Projected regression method for solving Fredholm integral equations arising in the analytic continuation problem of quantum physics

Arsenault, Louis-François; Neuberg, Richard; Hannah, Lauren A.; Millis, Andrew J.

doi:10.1088/1361-6420/aa8d93

Cited by 47 publications

(43 citation statements)

References 31 publications

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“…We have seen that the average spectrum method is not the parameter free method suggested by the deceptively written functional integral (13): We have to choose a grid density ρ(x), which acts as a default model, and a number N of grid points, which acts as a regularization parameter. The reason for this is that the naive discretization (19) does not converge to a well defined functional integral. Instead we have to sample the components of the models we are integrating over from distributions that are consistent for different discretizations.…”

Section: Discussionmentioning

confidence: 99%

“…The straightforward method for evaluating (19) is to perform a random walk in the space of non-negative vectors f , updating a single component, f n → f n , at a time. Detailed balance is fulfilled if we sample f n from the conditional distribution ∝ exp(−χ 2 (f ; f n )/2) with…”

Section: A Components Samplingmentioning

confidence: 99%

“…We can understand this by considering the limit where the data contains no information about the model except a sum rule f n = 1 to keep the result finite. Then (19)…”

Section: We Express the Integral Equation In The New Variablementioning

confidence: 99%

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

Ghanem

Koch

2020

Phys. Rev. B

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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.

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Section: Discussionmentioning

confidence: 99%

Section: A Components Samplingmentioning

confidence: 99%

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

Ghanem

Koch

2020

Phys. Rev. B

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“…In that sense, our framework directly integrates the projection step employed in Ref. 17. We show that by applying standard training techniques, we are able to reach accuracy comparable with the MaxEnt approach while keeping lower standard deviations in the error distribution.…”

mentioning

confidence: 87%

Artificial Neural Network Approach to the Analytic Continuation Problem

et al. 2020

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Inverse problems are encountered in many domains of physics, with analytic continuation of the imaginary Green's function into the real frequency domain being a particularly important example. However, the analytic continuation problem is ill-defined and currently no analytic transformation for solving it is known. We present a general framework for building an artificial neural network that efficiently solves this task with a supervised learning approach, provided that the forward problem is sufficiently stable for creating a large training dataset of physically relevant examples. By comparing with the commonly used maximum entropy approach, we show that our method can reach the same level of accuracy for low-noise input data, while performing significantly better when the noise strength increases. We also demonstrate that adding an unsupervised denoising step significantly improves the accuracy of the maximum entropy predictions for noisy inputs. The computational cost of the proposed neural network appoach is reduced by almost three orders of magnitude compared to the maximum entropy method.Numerical simulations are playing an extensive role in a growing number of scientific disciplines. Most commonly, numerical simulations approximate a process or a field on a discretized map, taking as input an equation describing the model as well as initial and boundary conditions [1]. Problems falling under this definition are known as direct or forward problems. However, in a number of situations it is required to reconstruct an approximation of the input data or the model that generated it given the observable data. One particularly important example, known as the Fredholm integral equation of the first kind, takes the following form:where g(s, t) is the available quantity, f (s) is the quantity of interest and k(t, s) is the kernel. Such problem definitions are known as inverse problems and are mostly illposed [2]. Noise affecting the data may lead to arbitrarily large errors in the solutions. Formally speaking, one is interested in operators k• that are ill-conditioned or degenerated. Therefore, the formal inversion f = k −1 • g is not a stable operation [1]. One popular approach for solving such problems is to construct regularizing algorithms that converge to a "pseudo-solution" [3]. In this work, we will focus on a particularly important case of the analytic continuation problem in quantum many-body physics. The analytic continuation problem seeks to recover the electron single-particle spectral density in the real frequency domain A(ω) from the fermionic Green's function G(τ ) in imaginary time domain. These quantities are related by the following equation:G(τ ) = − e −ωτ 1 + e − ωβ A(ω)dω.(2) *

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“…Fortunately, there has been extensive research in the last several years using artificial neural networks and other machine learning algorithms to attack such inverse problems [23][24][25]. The setting of such approaches are data driven -meaning that a large number of forward problems are first solved in order to produce a set of training data.…”

Section: A Neural Network For the Inverse Problemmentioning

confidence: 99%

Machine learning design of a trapped-ion quantum spin simulator

Teoh

Drygala

Melko

et al. 2020

Quantum Sci. Technol.

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Trapped ions have emerged as one of the highest quality platforms for the quantum simulation of interacting spin models of interest to various fields of physics. In such simulators, two effective spins can be made to interact with arbitrary strengths by coupling to the collective vibrational or phonon states of ions, controlled by precisely tuned laser beams. However, the task of determining laser control parameters required for a given spin-spin interaction graph is a type of inverse problem, which can be highly mathematically complex. In this paper, we adapt a modern machine learning technique developed for similar inverse problems to the task of finding the laser control parameters for a number of interaction graphs. We demonstrate that typical graphs, forming regular lattices of interest to physicists, can easily be produced for up to 50 ions using a single GPU workstation. The scaling of the machine learning method suggests that this can be expanded to hundreds of ions with moderate additional computational effort.

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Projected regression method for solving Fredholm integral equations arising in the analytic continuation problem of quantum physics

Cited by 47 publications

References 31 publications

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

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

Artificial Neural Network Approach to the Analytic Continuation Problem

Machine learning design of a trapped-ion quantum spin simulator

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