Method Of Analytic Continuation Research Articles

Zhang neural networks: an introduction to predictive computations for discretized time-varying matrix problems

This paper wants to increase our understanding and computational know-how for time-varying matrix problems and Zhang Neural Networks. These neural networks were invented for time or single parameter-varying matrix problems around 2001 in China and almost all of their advances have been made in and most still come from its birthplace. Zhang Neural Network methods have become a backbone for solving discretized sensor driven time-varying matrix problems in real-time, in theory and in on-chip applications for robots, in control theory and other engineering applications in China. They have become the method of choice for many time-varying matrix problems that benefit from or require efficient, accurate and predictive real-time computations. A typical discretized Zhang Neural Network algorithm needs seven distinct steps in its initial set-up. The construction of discretized Zhang Neural Network algorithms starts from a model equation with its associated error equation and the stipulation that the error function decrease exponentially fast. The error function differential equation is then mated with a convergent look-ahead finite difference formula to create a distinctly new multi-step style solver that predicts the future state of the system reliably from current and earlier state and solution data. Matlab codes of discretized Zhang Neural Network algorithms for time varying matrix problems typically consist of one linear equations solve and one recursion of already available data per time step. This makes discretized Zhang Neural network based algorithms highly competitive with ordinary differential equation initial value analytic continuation methods for function given data that are designed to work adaptively. Discretized Zhang Neural Network methods have different characteristics and applicabilities than multi-step ordinary differential equations (ODEs) initial value solvers. These new time-varying matrix methods can solve matrix-given problems from sensor data with constant sampling gaps or from functional equations. To illustrate the adaptability of discretized Zhang Neural Networks and further the understanding of this method, this paper details the seven step set-up process for Zhang Neural Networks and twelve separate time-varying matrix models. It supplies new codes for seven of these. Open problems are mentioned as well as detailed references to recent work on discretized Zhang Neural Networks and time-varying matrix computations. Comparisons are given to standard non-predictive multi-step methods that use initial value problems ODE solvers and analytic continuation methods.

Adhesive contact between two-dimensional anisotropic elastic bodies

Adhesion plays a vital role in the design of smart and intelligent high-tech devices such as modern optical, microelectromechanical, and biomedical systems. However, in the literature, adhesive contact is mostly considered for contact of rigid substrates and transversely isotropic and isotropic elastic materials. The composite materials are increasingly used in the mart and intelligent high-tech devices. Since the composite materials are generally anisotropic and contact bodies are all deformable, it is more practical to consider the adhesive contact of two anisotropic elastic materials. In this paper, an adhesive contact model of anisotropic elastic bodies is established, and the closed-form solutions for two-dimensional adhesive contact of two anisotropic elastic bodies are derived. The full-field solutions and the relation for the contact region and applied force are developed using the Stroh complex variable formalism, the analytical continuation method, and concepts of the JKR adhesive model. We will show that the frictionless contact of two anisotropic elastic materials is just a special case of the present contact problem, and its solutions can be obtained by setting the work of adhesion equal to zero. In addition, we also show that our present solutions are valid for the problems of indentation by a rigid punch on an elastic half-space through a proper placement of the contact radius and the corresponding material constant. Numerical results are provided to demonstrate the accuracy, applicability, and versatility of the developed solutions.

Stochastic pole expansion method for analytic continuation of the Green's function

Stochastic pole expansion method for analytic continuation of the Green's function

Stieltjes functions and spectral analysis in the physics of sea ice

Abstract. Polar sea ice is a critical component of Earth’s climate system. As a material, it is a multiscale composite of pure ice with temperature-dependent millimeter-scale brine inclusions, and centimeter-scale polycrystalline microstructure which is largely determined by how the ice was formed. The surface layer of the polar oceans can be viewed as a granular composite of ice floes in a sea water host, with floe sizes ranging from centimeters to tens of kilometers. A principal challenge in modeling sea ice and its role in climate is how to use information on smaller-scale structures to find the effective or homogenized properties on larger scales relevant to process studies and coarse-grained climate models. That is, how do you predict macroscopic behavior from microscopic laws, like in statistical mechanics and solid state physics? Also of great interest in climate science is the inverse problem of recovering parameters controlling small-scale processes from large-scale observations. Motivated by sea ice remote sensing, the analytic continuation method for obtaining rigorous bounds on the homogenized coefficients of two-phase composites was applied to the complex permittivity of sea ice, which is a Stieltjes function of the ratio of the permittivities of ice and brine. Integral representations for the effective parameters distill the complexities of the composite microgeometry into the spectral properties of a self-adjoint operator like the Hamiltonian in quantum physics. These techniques have been extended to polycrystalline materials, advection diffusion processes, and ocean waves in the sea ice cover. Here we discuss this powerful approach in homogenization, highlighting the spectral representations and resolvent structure of the fields that are shared by the two-component theory and its extensions. Spectral analysis of sea ice structures leads to a random matrix theory picture of percolation processes in composites, establishing parallels to Anderson localization and semiconductor physics and providing new insights into the physics of sea ice.

Nevanlinna.jl: A Julia implementation of Nevanlinna analytic continuation

We introduce a Julia implementation of the recently proposed Nevanlinna analytic continuation method. The method is based on Nevanlinna interpolants and inherently preserves the causality of a response function due to its construction. For theoretical calculations without statistical noise, this continuation method is a powerful tool to extract real-frequency information from numerical input data on the Matsubara axis. This method has been applied to first-principles calculations of correlated materials. This paper presents its efficient and full-featured open-source implementation of the method including the Hamburger moment problem and smoothing.

Codebase release 1.0 for Nevanlinna.jl

We introduce a Julia implementation of the recently proposed Nevanlinna analytic continuation method. The method is based on Nevanlinna interpolants and inherently preserves the causality of a response function due to its construction. For theoretical calculations without statistical noise, this continuation method is a powerful tool to extract real-frequency information from numerical input data on the Matsubara axis. This method has been applied to first-principles calculations of correlated materials. This paper presents its efficient and full-featured open-source implementation of the method including the Hamburger moment problem and smoothing.

Casimir free energy for massive scalars: A comparative study of various approaches

We compute the Casimir thermodynamic quantities for a massive real scalar field between two parallel plates with the Dirichlet boundary conditions, using three different general approaches and present explicit solutions for each. The Casimir thermodynamic quantities include the Casimir Helmholtz free energy, pressure, energy, and entropy. The three general approaches that we use are based on the fundamental definition of Casimir thermodynamic quantities, the analytic continuation method, and the zero temperature subtraction method. Within the analytic continuation approach, we use two distinct methods which are based on the utilization of the zeta function and the Schlömilch summation formula. We include the renormalized versions of the latter two approaches as well, whereas the first approach does not require one. Within each general approach, we obtain the same results in a few different ways to ascertain the selected cancellations of infinities have been done correctly. We show that, as expected, the results based on the zeta function and the Schlömilch summation formula are equivalent. We then do a comparative study of the three different general approaches and their results and show that they are in principle not equivalent to each other, and they yield equivalent results only in the massless case. In particular, we show that the Casimir energy calculated only by the first approach has all three properties of going to zero as the temperature, mass of the field or the distance between the plates increases. Moreover, we show that in this approach the Casimir entropy reaches a positive constant in the high temperature limit, which can explain the linear term in the Casimir free energy. We use the results obtained by the fundamental approach as our reference for the correct results.

ACFlow: An open source toolkit for analytic continuation of quantum Monte Carlo data

The purpose of analytic continuation is to establish a real frequency spectral representation of single-particle or two-particle correlation function (such as Green's function, self-energy function, spin and charge susceptibilities) from noisy data generated in finite temperature quantum Monte Carlo simulations. It requires numerical solutions of a family of Fredholm integral equations of the first kind, which is indeed a challenging task. In this paper, an open source toolkit (dubbed ACFlow) for analytic continuation of quantum Monte Carlo data is presented. We first give a short introduction to the analytic continuation problem. Next, three popular analytic continuation algorithms, including the maximum entropy method, the stochastic analytic continuation, and the stochastic optimization method, as implemented in this toolkit are reviewed. And then we elaborate on the major features, implementation details, basic usage, inputs, and outputs of this toolkit. Finally, four representative examples, including analytic continuations of Matsubara self-energy function, Matsubara and imaginary time Green's functions, and current-current correlation function, are shown to demonstrate the usefulness and flexibility of the ACFlow toolkit. Program summaryProgram Title: ACFlowCPC Library link to program files:https://doi.org/10.17632/th6w74gwjm.1Developer's repository link:https://github.com/huangli712/ACFlowLicensing provisions: GNU General Public License Version 3Programming language: JuliaNature of problem: Most of the quantum Monte Carlo methods work on imaginary axis. In order to extract physical observables and compare them with the experimental results, analytic continuation must be done in the post-processing stage to convert the quantum Monte Carlo simulated data from imaginary axis to real axis.Solution method: Three well-established analytic continuation methods, including the maximum entropy method, the stochastic analytic continuation (both A. W. Sandvik's and K. S. D. Beach's algorithms), and the stochastic optimization method, have been implemented in the ACFlow toolkit.Additional comments including restrictions and unusual features: The ACFlow toolkit is written in pure Julia language. It is highly optimized and parallelized. It can be executed interactively in a Jupyter notebook environment.

Bosonic Nevanlinna Analytic Continuation

Analytical continuation (AC) connects theoretical calculations and experimentally measurable quantities. The recently proposed Nevanlinna AC method is capable of accurately reproducing the sharp features of spectral functions at high frequencies while maintaining the causality of the response function. However, their use is currently limited to fermions. Here, we present an extension of this method to bosons using the hyperbolic tangent trick, allowing us to transform bosons into auxiliary fermions to which the Nevanlinna analytic continuation can be applied.

Analytic continuation of harmonic sums with purely imaginary indices near the integer values

We present a simple algebraic method for the analytic continuation of harmonic sums with integer real or purely imaginary indices near negative and positive integers. We provide a MATHEMATICA code for exact expansion of harmonic sums in a small parameter near these integers. As an application, we consider the analytic continuation of the anomalous dimension of twist-1 operators in the ABJM model, which contains nested harmonic sums with purely imaginary indices. We found that in the BFKL-like limit the result has the same single-logarithmic behavior as in [Formula: see text] SYM and QCD, however, we did not find a general expression for the “BFKL Pomeron” eigenvalue in this model. For the slope function, we found full agreement with the expansion of the known general result and give predictions for the first three perturbative terms in the expansion of the next-to-slope function. The proposed method of analytic continuation can also be used for other generalization of nested harmonic sums.

Parameter-free analytic continuation for quantum many-body calculations

We develop a reliable parameter-free analytic continuation method for quantum many-body calculations. Our method is based on a kernel grid, a causal spline, a regularization using the second-derivative roughness penalty, and the L-curve criterion. We also develop the L-curve averaged deviation to estimate the precision of our analytic continuation. To deal with statistically obtained data more efficiently, we further develop a bootstrap-averaged analytic continuation method. In the test using the exact imaginary-frequency Green's function with added statistical error, our method produces the spectral function that converges systematically to the exact one as the statistical error decreases. As an application, we simulate the two-orbital Hubbard model for various electron numbers with the dynamical-mean field theory in the imaginary time and obtain the real-frequency self-energy with our analytic continuation method, clearly identifying a non-Fermi liquid behavior as the electron number approaches the half filling from the quarter filling. Our analytic continuation can be used widely and it will facilitate drawing clear conclusions from imaginary-time quantum many-body calculations.

Determination of asymptotic normalization coefficients for the channel $$^{16}$$O$$\rightarrow \alpha +^{12}$$C: excited state $$^{16}$$O($$0^+; 6.05$$ MeV)

Asymptotic normalization coefficients (ANC) determine the overall normalization of cross sections of peripheral radiative capture reactions. In the present paper, we treat the ANC $C$ for the virtual decay $^{16}$O$(0^+; 6.05$ MeV)$\to \alpha+^{12}$C(g.s.), the known values of which are characterized by a large spread $(0.29-1.65)\times 10^3$ fm$^{-1/2}$. The ANC $C$ is found by analytic continuation in the energy of the $\alpha^{12}$C $s$-wave scattering amplitude, known from the phase-shift analysis of experimental data, to the pole corresponding to the $^{16}$O bound state and lying in the unphysical region of negative energies. To determine $C$, two different methods of analytic continuation are used. In the first method, the scattering data are approximated by the sum of polynomials in energy in the physical region and then extrapolated to the pole. The best way of extrapolation is chosen on the basis of the exactly solvable model. Within the second approach, the ANC $C$ is found by solving the Schr\"odinger equation for the two-body $\alpha^{12}$C potential, the parameters of which are selected from the requirement of the best description of the phase-shift analysis data at a fixed experimental binding energy of $^{16}$O$(0^+; 6.05$ MeV) in the $\alpha+^{12}$C channel. The values of the ANC $C$ obtained within these two methods lie in the interval (886--1139) fm$^{-1/2}$.

Exponential asymptotics of woodpile chain nanoptera using numerical analytic continuation

AbstractTraveling waves in woodpile chains are typically nanoptera, which are composed of a central solitary wave and exponentially small oscillations. These oscillations have been studied using exponential asymptotic methods, which typically require an explicit form for the leading‐order behavior. For many nonlinear systems, such as granular woodpile chains, it is not possible to calculate the leading‐order solution explicitly. We show that accurate asymptotic approximations can be obtained using numerical approximation in place of the exact leading‐order behavior. We calculate the oscillation behavior for Toda woodpile chains, and compare the results to exponential asymptotics based on previous methods from the literature: long‐wave approximation and tanh‐fitting. We then use numerical analytic continuation methods based on Padé approximants and the adaptive Antoulas–Anderson (AAA) method. These methods are shown to produce accurate predictions of the amplitude of the oscillations and the mass ratios for which the oscillations vanish. Exponential asymptotics using an AAA approximation for the leading‐order behavior is then applied to study granular woodpile chains, including chains with Hertzian interactions—this method is able to calculate behavior that could not be accurately approximated in previous studies.

Perturbative steady states of completely positive quantum master equations.

The Lindblad form guarantees complete positivity of a Markovian quantum master equation(QME). However, its microscopic derivation for a quantum system weakly interacting with a thermal bath requires several approximations, which may result in inaccuracies in the QME. Recently, various Lindbladian QMEs were derived without resorting to the secular approximation from the Redfield equation which does not guarantee the complete positivity. Here we explicitly calculate, in a perturbative manner, the equilibrium steady states of these Lindbladian QMEs. We compare the results with the steady state of the Redfield equation obtained from an analytic continuation method, which coincides with the so-called mean force Gibbs (MFG) state. The MFG state is obtained by integrating out the bath degrees of freedom for the Gibbs state of the total Hamiltonian. We explicitly show that the steady states of the Lindbladian QMEs are different from the MFG state. Our results indicate that manipulations of the Redfield equationneeded to enforce complete positivity of a QME drives its steady state away from the MFG state. We also find that, in the high-temperature regime, both the steady states of the Lindbladian QMEs and MFG state reduce to the same Gibbs state of a system Hamiltonian under certain conditions.

Analytic continuation of multiple polylogarithms in positive characteristic

Our aim of this paper is to propose a method of analytic continuation of Carlitz multiple (star) polylogarithms to the whole space by using Artin-Schreier equation and present a treatment of their branches by introducing the notion of monodromy modules. As applications of this method, we obtain (1) a method of continuation of the logarithms of higher tensor powers of Carlitz module, (2) the orthogonal property (Chang-Mishiba functional relations), (3) a branch independency of the Eulerian property.

A full field solution for an anisotropic elastic plate with a hole perturbed from an ellipse

A full-field solution for an infinite anisotropic plate containing a hole perturbed from an ellipse subjected to uniform loading at infinity is derived with Stroh formalism. With perturbation technique, the solution is expanded into a series with reference to the solution of the elliptical hole problem. Through the traction-free condition on the hole boundary, the unknown coefficients of the series are solved using the method of analytical continuation. The explicit full-field solution up to the first order and its corresponding explicit expression of the hoop stress along the hole boundary is presented. Numerical examples with different hole shapes (triangle, quadrilateral, oval, and pentagon), material types (isotropic, orthotropic, and anisotropic), and loading types (in-plane stresses and anti-plane shear) are provided. The results along the hole boundary and in the full field are verified with existing solution and commercial finite element software ANSYS. Through this verification, we conclude that although the hoop stress along the hole boundary provided by the existing analytical solutions is correct, their associated full-field solutions are incorrect because of the non-conformal mapping functions. The solutions presented in this paper are the first verified correct full-field analytical solutions published in the literature.

TRIQS/SOM 2.0: Implementation of the stochastic optimization with consistent constraints for analytic continuation

TRIQS/SOM 2.0: Implementation of the stochastic optimization with consistent constraints for analytic continuation

A unified full field solution for indentation of an anisotropic piezoelectric half-plane by multiple rigid punches

Consider the indentation of an anisotropic piezoelectric half-plane by multiple rigid punches. By using the Stroh’s formalism for anisotropic piezoelasticity and the analytical continuation method for complex function’s manipulation, a unified full-field solution is obtained for four different contact conditions including adhesive contact with conducting punches, adhesive contact with insulating punches, frictional contact with conducting punches, and frictional contact with insulating punches. To demonstrate the generality and applicability of the unified general solution, the closed-form solutions for some benchmark problems such as indentation by one flat-ended punch, one parabolic punch, or multiple flat-ended punches are then derived. With the numerical results evaluated by the closed-form solutions, the effects of anisotropy, poling direction, frictional coefficient, contact condition, applied loads, and punch interaction are also studied and discussed.

Diagrammatic quantum Monte Carlo toward the calculation of transport properties in disordered semiconductors.

A new diagrammatic quantum Monte Carlo approach is proposed to deal with the imaginary time propagator involving both dynamic disorder (i.e., electron-phonon interactions) and static disorder of local or nonlocal nature in a unified and numerically exact way. The establishment of the whole framework relies on a general reciprocal-space expression and a generalized Wick's theorem for the static disorder. Since the numerical cost is independent of the system size, various physical quantities, such as the thermally averaged coherence, Matsubara one-particle Green's function, and current autocorrelation function, can be efficiently evaluated in the thermodynamic limit (infinite in the system size). The validity and performance of the proposed approach are systematically examined in a broad parameter regime. This approach, combined with proper numerical analytic continuation methods and first-principles calculations, is expected to be a versatile tool toward the calculation of various transport properties, such as mobilities in realistic semiconductors involving multiple electronic energy bands, high-frequency optical and low-frequency acoustic phonons, different forms of dynamic and static disorders, and anisotropy.

Partial classicality of Hilbert modular forms

Let F be a totally real field and p a rational prime unramified in F. We prove a partial classicality theorem for overconvergent Hilbert modular forms: when the slope is small compared to a subset of weights, an overconvergent form is partially classical. We use the method of analytic continuation.