I wrote a blog post on how a piece of pure mathematics - the development of the landscape function in PDE - played a part in realizing noticeable savings in household energy bills due to improved LED lighting technology: terrytao.wordpress.com/2025/02/23/c... - ThreadSky

teorth.bsky.social • 1 day ago

I wrote a blog post on how a piece of pure mathematics - the development of the landscape function in PDE - played a part in realizing noticeable savings in household energy bills due to improved LED lighting technology: https://terrytao.wordpress.com/2025/02/23/closing-the-green-gap-from-the-mathematics-of-the-landscape-function-to-lower-electricity-costs-for-households/

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makejakestop.bsky.social•1 day ago

📌

antonakhmerov.org•1 day ago

I'm surprised that using the landscape method makes enough of a performance difference compared to standard iterative sparse eigensolvers. Plus I'd imagine that the imprecision of the model introduces more bias than the ability to find the spectrum in a bigger problem or with more statistics.

teorth.bsky.social•1 day ago

According to https://doi.org/10.1103/PhysRevB.95.144206 , the computational performance speedup here is of the order of three orders of magnitude (see Tables V, VI), and the agreement with empirical values is superior (see page 8).

antonakhmerov.org•1 day ago

Thanks, I now checked the reference.

The numerics details are minimal. They use a direct solver PARDISO for the LU problem, and compare it to ARPACK, seemingly without using shift-invert for the eigenvalue problem. The second part is only a guess, the manuscript doesn't tell enough.

antonakhmerov.org•1 day ago

On the other hand, I always had an impression that using ARPACK with shift-invert to obtain a small fraction of the spectrum should be comparable to getting the sparse LU (and therefore to using LL) because applying sparse LU is much cheaper.

antonakhmerov.org•1 day ago

A while back I implemented a comparison of diagonalization with LL: https://mybinder.org/v2/git/https%3A%2F%2Fgitlab.kwant-project.org%2Fqt%2Flocalization_landscape.git/master?urlpath=lab/tree/index.ipynb.

For a 50x50x50 cube and 40 lowest eigenvectors—I think not too far from their system size—landscape is only 5x faster than eigenvectors on my laptop.

antonakhmerov.org•1 day ago

This advantage becomes even more uncertain because ARPACK computes eigenenergies close to machine precision, and LL gives an estimate that may off by some 10%.

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