OVERCOMING HOLDER CONTINUITY IN ILL-POSED CONTINUATION PROBLEMS

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 parallel-to u1(t) - u2(t) parallel-to less-than-or-equal-to 2M1-t epsilon(t), 0 less-than-or-equal-to t less-than-or-equal-to 1, for the difference of any two continuations, where t is the continuation variable, M is an a priori bound on parallel-to u(0) parallel-to, and epsilon is an upper bound on the norm of the error in the continuation data at t = 1. For small t > 0, such Holder-continuous dependence on the data is not useful at the levels of data error epsilon typically found in practice, and noise contamination as t down 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 parallel-to u1(t) - u2(t) parallel-to less-than-or-equal-to 2GAMMA1-t epsilon, 0 less-than-or-equal-to t less-than-or-equal-to 1, with GAMMA much-less-than 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 L2(R2) is constructed for which the SECB deblurred image is the unique minimizer. 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 x 512 image, SECB restoration requires about 20 seconds of cpu time on current desktop workstations. To illustrate the theory, a sharp 512 x 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.