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The LOBPCG algorithm finds eigenpairs of the generalized eigenproblem $$ Ax = \lambda B x $$ where $B$ is symmetric and positive-definite, $A$ is symmetric. One of the features that makes LOBPCG so interesting is the fact that one can use preconditioners of $A$ to speed up convergence.

When applying LOBPCG to the FEM-discretized Poisson problem (with $B=I$), and converging the smallest 6 eigenvalues, the convergence behavior I'm getting is

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When using an ML preconditioner, convergence speed increases dramatically:

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Now, let's try the same thing for the 6 largest eigenpairs.

Without preconditioner:

enter image description here

With ML preconditioner:

enter image description here

It seems that the preconditioner slows down the convergence! Is this something one would expect? I couldn't find anything in literature about it.

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    $\begingroup$ Framing question -- why do you want the largest eigenvalues? The smaller / mid-range eigenvalues of a FE / FD discretization of an elliptic operator are probably close to those of the true operator, but I'd expect that the largest ones aren't. $\endgroup$
    – Daniel Shapero
    Jan 23 at 19:09
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    $\begingroup$ @DanielShapero True. I just read in LOBPCG articles that the algorithm should be able to approximate both small and large eigenpairs, and I was wondering if perhaps there's something I'm doing wrong. Small-magnitude eigenvalue are indeed of greater interest though. $\endgroup$
    – Nico Schlömer
    Jan 23 at 20:18

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I would not be surprised by this, my understanding is that LOPCG is specifically designed to seek small eigenpairs (at least, that's what I have used it for). I find it a novel algorithm, because it does this without invoking inverse iteration (ie it does not require a solution operator for A). If you instead want large eigenpairs (imo, an easier problem), I think you'd be better served using the Lanczos method, though in the generalized case you might still have to apply the "cholesky trick" in order to keep everything symmetric.

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