divergent terms is often estimated in some indirect ways with the accuracy to the leading order. In this paper, by using an analytical continuation method to solve the divergence problem, we obtain the analytical expressions of critical temperature and condensate fraction for Bose gases in a 2D anisotropic box and harmonic trap, respectively. They are consistent with or better than previous studies. Then, we further consider the nonvanishing chemical potential, and obtain the expressions of chemical potential and more precise condensate fraction. These results agree with the numerical calculation well, especially for the case of harmonic traps. The comparison between the grand canonical and canonical ensembles shows that our calculation in the grand canonical ensemble is reliable.

TRIQS/SOM: Implementation of the stochastic optimization method for analytic continuation

We present the TRIQS/SOM analytic continuation package, an efficient implementation of the Stochastic Optimization Method proposed by Mishchenko et al. (2000). TRIQS/SOM strives to provide a high quality open source (distributed under the GNU General Public License version 3) alternative to the more widely adopted Maximum Entropy continuation programs. It supports a variety of analytic continuation problems encountered in the field of computational condensed matter physics. Those problems can be formulated in terms of response functions of imaginary time, Matsubara frequencies or in the Legendre polynomial basis representation. The application is based on the TRIQS C++/Python framework, which allows for easy interoperability with TRIQS-based quantum impurity solvers, electronic band structure codes and visualization tools. Similar to other TRIQS packages, it comes with a convenient Python interface. Program summaryProgram Title:TRIQS/SOMProgram Files doi:http://dx.doi.org/10.17632/fcjzyhrwpw.1Licensing provisions: GNU General Public License (GPLv3)Programming language:C++/PythonExternal routines/libraries:TRIQS 1.4.2, Boost >=1.58, cmake.Nature of problem: Quantum Monte Carlo methods (QMC) are powerful numerical techniques widely used to study quantum many-body effects and electronic structure of correlated materials. Obtaining physically relevant spectral functions from noisy QMC results in the imaginary time/Matsubara frequency domains requires solution of an ill-posed analytic continuation problem as a post-processing step.Solution method: We present an efficient C++/Python open-source implementation of the stochastic optimization method for analytic continuation.Additional comments: Project homepage http://krivenko.github.io/som/.

Analytical solutions to a coated triangle hole embedded in an infinite plate under a remote uniform heat flow

A general solution for a remote uniform heat flow interacting with a coated triangle hole embedded in an infinite plate is provided in this article. Based on the method of analytical continuation in conjunction with the alternation technique, the solutions to the heat conduction and thermoelastic problems are derived. A rapidly convergent series solution for both temperature and stress field is obtained in an elegant form which is expressed in terms of an explicit general form of the corresponding homogeneous complex potential. Numerical results of interfacial stresses are evaluated to provide a better understanding of an optimum design for various material properties and geometric parameters. The present proposed method can be further extended to deal with the corresponding thermoelastic problem with any shape of holes.

Numerical analytic continuation of Euclidean data

In this work we present a direct comparison of three different numerical analytic continuation methods: the Maximum Entropy Method, the Backus–Gilbert method and the Schlessinger point or Resonances Via Padé method. First, we perform a benchmark test based on a model spectral function and study the regime of applicability of these methods depending on the number of input points and their statistical error. We then apply these methods to more realistic examples, namely to numerical data on Euclidean propagators obtained from a Functional Renormalization Group calculation, to data from a lattice Quantum Chromodynamics simulation and to data obtained from a tight-binding model for graphene in order to extract the electrical conductivity.