By Aaron Cinzori and Patricia Lamm
Department of Mathematics
Meadville, PA 16335
We examine a new discrete method for regularizing ill-posed Volterra
problems. Unlike many classical regularization techniques (such as
Tikhonov regularization), this method preserves the Volterra (causal)
nature of the problem allowing the regularized solution to be produced
sequentially in near real time as data arrives. We analyze the method
and for an important class of Volterra problems, prove that
regularized solutions converge to the true solution at the best
possible rate with respect to error in the data.
In fact, the future polynomial regularization method discussed here may be applied to quite general operator equations provided that the operator may be discretized by a lower-triangular matrix of Toeplitz type. This enlarges the class of operator equations that may be approximated using the method, but also introduces degenerate situations in which the future polynomial method is no more regularizing than an ordinary discretization method. We characterize these degenerate cases and argue that we are unlikely to see them for the problems of interest here. In particular, such degeneracies cannot occur for the class of Volterra problems for which we are able to prove the future polynomial method converges.
Finally we present numerical evidence that this method works well in the recovery of sharp and discontinuous features in the true solution, features that can be oversmoothed by classical regularization techniques.
2nd Forum On Numerics & Modeling for
Partial Differential Equations
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