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We have the following generalized eigenvalue (set of) problem(s)

$$[K_R(\kappa)]\{u_R\} = \omega^2[M_R(\kappa)]\{u_R\}\quad \forall \kappa \in [\kappa_0, \kappa_1]$$

with

\begin{align} &K_R(\kappa) = T^H(\kappa) K T(\kappa)\, ,\\ &M_R(\kappa) = T^H(\kappa) M T(\kappa)\, ,\\ &u = T(\kappa) u_R\, . \end{align}

Where $K$ and $M$ are sparse and symmetric and come from a PDE, and $T(\kappa)$ is also sparse, non-square, complex and represents a set of multipoint constraints. Forming the matrix $T$ is relatively cheap compared with the other matrices since they have a fixed structure and are sparse.

If we consider a regular mesh with $n^2$ nodes, we have that:

\begin{align} &K\in \mathbb{R}^{n^2\times n^2}\, ,&M\in \mathbb{R}^{n^2\times n^2}\, ,\\ &u\in \mathbb{C}^{n^2}\, , & &\\ &K_R\in \mathbb{C}^{(n^2 - 2n + 1)\times (n^2 - 2n + 1)}\, ,&M_R\in \mathbb{C}^{(n^2 - 2n + 1)\times (n^2 - 2n + 1)}\, ,&\\ &u_R\in \mathbb{C}^{(n^2 - 2n + 1)}\, , & &\\ &T(\kappa) \in \mathbb{C}^{(2n -1)\times(n^2 - 2n + 1)}\, .& & \end{align}

We normally handle the problem in one of the following ways:

  1. Assemble $K$ and $M$ and for each value of $\kappa$ form the product matrices $K_R$ and $M_R$. The main advantage of this method is that we have to assemble once, but then we loose the sparse nature of the problem.
  2. Assemble $K_R$ and $M_R$ for each $\kappa$ value, conserving the sparse nature of the problem.

Question

Is there any way of solving the generalized eigenvalue problem

$$[T^H(\kappa) K T(\kappa)]\{u_R\} = \omega^2[T^H(\kappa) M T(\kappa)]\{u_R\}\quad \forall \kappa \in [\kappa_0, \kappa_1]\, ,$$

conserving the sparse nature of the system and assembling the matrices $K$ and $M$ only once?

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  • $\begingroup$ I think I'm missing something, but can't you use an iterative method for the eigenproblem, can't you work with the product of matrices as it is written? $\endgroup$ – VorKir Mar 23 at 3:47
  • $\begingroup$ @VorKir, I think that you are right. I think I could just define the linear operators and pass that to the solver. $\endgroup$ – nicoguaro Mar 23 at 15:42
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As suggested by @VorKir, it is enough to write the product of matrices as a linear operator for the iterative solver. Forming the matrices $T(\kappa)$ and creating the linear operator for each value of $\kappa$ is really fast compared with assembling the whole matrix and solving the eigenvalue problem for well-posed problems.

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