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I work on an inverse problem for my Ph.D. research, which for simplicity's sake we'll say is determining $\beta$ in

$L(\beta)u \equiv -\nabla\cdot(k_0e^\beta\nabla u) = f$

from some observations $u^o$; $k_0$ is a constant and $f$ is known. This is typically formulated as an optimization problem for extremizing

$J[u, \lambda; \beta] = \frac{1}{2}\int_\Omega(u(x) - u^o(x))^2dx + \int_\Omega\lambda(L(\beta)u - f)dx$

where $\lambda$ is a Lagrange multiplier. The functional derivative of $J$ with respect to $\beta$ can be computed by solving the adjoint equation

$L(\beta)\lambda = u - u^o.$

Some regularizing functional $R[\beta]$ is added to the problem for the usual reasons.

The unspoken assumption here is that the observed data $u^o$ are defined continuously throughout the domain $\Omega$. I think it might be more appropriate for my problem to instead use

$J[u, \lambda; \beta] = \sum_{n = 1}^N\frac{(u(x_n) - u^o(x_n))^2}{2\sigma_n^2} + \int_\Omega\lambda(L(\beta)u - f)dx$

where $x_n$ are the points at which the measurements are taken and $\sigma_n$ is the standard deviation of the $n$-th measurement. The measurements of this field are often spotty and missing chunks; why interpolate to get a continuous field of dubious fidelity if that can be avoided?

This gives me pause because the adjoint equation becomes

$L(\beta)\lambda = \sum_{n = 1}^N\frac{u(x_n) - u^o(x_n)}{\sigma_n^2}\delta(x - x_n)$

where $\delta$ is the Dirac delta function. I'm solving this using finite elements, so in principle integrating a shape function against a delta function amounts to evaluating the shape function at that point. Still, the regularity issues probably shouldn't be dismissed out of hand. My best guess is that the objective functional should be defined in terms of the finite element approximation to all the fields, rather than in terms of the real fields and then discretized after.

I can't find any comparisons of assuming continuous or pointwise measurements in inverse problems in the literature, either in relation to the specific problem I'm working on or generally. Often pointwise measurements are used without any mention of the incipient regularity issues, e.g. here. Is there any published work comparing the assumptions of continuous vs. pointwise measurements? Should I be concerned about the delta functions in the pointwise case?

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The measurements of this field are often spotty and missing chunks; why interpolate to get a continuous field of dubious fidelity if that can be avoided?

You're perfectly right - most of the time, interpolation to a continuous field covering the entire domain is not an option. Think about weather prediction problems, where measurements (point-sources) are available only at selected domain locations. I'd say point-wise data is more the norm than the exception when you consider "real-life" inverse problems.

My best guess is that the objective functional should be defined in terms of the finite element approximation to all the fields (discretize-then-optimize), rather than in terms of the real fields and then discretized after (optimize-then-discretize).

The two approaches are not equivalent (except for very simple problems). There is a vast body of literature comparing the two approaches (each with its advantages and drawbacks). I'd point you towards Max Gunzburger's monograph (in particular the end of chapter 2).

Is there any published work comparing the assumptions of continuous vs. pointwise measurements? Should I be concerned about the delta functions in the pointwise case?

You can represent your source terms exactly - namely, your source term will be modeled as a (discrete approximation to a) Dirac distribution [Arraya et al., 2006], or you can approximate the source term by some regularized function (as is done, for example, in the immersed boundary method). Have a look (for starters) at this recent paper by Hosseini et al. (and references therein).

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To expand on @GoHokies's answer: If you're interested in regularity questions, you can also ask what "point measurements" really are. In physical practice, you cannot measure anything at a "point". Rather, you are always going to get some kind of average over some kind of space-time chunk: a thermometer is not a point but an extended object, and it takes time to adjust to the temperature of the medium around it; a concentration measurement device needs a finite sample size; etc.

What this means mathematically is that the delta functions in your functional are, really, averages over sufficiently small areas and/or time intervals. Consequently, the right hand sides in the dual equation are also finite, and no regularity issues arise.

Of course, in practice, you will typically not be able to resolve the small space or time intervals on which you measure with a finite element mesh. That is, on length scales you can resolve, the right hand side does look singular, and consequently so does the solution. But, since you're already introducing a discretization error, you can also regularize the characteristic function of the volume over which you measure by a discrete approximation with the same weight; if you do it right, you will introduce an error that is no larger than the discretization error, at the benefit of receiving a perfectly nice right hand side function for the (discrete) dual equation.

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