COMPUTATIONAL AND APPLIED MATHEMATICS PROSEMINAR ARIZONA STATE UNIVERSITY

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COMPUTATIONAL AND APPLIED MATHEMATICS PROSEMINAR
ARIZONA STATE UNIVERSITY

• ON ITERATIVELY REGULARIZED NUMERICAL PROCEDURES
  FOR NONLINEAR ILL-POSED PROBLEMS

Alexandra Smirnova Dept of
Mathematics Statistics Georgia State
University E-mail asmirnova_at_gsu.edu
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STATEMENT OF THE PROBLEM
Consider a nonlinear operator equation
on a pair of Hilbert spaces, or a more general
problem of minimizing the functional
where
is a solution to the equation with exact data,
Suppose that
maybe non-unique, and the linear operator
is not boundedly invertible in a neighborhood of
the root, i.e., the problem is ill-posed
(unstable, irregular)
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EXAMPLE I
Nonlinear integral equation of the first kind
Theorem 1. If
is twice continuously differentiable wrt u on
and
then
Under the above assumptions
is a compact operator, which
cannot be boundedly invertible in an infinite
dimensional Hilbert space
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INVERSE GRAVIMETRY PROBLEM
is the unknown function, which describes the
interface
between two media of different densities
is the measured data (the gravitational anomaly)
The Frechet derivative of F(x) is given by the
formulas
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The two interfaces reconstructed from the real
gravity fields in the Ural mountains, Russia.
The reconstruction was done by the research group
of V.Vasin, Ural State University. The data is
provided by Institute of Geophysics (RAS).
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ILL-POSEDNESS OF THE PROBLEM
?2.8, 19.6 X 0.0, 7.6 (km x km)
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EXAMPLE 2

• Numerous nonlinear inverse problems in PDEs give
  rise to either irregular problems or to such
  problems whose regularity is extremely difficult
  to investigate.
• The inverse problem in optical tomography is
  exponentially ill-posed, i.e., small errors in
  the data collected at the boundary of the body
  grow exponentially fast as one proceeds in the
  interior.

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OPTICAL TOMOGRAPHY

Optical Tomography is the technique of using
light in the near-infrared range (wavelength from
700 to 1200 nm) for imaging specific parts of the
body to obtain information about tissue
abnormalities, such as breast or brain tumors.
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MATHEMATICAL MODEL
If we let O be the domain under consideration
with surface ??, the illumination of the tissue
can be modeled as the diffusion approximation
(S.R.Arridge 1999, F.Natterer and F.Wubbeling
2001)

With the associated initial
and boundary conditions
In time-independent case, the weak forward
problem corresponding to the above equation is
to find u(x) in H(O), such that for all v(x) in
H(O), the following variational equation is
satisfied,
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Forward problem given distribution of sources fj
at ??, and a vector q (Dµa) in ?, where D is
the coefficient of diffusion and µa is the
coefficient of absorption, find the photon
density function u on ??.
Inverse problem given distribution of sources
fj, and some observable data z on ??, find q
(Dµa) in ?
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MINIMIZATION PROBLEM

• In general, measurement of u(q) is not be
  possible, only some observable part Cu(q) of the
  actual state u(q) can be measured

In this abstract setting, the objective of the
inverse or parameter estimation problem is to
choose a parameter q, that minimizes the cost
functional
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ILL-POSEDNESS OF THE PROBLEM
?0,L. Diffusion approximation with constant
background, the Strum Louiville equation
The Rubin boundary condition
The inverse problem is to estimate the scalar q
from the data z measured at x 0 or x L.
The solution for
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Typical observation operator
The cost functional
Solid curve
Broken curve
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EXAMPLE 3
M.J.Feigenbaum, 1983
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Other systems, for which doubling of the period
occurs with the same constants
Hennons process
The Lorenz Equations
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Briggs, 1991
The Feigenbaum constants are computed for z
2,3,,12.
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ILL-POSEDNESS OF THE PROBLEM
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ALGORITHMS

• Consider Gauss-Newton scheme for minimizing the
  functional

generates Iteratively Regularized Gauss-Newtons
scheme first proposed by A.Bakushinsky in 1992.
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STOPPING RULES

• Source type condition

In BK05 it is shown that for
In BNS97
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GENERALIZED DISCREPANCY PRINCIPLE

• Consider arbitrary nonlinear operator F such that

Generalized discrepancy principle BS05
will guarantee
under the basic assumption that
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CONVERGENCE ANALYSIS
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CONVERGENCE RATES
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METHOD I
()
As it follows from () the largest value of p,
for which condition
holds, is p 1. Thus the convergence rate
will be the best possible (the saturation
phenomena).
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METHOD II
Then
The process is saturation free
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METHOD III
The best convergence rate for this algorithm is
This convergence rate will be achieved if
condition
satisfied with p M.
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METHOD IV

• An iteration of the basic scheme with

The process is saturation free
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METHOD V
to have
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LINE SEARCH STRATEGY

• Backtracking strategy with

until the
following two requirements are simultaneously
fulfilled
HNV00
which is the Armijo-Goldstein strategy and
which is the Wolfe type strategy.
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Feigenbaum's constants
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References