  A randomized misfit approach is presented for the efficient solution of
large-scale PDE-constrained inverse problems with high-dimensional data. The
purpose of this paper is to offer a theory-based framework for random
projections in this inverse problem setting. The stochastic approximation to
the misfit is analyzed using random projection theory. By expanding beyond mean
estimator convergence, a practical characterization of randomized misfit
convergence can be achieved. The theoretical results developed hold with any
valid random projection in the literature. The class of feasible distributions
is broad yet simple to characterize compared to previous stochastic misfit
methods. This class includes very sparse random projections which provide
additional computational benefit. A different proof for a variant of the
Johnson-Lindenstrauss lemma is also provided. This leads to a different
intuition for the $O(\epsilon^{-2})$ factor in bounds for Johnson-Lindenstrauss
results. The main contribution of this paper is a theoretical result showing
the method guarantees a valid solution for small reduced misfit dimensions. The
interplay between Johnson-Lindenstrauss theory and Morozov's discrepancy
principle is shown to be essential to the result. The computational cost
savings for large-scale PDE-constrained problems with high- dimensional data is
discussed. Numerical verification of the developed theory is presented for
model problems of estimating a distributed parameter in an elliptic partial
differential equation. Results with different random projections are presented
to demonstrate the viability and accuracy of the proposed approach.
