[Hadstruct] Lanczos for ground state energy

[Joseph Karpie]

Thanks Savvas for letting me know I sent the wrong link. You can find all my notes on overleaf from the links in https://github.com/JeffersonLab/HadStruc/blob/master/overleaf
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From: Joseph Karpie <jkarpie at jlab.org>
Sent: Tuesday, September 3, 2024 6:04 PM
To: Joseph Karpie via Hadstruct <hadstruct at jlab.org>
Subject: Re: Lanczos for ground state energy

Hey,

I have made some notes on the way the algorithm works and derived the  block Lanczos version for using multiple interpolators. https://www.overleaf.com/1179645165mtwnjtssbnjf#b223e It is a pretty cute algorithm where you just assume there exist vectors which when used to make inner products with a matrix result in a tridiagonal structure. You then just iteratively find what the vectors are and realize that you actually made an orthogonal basis which preserved information on the eigenvalues.

After we get through any proper physics, I'd like to discuss a little tomorrow about prospects of this direction. I know some of you are worried about this algorithm being biased, and I'd like to get a criteria for what we think "working" is. In the original papers, they find that in the eigenspace pruning step each bootstrap sample returns at least 3 values and 85% given 4. Does that mean we can really trust those 3 values and the fourth might be biased? Or should we worry they are all spoiled? Wagman and Hackett argue the former, and that is my intuition as well.

Let's ignore that I currently can't get such consistent results stably. I am looking for a bug in my 100 digit numerical precision implementation, since they claim it's important yet I get identical results with/without it.

Best,
Joe
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From: Joseph Karpie <jkarpie at jlab.org>
Sent: Thursday, August 22, 2024 5:30 PM
To: Joseph Karpie via Hadstruct <hadstruct at jlab.org>
Subject: Re: Lanczos for ground state energy

I have made a little more progress for the curious. First, taking only the real component is quite important. It stabilizes the complexity of the evalues significantly of course. I now take the real evalues and those that pass a very stringent CW test. For comparing to the effective mass, I take the E_0 from all remaining evalues whose jackknife samples are within the plot windows. You can see them plotted between an estimate of mean/error from standard deviation and from a maybe still incorrect 60% confidence window the authors suggest.

With the larger momentum pz=2,3 where we may have wished we used momentum smearing, you can begin to see the advantage creep in. By tmax=21, the effective mass is lost, but the Lanczos result sometimes is quite stable. There are still occasional cases you can see where the spread of the energies I get are quite wild. I haven't quite understood this yet.

It would be nice to discuss more what problems can come in from the noise or another problem which may bias this. Take a look athttps://arxiv.org/pdf/2407.21777 which as more details then the original.
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From: Joseph Karpie
Sent: Tuesday, August 20, 2024 5:05 PM
To: Joseph Karpie via Hadstruct <hadstruct at jlab.org>
Subject: Lanczos for ground state energy

Hey,

I have started testing the Lanczos idea for analyzing 2pt functions. I put basic formula and results into slides https://docs.google.com/presentation/d/1uaQpQhQawA7VGA--2cjqrsSCSPoB6ORvl2M9iQZD4dQ/edit?usp=sharing I'm happy with the results given how roughly they are made. In the plot of evalues, and considering physical evalues must be positive and real, you can tell most of them are garbage. I can explain the dirty filtering procedure I did tomorrow morning. I hope to actually do the more professional diagnostics in a next attempt. Unfortunately without going to longer times, this data doesn't actually reach the regime where the advantage is significant.

Best,
Joe

--
Joe Karpie
Pronouns: He/His/Him
Postdoctoral Fellow
Theoretical and Computational Physics Center
Jefferson Lab

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