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I'm interested in solving an inverse coefficient problem for a PDE. Let's say the field to be estimated is $\theta$. The conventional approach would be to use a finite element discretization for $\theta$ and add a Tikhonov regularization term to the objective functional in order to penalize unphysical values. Instead, I'd like to expand $\theta$ in a series of orthogonal functions $\{\psi_m\}_{m = 1}^M$:

$$\theta = \sum_m\hat\theta_m\psi_m$$

where the number $M$ of basis functions is much smaller than the number of finite element basis functions used to approximate the solution of the PDE. I want to do this because it's easier to count degrees of freedom and then apply tricks like the Akaike information criterion, which are not obviously applicable when you use Tikhonov regularization.

One obvious choice would be to use the first $M$ eigenfunctions of the Laplace operator. But if we do that, we have to decide on the boundary conditions -- Dirichlet, Neumann, or Robin. What if $\theta$ and its normal derivative have non-trivial variations along the boundary of the domain, to such a degree that using either Dirichlet or Neumann boundary conditions is inappropriate? If the domain $\Omega$ were nice, we could use wavelets, but my understanding is that constructing wavelets on arbitrary-shaped domains is very messy. By contrast, expanding in eigenfunctions is easy to jury-rig on top of an existing finite element code.

What are some good ways to pick the basis functions that do not make assumptions about the boundary behavior? Most of what I was able to find via Google scholar was about constructing reduced-order models in a data-driven way, whereas I'm trying to pick a basis a priori. The best I can come up with is to use both the Dirichlet and Neumann eigenfunctions and either sacrifice orthogonality or use Gram-Schmidt to orthogonalize them. I also considered using eigenfunctions of the biharmonic operator but in that case you end up picking even more boundary conditions. If this has been addressed somewhere in the literature I'd greatly appreciate any references.

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  • $\begingroup$ Say you have found such a set of functions and pick the first M. The boundary conditions can be such that the M+1 function alone solves the problem and the first M are useless. What this implies is that you must say something about the boundary conditions. $\endgroup$ Jul 21, 2022 at 5:29

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From a mathematical perspective, the eigenfunctions of the Laplace operator span the space $L^2(\Omega)$ regardless of whether you choose Dirichlet or Neumann boundary conditions. As a consequence, you can use them also as a basis to expand functions that have nonzero boundary conditions (either Dirichlet or Neumann).

From a practical perspective, you will probably take a relatively small number of basis functions and in that case you will get large approximation errors at the boundary and might want to choose a better basis. It would probably be worthwhile to incorporate knowledge about what boundary conditions the "true" coefficient satisfies when defining the basis.

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