MC 6496

## Speaker

Dr.
Davide
La
Torre

University
of
Milan,
Italy
and Khalifa
University,
Abu
Dhabi,
UAE

## Title

Inverse Problems for Differential Equations via the "Collage Theorem”: Recent Results and Future Directions

## Abstract

Many
inverse
problems
may
be
recast
as
the
approximation
of
a
target
element
x
in
a
complete
metric
space
(X,d)
by
the
fixed
point
x*
of
a
contraction
mapping
T:X->X.
Thanks
to
a
simple
consequence
of
Banach's
Fixed
Point
Theorem
known
as
the
**
Collage
Theorem**,
most
practical
methods
of
solving
the
inverse
problem
for
fixed
point
equations
seek
an
operator T
for
which
the
collage
distance d(x,Tx)
is
as
small
as
possible.
This
is
the
essence
of
the
method
of
collage
coding
which
has
been
the
basis
of
most,
if
not
all,
fractal
image
coding
and
compression
methods
[11].

This method may be applied in other situations where contractive mappings are encountered: These ideas have been extended to inverse problems for Initial Value Problems [6], [8], [10], [14],[15].

In
a
manner
analogous
to
the
Collage
Theorem,
the ** Generalized
Collage
Theorem** has
been
formulated
for
solving
Boundary
Value
Problems
[7],
[12],[13],
replacing
the
minimization
of
the
true
error
by the
minimization
of
something
akin
to
the
collage
distance.
In
place
of
Banach's
fixed
point
theorem
for
contraction
maps
on
a
complete
metric
space,
we
have
appealed
to
the
Lax-Milgram
representation
theorem.

These results have been recently extended to a wider class of partial differential equations [3],[9] and boundary valued problems by considering not only Hilbert but also reflexive Banach spaces, and even replacing the primal variational formulation of such a problem with a more general constrained variational one [1],[5]. Other recent extensions of the Generalized Collage Theorem deal with inverse problems for vector-valued Lax-Milgram systems [2] and inverse problems on perforated domains [4].

**
Main
references:**

[1] M.I. Berenguer, H. Kunze, D. La Torre, M. Ruiz Galan (2016) Galerkin method for constrained variational equations and a collage-based approach to related inverse problems, Journal of Computational and Applied Mathematics, 292, 67–75.

[2] H. Kunze, D. La Torre, K. Levere, M. Ruiz Galan (2015) Inverse problems via the ”Generalized Collage Theorem“ for vector-valued Lax Milgram-based variational problems (2015), Mathematical Problems in Engineering, Volume 2015, Article ID 764643, 8 pages.

[3] K.Levere, H.Kunze, D. La Torre (2015), A collage-based approach to solving inverse problems for second-order nonlinear hyperbolic PDEs, Communications in Nonlinear Science and Numerical Simulations, 29 (1–3), 283–299.

[4] H.Kunze, D. La Torre (2015), Collage-type approach to inverse problems for elliptic PDEs on perforated domains (2015), Electronic Journal of Differential Equations, 48, 1-11.

[5] M.I. Berenguer, H. Kunze, D. La Torre, M. Ruiz Galan (2015), Galerkin schemes and inverse boundary value problems in reflexive Banach spaces (2015), Journal of Computational and Applied Mathematics, 275, 100-112.

[6] H.Kunze, D. La Torre, F.Mendivil, M.Ruiz Galan, R.Zaki (2014), Fractal-based methods and inverse problems for differential equations: current state-of-the-art (2014), Mathematical Problems in Engineering, Volume 2014, Article ID 737694, 11 pages.

[7] V.Capasso, H.Kunze, D. La Torre, E.R.Vrscay (2014), Solving inverse problems for differential equations by a ``generalized collage'' method and application to a mean field stochastic model (2014), Nonlinear Analysis: Real World Applications. 15, 276-289.

[8] V.Capasso, H.Kunze, D. La Torre, E.R.Vrscay (2013), Solving inverse problems for biological models using the collage method for differential equations, Journal of Mathematical Biology, 67, 1, 25-38.

[9] K.Levere, H.Kunze, D. La Torre, A collage-based approach to solving inverse problems for second-order nonlinear parabolic PDEs (2013), Journal of Mathematical Analysis and Applications, 406, 1, 120-133.

[10] H.Kunze, D. La Torre, E.R.Vrscay (2012), Solving inverse problems for DEs using the collage theorem and entropy maximization, Applied Mathematics Letters, 25, 2306-2311.

[11] H.Kunze, D. La Torre, F.Mendivil, E.R.Vrscay (2012), Fractal-based methods in analysis, Springer, ISBN 1461418909.

[12] H.Kunze, D. La Torre, E.R. Vrscay (2010), Solving inverse problems for variational equations using "generalized collage methods," with applications to boundary value problems, (2010), Nonlinear Analysis Real World Applications, 11 (5), 3734-3743.

[13] H.Kunze, D. La Torre, E.R. Vrscay (2009), A generalized collage method based upon the Lax-Milgram functional for solving boundary value inverse problems, Nonlinear Analysis: Theory, Methods \& Applications, 71 (12), e1337-e1343.

[14] H.Kunze, D. La Torre, E.R. Vrscay (2009), Inverse problems for random differential equations using the collage method for random contraction mappings, Journal of Computational and Applied Mathematics, 223 (2), 853-861.

[15] H.Kunze, D. La Torre, E.R. Vrscay (2007), Random fixed point equations and inverse problems using "the collage method" for contraction mappings, Journal of Mathematical Analysis and Applications, 334 (2), 1116--1129.