This title appears in the Scientific Report :
2013
Please use the identifier:
http://hdl.handle.net/2128/5590 in citations.
Stochastic Mode Sampling (SMS) -- An Efficient Approach to the Analytic Continuation Problem
Stochastic Mode Sampling (SMS) -- An Efficient Approach to the Analytic Continuation Problem
Analytic continuation is a recurring problem in different contexts of condensed matter physics. Typically we need to find a non-negative function, like spectral function or optical conductivity, using data from Quantum Monte Carlo (QMC) simulations. The relation between the data and the desired func...
Saved in:
Personal Name(s): | Ghanem, Khaldoon (Corresponding author) |
---|---|
Contributing Institute: |
GRS; GRS |
Imprint: |
2013
|
Physical Description: |
100 p. |
Dissertation Note: |
RWTH Aachen, Masterarbeit, 2013 |
Document Type: |
Master Thesis |
Research Program: |
Exploratory materials and phenomena |
Subject (ZB): | |
Link: |
OpenAccess |
Publikationsportal JuSER |
Analytic continuation is a recurring problem in different contexts of condensed matter physics. Typically we need to find a non-negative function, like spectral function or optical conductivity, using data from Quantum Monte Carlo (QMC) simulations. The relation between the data and the desired function can be formulated as a Fredholm integral equation of first kind which is an ill-posed problem with no unique solution in the presence of noise. What is special about these particular Fredholom integral equations is the non-negativity of the solution. Utilizing this property does not only make sure we get a physically-acceptable solution, but it also provides additional information that helps improving its quality.One class of methods that solve the problem using only the non-negativity of the solution as a priori knowledge, is the Stochastic Sampling. These methods use Bayesian inference to derive a probability distribution of the solution and use the mean as an estimator. To get the mean, they usually sample the solution space directly, but unfortunately this leads to large correlation time due to the high correlations between the different components of the solution.In this thesis, we propose a new stochastic sampling method, Stochastic Mode Sampling (SMS), where instead of sampling the solution’s components directly, we sample the right singular vectors (modes) of the kernel of the integral equation using Gibbs sampling. In this basis, the sampled quantities are statistically uncorrelated, but they are coupled through the non-negativity constraint. The efficiency of our method depends on this coupling, so we also show how to modify the kernel and choose the grid such that the coupling is minimized. Using the proper modification and grid, the SMS method has much less correlation times than earlier stochastic sampling methods. Besides, since the modes are ordered naturally according to their relevance to the data (using singular values), the SMS method provides a convenient way of trading-off between quality and speed by simply limiting the modes included in the sampling. |