SciPost Submission Page
Fast and stable determinant quantum Monte Carlo
by Carsten Bauer
This is not the current version.
|As Contributors:||Carsten Bauer|
|Arxiv Link:||https://arxiv.org/abs/2003.05286v2 (pdf)|
|Date submitted:||2020-03-14 01:00|
|Submitted by:||Bauer, Carsten|
|Submitted to:||SciPost Physics|
We assess numerical stabilization methods employed in fermion many-body quantum Monte Carlo simulations. In particular, we empirically compare various matrix decomposition and inversion schemes to gain control over numerical instabilities arising in the computation of equal-time and time-displaced Green's functions within the determinant quantum Monte Carlo (DQMC) framework. Based on this comparison, we identify a procedure based on pivoted QR decompositions which is both efficient and accurate to machine precision. The Julia programming language is used for the assessment and implementations of all discussed algorithms are provided in the open-source software library StableDQMC.jl [http://github.com/crstnbr/StableDQMC.jl].
Ontology / TopicsSee full Ontology or Topics database.
Submission & Refereeing History
You are currently on this page
Reports on this Submission
Anonymous Report 1 on 2020-4-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2003.05286v2, delivered 2020-04-08, doi: 10.21468/SciPost.Report.1615
1) Provides a systematic analysis of stabilization problems, and how to mitigate them, related to the determinant QMC method.
1) It is not clear how general the results are.
The paper considers the propagation of errors du to finite numerical precision in computing products of a large number of matrices, as appears in determinant QMC calculations. The stabilization problem and how to mitigate it has a long history, and here the author attempts to test several stabilization technique and compare them, both as to how well they work and as to their computational costs. The main conclusion is that the QR method is overall much better than the three SVD methods considered.
A potential limitation of the approach is that only a product of the same matrix is considered, which corresponds to a time-independent bosonic field in the determinant QMC simulation. In reality, the field of course fluctuates, and it is quite likely that the stabilization problems will be even worse for the typical contributing configurations, at least for some models (e.g., the Hubbard model with large U).
The paper is very technical, and it is not clear how general the results are. Nevertheless, the results and testing methodology may be interesting to researchers working on the technical aspects of the determinant QMC method. I recommend the paper to be published after revisions as listed below:
The illustration of the beta dependence of the singular values in Fig. 1 and some later figures requires some more explanation. I think the upper green line and lower orange line in Fig. 1 represents the smallest and largest singular values. But exactly what is the jagged data set that suddenly starts to increase with beta, taking off from the lower bound, e.g., around beta=10 in Fig. 1(a). Is that "curve" showing that the actual lowest singular value in the unstable numerical calculation? If so, it should be explained why there are no longer any small singular values. Perhaps it is somehow obvious that the destabilization causes the singular values to grow, but at least to me its not completely clear why.
As I mentioned above, the limitation to a product of the same matrix may give misleading results. Ideally, the author should at least show some results from an actual simulation, e.g., for the Hubbard model. otherwise it is not at all clear what the value is of this work. For instance, one could imagine that the QR method could work less well if the stabilization problem is even worse, as it may be for large U in typical Hubbard configurations.