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My question is related to my issue from yesterday (Initializing very large matrices). It was recommended to use SparseArray to initialize large matrices efficiently. However, I struggle to do so with my current issue. I want to initialize a matrix that approximates the Laplace operator in 3D with periodic boundary conditions. My ansatz for this is to calculate Kronecker products from the identity matrix $I$ and the matrix $A$, $$\Delta_{3D}=I\otimes I \otimes A+I\otimes A\otimes I+A\otimes I \otimes I\,\, ,$$ where the matrix $A$ is the 1D-Laplace operator with periodic boundary conditions defined through:

A = SparseArray[{Band[{1, 1}] -> -2, Band[{2, 1}] -> 1, Band[{1, 2}] -> 1, 
      {M, 1} -> 1, {1, M} -> 1}, {M, M}];

My general goal is to calculate the smallest eigenvalues of such a big matrix. I am using the following code, but when I run it on my (maybe too old) laptop, Mathematica crashes.

M = 128;

A = SparseArray[{Band[{1, 1}] -> -2, Band[{2, 1}] -> 1, 
    Band[{1, 2}] -> 1, {M, 1} -> 1, {1, M} -> 1}, {M, M}];
Ident = IdentityMatrix[M];
Laplace = KroneckerProduct[Ident, Ident, A] + 
   KroneckerProduct[Ident, A, Ident] + 
   KroneckerProduct[A, Ident, Ident];

AbsoluteTiming[Eigenvalues[1.0*Laplace, -2, 
   Method -> {"Arnoldi", "Criteria" -> "Magnitude", 
     "MaxIterations" -> 10000}] // Chop]

I would be glad for any suggestions that make my code working (or even working faster and more efficiently).

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  • $\begingroup$ If you only want the smallest eigenvalue of a large matrix, iterative methods that only calculate a few eigenvalues are better than calculate all eigenvalues. See e.g: [link] (en.wikipedia.org/wiki/…) $\endgroup$ Commented Oct 8, 2024 at 11:11
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    $\begingroup$ This is actually my plan. Thats why i use the Arnoldi method for the eigenvalues. By putting the -2 in the eigenvalue command i tell mathematica to only calculate the smallest 2 eigenvalues $\endgroup$ Commented Oct 8, 2024 at 11:20
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    $\begingroup$ I can confirm that. I ran this on my M1 Max and after quite some time the kernel crashed. I don't think your hardware is the problem. My machine has 64GB of RAM and it was not even half full. I think you should file a bug report with the Wolfram User Support. Make sure to include the code. You can also send them a link to this post instead. $\endgroup$ Commented Oct 8, 2024 at 12:00
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    $\begingroup$ To solve your problem I wonder whether you really have to/want to solve this numerically. The eigenfunctions of the Laplacian on the 3D torus $\mathbb{R}^3 / \mathbb{Z}^3$ are well-known. They are of the form $\mathrm{e}^{\mathrm{i} \pi (n_1 x_1 + n_2 x_2 + n_3 x_3)}$, where $n_1$, $n_2$, $n_3$ are natural numbers (or zero). $\endgroup$ Commented Oct 8, 2024 at 12:03
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    $\begingroup$ Yes, thats true. My real task looks a little bit different, but i comprehensed my issue. If you care: It is about the BdG- formalism in quantum mechanics. But thanks for help. I will send to the Wolfram user support. $\endgroup$ Commented Oct 8, 2024 at 13:01

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