Overcoming Hölder Continuity in Ill-Posed Continuation Problems

Abstract

Many ill-posed continuation problems in partial differential equations obey a logarithmic convexity inequality and can be stabilized in an appropriate Banach space by imposing an a priori bound on the solutions. In the simplest cases, such an inequality leads to the sharp stability estimate $\| {u_1 (t) - u_2 (t)} \| \leq 2M^{1 - t} \epsilon ^t , \quad 0 \leq t \leq 1$, for the difference of any two continuations, where t is the continuation variable, M is an a priori bound on $\| {u(0)} \|$, and $\epsilon $ is an upper bound on the norm of the error in the continuation data at $t = 1$. For small $t > 0$, such Hölder-continuous dependence on the data is not useful at the levels of data error $\epsilon $ typically found in practice, and noise contamination as $t \downarrow 0$ is a characteristic feature of many stabilized ill-posed computations. The present paper analyzes the effects of prescribing a physically motivated supplementary constraint, the so-called slow evolution from the continuation boundary (SECB) constraint. When the SECB constraint is applicable, there results the improved stability estimate $\| {u_1 (t) - u_2 (t)} \| \leq 2\Gamma ^{1 - t} \varepsilon ,\, 0 \leq t \leq1$, with $\Gamma \ll {M/\epsilon }$ typically. This theoretical result is valid for a large class of ill-posed continuation problems. The computational significance of this result is demonstrated in the latter half of the paper. An important class of image deblurring problems is reformulated as a backwards-in-time continuation problem for a generalized diffusion equation. A quadratic functional on $L^2 (R^2 )$ is constructed for which the SECB deblurred image is the unique minimizes. An explicit formula is then obtained for SECB restoration in the Fourier transform domain, leading to a fast, practical, numerical restoration procedure involving fast Fourier transform (FFT) algorithms. For a $512 \times 512$ image, SECB restoration requires about 20 seconds of cpu time on current desktop workstations. To illustrate the theory, a sharp $512 \times 512$ image is artificially blurred in the presence of noise. The blurred noisy image is then deblurred using the SECB method, as well as the Tikhonov–Miller, Backward Beam, and L-curve methods. Based on qualitative and quantitative comparisons between the four deblurring procedures, it is verified that the SECB constraint sharply reduces noise contamination.