# American Institute of Mathematical Sciences

March  2020, 2(1): 1-17. doi: 10.3934/fods.2020001

## Stochastic gradient descent algorithm for stochastic optimization in solving analytic continuation problems

 1 Department of Mathematics, Florida State University, Tallahassee, Florida, USA 2 Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA

* Corresponding author: Feng Bao

Published  February 2020

Fund Project: The first author is supported by NSF grant DMS-1720222

We propose a stochastic gradient descent based optimization algorithm to solve the analytic continuation problem in which we extract real frequency spectra from imaginary time Quantum Monte Carlo data. The procedure of analytic continuation is an ill-posed inverse problem which is usually solved by regularized optimization methods, such like the Maximum Entropy method, or stochastic optimization methods. The main contribution of this work is to improve the performance of stochastic optimization approaches by introducing a supervised stochastic gradient descent algorithm to solve a flipped inverse system which processes the random solutions obtained by a type of Fast and Efficient Stochastic Optimization Method.

Citation: Feng Bao, Thomas Maier. Stochastic gradient descent algorithm for stochastic optimization in solving analytic continuation problems. Foundations of Data Science, 2020, 2 (1) : 1-17. doi: 10.3934/fods.2020001
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##### References:
Example 1. True spectrum
Example 1. (a) FESOM samples; (b) FESOM estimation
Example 1. Estimated spectrum learned from FESOM samples
Example 2. True spectrum
Example 2. (a) FESOM estimation; (b) Estimated spectrum learned from FESOM samples
Example 2. Comparison between SGD and MaxEnt
True spectrum
Example 3. Estimations for the spectrum
Example 3. Spectrum with fine feature in positive frequency region
Example 3. (a) MaxEnt estimation for $A_2$; (b) Comparison of MaxEnt in estimating $A_1$ (red) and $A_2$ (blue)
Example 3. (a) SGD estimation for $A_1$; (b) SGD estimation for $A_2$
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