Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1912.01379 · Physical review. B./Physical review. B

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

Khaldoon Ghanem, Erik Koch

Published Feb 10, 2020• 26 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.101.085111

Related papers

Extending the average spectrum method: Grid point sampling and density averaging

Khaldoon Ghanem, Erik Koch

Analytic continuation of imaginary time or frequency data to the real axis is a crucial step in extracting dynamical properties from quantum Monte Carlo simulations. The average spectrum method provides an elegant solution by integrating over all nonnegative spectra weighted by how well they fit the data. In a recent paper, we found that discretizing the functional integral, as in Feynman's path-integrals, does not have a well-defined continuum limit. Instead, the limit depends on the discretization grid whose choice may strongly bias the results. In this paper, we demonstrate that sampling the grid points, instead of keeping them fixed, also changes the functional integral limit and rather helps to overcome the bias considerably. We provide an efficient algorithm for doing the sampling and show how the density of the grid points acts now as a default model with a significantly reduced biasing effect. The remaining bias depends mainly on the width of the grid density, so we go one step further and average also over densities of different widths. For a certain class of densities, including Gaussian and exponential ones, this width averaging can be done analytically, eliminating the need to specify this parameter without introducing any computational overhead.

arXiv:2004.01155•Physical review. B./Physical review. B•Jul 6, 2020•20 citations

Analytical continuation of spectral data from imaginary time axis to real frequency axis using statistical sampling

Kiamars Vafayi, O. Gunnarsson

We present a method for performing analytical continuation of spectral data from imaginary time to real frequencies based on a statistical sampling method. Compared with the maximum entropy method (MEM), an advantage is that no default model needs to be introduced. For the problems studied here, the statistical sampling method gives comparable or slightly better results than MEM using quite accurate default models.

Physical Review B•Jul 19, 2007•46 citations

Parameter-free analytic continuation for quantum many-body calculations

Mancheon Han, Hyoung Joon Choi

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.

arXiv:2301.00129•Physical review. B./Physical review. B•Dec 29, 2022•2 citations

Sparse modeling approach to analytical continuation of imaginary-time quantum Monte Carlo data

Junya Otsuki, Masayuki Ohzeki, Hiroshi Shinaoka +1 more

A data-science approach to solving the ill-conditioned inverse problem for analytical continuation is proposed. The root of the problem lies in the fact that even tiny noise of imaginary-time input data has a serious impact on the inferred real-frequency spectra. By means of a modern regularization technique, we eliminate redundant degrees of freedom that essentially carry the noise, leaving only relevant information unaffected by the noise. The resultant spectrum is represented with minimal bases and thus a stable analytical continuation is achieved. This framework further provides a tool for analyzing to what extent the Monte Carlo data need to be accurate to resolve details of an expected spectral function.

arXiv:1702.03056•Physical review. E•Jun 21, 2017•107 citations

Analytical continuation of imaginary axis data using maximum entropy

O. Gunnarsson, M. W. Haverkort, Giorgio Sangiovanni

We study the maximum entropy (MaxEnt) approach for analytical continuation of spectral data from imaginary times to real frequencies. The total error is divided in a statistical error, due to the noise in the input data, and a systematic error, due to deviations of the default function, used in the MaxEnt approach, from the exact spectrum. We find that the MaxEnt approach in its classical formulation can lead to a nonoptimal balance between the two types of errors, leading to an unnecessary large statistical error. The statistical error can be reduced by splitting up the data in several batches, performing a MaxEnt calculation for each batch and averaging. This can outweigh an increase in the systematic error resulting from this approach. The output from the MaxEnt calculation can be used as a default function for a new MaxEnt calculation. Such iterations often lead to worse results due to an increase in the statistical error. By splitting up the data in batches, the statistical error is reduced and the increase resulting from iterations can be outweighed by a decrease in the systematic error. Finally we consider a linearized version to obtain a better understanding of the method.

arXiv:1001.4351•Physical Review B•Apr 9, 2010•47 citations

Robust analytic continuation combining the advantages of the sparse modeling approach and the Padé approximation

Yuichi Motoyama, Kazuyoshi Yoshimi, Junya Otsuki

Analytic continuation (AC) from the imaginary-time Green's function to the spectral function is a crucial process for numerical studies of the dynamical properties of quantum many-body systems. This process, however, is an ill-posed problem; that is, the obtained spectrum is unstable against the noise of the Green's function. Though several numerical methods have been developed, each of them has its own advantages and disadvantages. The sparse modeling (SpM) AC method, for example, is robust against the noise of the Green's function but suffers from unphysical oscillations in the low-energy region. We propose a method that combines the SpM AC with the Pad\'e approximation. This combination, called SpM-Pad\'e, inherits robustness against noise from SpM and low-energy accuracy from Pad\'e, compensating for the disadvantages of each. We demonstrate that the SpM-Pad\'e method yields low-variance and low-biased results with almost the same computational cost as that of the SpM method.

arXiv:2109.08370•Physical review. B./Physical review. B•Jan 24, 2022•15 citations
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.
Analytic continuation of imaginary time or frequency data to the real axis is a crucial step in extracting dynamical properties from quantum Monte Carlo simulations. The average spectrum method provides an elegant solution by integrating over all nonnegative spectra weighted by how well they fit the data. In a recent paper, we found that discretizing the functional integral, as in Feynman's path-integrals, does not have a well-defined continuum limit. Instead, the limit depends on the discretization grid whose choice may strongly bias the results. In this paper, we demonstrate that sampling the grid points, instead of keeping them fixed, also changes the functional integral limit and rather helps to overcome the bias considerably. We provide an efficient algorithm for doing the sampling and show how the density of the grid points acts now as a default model with a significantly reduced biasing effect. The remaining bias depends mainly on the width of the grid density, so we go one step further and average also over densities of different widths. For a certain class of densities, including Gaussian and exponential ones, this width averaging can be done analytically, eliminating the need to specify this parameter without introducing any computational overhead.
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.
A data-science approach to solving the ill-conditioned inverse problem for analytical continuation is proposed. The root of the problem lies in the fact that even tiny noise of imaginary-time input data has a serious impact on the inferred real-frequency spectra. By means of a modern regularization technique, we eliminate redundant degrees of freedom that essentially carry the noise, leaving only relevant information unaffected by the noise. The resultant spectrum is represented with minimal bases and thus a stable analytical continuation is achieved. This framework further provides a tool for analyzing to what extent the Monte Carlo data need to be accurate to resolve details of an expected spectral function.
We study the maximum entropy (MaxEnt) approach for analytical continuation of spectral data from imaginary times to real frequencies. The total error is divided in a statistical error, due to the noise in the input data, and a systematic error, due to deviations of the default function, used in the MaxEnt approach, from the exact spectrum. We find that the MaxEnt approach in its classical formulation can lead to a nonoptimal balance between the two types of errors, leading to an unnecessary large statistical error. The statistical error can be reduced by splitting up the data in several batches, performing a MaxEnt calculation for each batch and averaging. This can outweigh an increase in the systematic error resulting from this approach. The output from the MaxEnt calculation can be used as a default function for a new MaxEnt calculation. Such iterations often lead to worse results due to an increase in the statistical error. By splitting up the data in batches, the statistical error is reduced and the increase resulting from iterations can be outweighed by a decrease in the systematic error. Finally we consider a linearized version to obtain a better understanding of the method.
Analytic continuation (AC) from the imaginary-time Green's function to the spectral function is a crucial process for numerical studies of the dynamical properties of quantum many-body systems. This process, however, is an ill-posed problem; that is, the obtained spectrum is unstable against the noise of the Green's function. Though several numerical methods have been developed, each of them has its own advantages and disadvantages. The sparse modeling (SpM) AC method, for example, is robust against the noise of the Green's function but suffers from unphysical oscillations in the low-energy region. We propose a method that combines the SpM AC with the Pad\'e approximation. This combination, called SpM-Pad\'e, inherits robustness against noise from SpM and low-energy accuracy from Pad\'e, compensating for the disadvantages of each. We demonstrate that the SpM-Pad\'e method yields low-variance and low-biased results with almost the same computational cost as that of the SpM method.