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A mollification method for ill-posed problems 总被引:3,自引:0,他引:3
Summary.
A mollification method for a class of ill-posed
problems is suggested. The idea of the method is very simple and
natural: if the data are given inexactly then we try to find a
sequence of ``mollification operators" which map the improper
data into well-posedness classes of the problem (mollify the
improper data). Within these mollified data our problem becomes
well-posed. And when these facts are in hand we try to obtain
error estimates and optimal or ``quasi-optimal" mollification
parameters. The
method is working not only for problems in Hilbert spaces, but
also for problems in Banach spaces. Applications of the
method to concrete problems, like numerical differentiation,
parabolic equations backwards in time, the Cauchy problem for the
Laplace equation,
one- and multidimensional non-characteristic Cauchy problems for
parabolic equations (in infinite or finite domains),... give us
very sharp stability estimates of H\"older continuous type. In
these cases the method is optimal in the sense that it gives the
same order of H\"older continuous dependence on the data as for
the regularized problems. Furthermore, the method may be
implemented numerically using fast Fourier transforms. For the
first time a uniform stability estimate of H\"older continuous
type of the solution of the heat equation backwards in time in the
space for all could
be established by our mollification method. A new simple sharp
pointwise estimate of H\"older type for the weak solution of a
non-characteristic Cauchy problem for parabolic equations in a
finite domain is established.
Received June 25, 1993 / Revised version received February
18, 1994 相似文献
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