Inverse problems for nonlocal / nonlinear PDEs
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Organizer(s): |
Name:
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Affiliation:
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Country:
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Barbara Kaltenbacher
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University of Klagenfurt
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Austria
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William Rundell
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Texas A&M University
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USA
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Introduction:
| | Fractional operators as components of partial differential equations have
been studied since the 1950’s. In the late 1960s it was realised that the use
of fractional order derivative damping terms in the wave equation restored a
dependence on frequency; a missing requirement from physical observations
that is not possible with integer order derivatives. Also dating from this
period was the concept of fractional powers of partial differential operators.
All such fractional operators are nonlocal; the value at a point depends
also on values in a domain that includes that point as opposed to the pure
pointwise situation of integer order derivatives. This paradigm has enormous
implications in modelling but in particular for inverse problems involving
such operators, especially those for which the "usual" inverse problem with
integer order derivatives is severely ill-conditioned. Due to work over the
last decade there are now many known examples of both fractional space
operators and fractional time operators where such a “history effect” reduces
the ill-conditioning significantly leading to much more tractable inversions.
However, there is often a price to be paid and the cost is in a more difficult
analysis due to the absence of some classical tools. This also shows up in the
difficulty of proving uniqueness of the inversion map.
The purpose of the special session is to bring together people working on
different aspects of this topic. |
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