Spelling suggestions: "subject:"illposedness"" "subject:"wellposedness""
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On use of inhomogeneous media for elimination of ill-posedness in the inverse problemFeroj, Md Jamil 17 April 2014 (has links)
This thesis outlines a novel approach to make ill-posed inverse source problem well-posed exploiting inhomogeneous media. More precisely, we use Maxwell fish-eye lens to make scattered field emanating from distinct regions of an object of interest more directive and concentrated onto distinct regions of observation. The object of interest in this thesis is a thin slab placed conformally to the Maxwell fish-eye lens. Focused Green’s function of the background medium results in diagonal dominance of the matrix to be inverted for inverse problem solution. Hence, the problem becomes well-posed. We have studied one-dimensional variation of a very thin dielectric slab of interest having conformal shape to the lens. This method has been tested solving the forward problem using both Mie series and using COMSOL.
Most common techniques for solving inverse problem are full non-linear inversion techniques, such as: distorted Born iterative method (DBIM) and contrast source inversion (CSI). DBIM needs to be regularized at every iteration. In some cases, it converges to a solution, and, in some cases, it does not. Diffraction tomography does not utilize regularization. It is a technique under Born approximation. It eliminates ill-posedness, but it works only for small contrast. Our proposed method works for high contrast and also provides well-posedness.
In this thesis, our objective is to demonstrate inverse source problem and inverse scattering problem are not inherently ill-posed. They are ill-posed because conventional techniques usually use homogeneous or non-focusing background medium. These mediums do not support separation of scattered field. Utilization of background medium for scattered field separation casts the inverse problem in well-posed form.
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An efficient method for an ill-posed problem [three dashes]band-limited extrapolation by regularizationChen, Weidong January 1900 (has links)
Doctor of Philosophy / Department of Mathematics / Robert B. Burckel / In this paper a regularized spectral estimation formula and a regularized iterative algorithm
for band-limited extrapolation are presented. The ill-posedness of the problem is taken into account. First
a Fredholm equation is regularized. Then it is transformed to a differential equation in the
case where the time interval is R. A fast algorithm to solve the differential equation
by the finite differences is given and a regularized spectral estimation formula is obtained.
Then a regularized iterative extrapolation algorithm is introduced and compared with the
Papoulis and Gerchberg algorithm. A time-frequency regularized extrapolation algorithm
is presented in the two-dimensional case. The Gibbs phenomenon is analyzed.
Then the time-frequency regularized extrapolation algorithm is applied to image restoration
and compared with other algorithms.
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Local Ill-Posedness and Source Conditions of Operator Equations in Hilbert SpacesHofmann, B., Scherzer, O. 30 October 1998 (has links) (PDF)
The characterization of the local ill-posedness and the local degree of nonlinearity are of particular importance for the stable solution of nonlinear ill-posed problems. We present assertions concerning the interdependence between the ill-posedness of the nonlinear problem and its linearization. Moreover, we show that the concept of the degree of nonlinearity com bined with source conditions can be used to characterize the local ill-posedness and to derive a posteriori estimates for nonlinear ill-posed problems. A posteriori estimates are widely used in finite element and multigrid methods for the solution of nonlinear partial differential equations, but these techniques are in general not applicable to inverse an ill-posed problems. Additionally we show for the well-known Landweber method and the iteratively regularized Gauss-Newton method that they satisfy a posteriori estimates under source conditions; this can be used to prove convergence rates results.
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Ill-posedness of parameter estimation in jump diffusion processesDüvelmeyer, Dana, Hofmann, Bernd 25 August 2004 (has links) (PDF)
In this paper, we consider as an inverse problem the simultaneous estimation
of the five parameters of a jump diffusion process from return observations of a
price trajectory. We show that there occur some ill-posedness phenomena in the
parameter estimation problem, because the forward operator fails to be injective
and small perturbations in the data may lead to large changes in the solution. We
illustrate the instability effect by a numerical case study. To overcome the difficulty
coming from ill-posedness we use a multi-parameter regularization approach that
finds a trade-off between a least-squares approach based on empircal densities and
a fitting of semi-invariants. In this context, a fixed point iteration is proposed that
provides good results for the example under consideration in the case study.
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Some stability results of parameter identification in a jump diffusion modelDüvelmeyer, Dana 06 October 2005 (has links) (PDF)
In this paper we discuss the stable solvability of the inverse problem of parameter identification in a jump diffusion model. Therefore we introduce the forward
operator of this inverse problem and analyze its properties. We show continuity of
the forward operator and stability of the inverse problem provided that the domain
is restricted in a specific manner such that techniques of compact sets can be exploited. Furthermore, we show that there is an asymptotical non-injectivity which
causes instability problems whenever the jump intensity increases and the jump
heights decay simultaneously.
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On multiplication operators occurring in inverse problems of natural sciences and stochastic financeHofmann, Bernd 07 October 2005 (has links) (PDF)
We deal with locally ill-posed nonlinear operator equations F(x) = y in L^2(0,1),
where the Fréchet derivatives A = F'(x_0) of the nonlinear forward operator F are
compact linear integral operators A = M ◦ J with a multiplication operator M
with integrable multiplier function m and with the simple integration operator J.
In particular, we give examples of nonlinear inverse problems in natural sciences
and stochastic finance that can be written in such a form with linearizations that
contain multiplication operators. Moreover, we consider the corresponding ill-posed
linear operator equations Ax = y and their degree of ill-posedness. In particular,
we discuss the fact that the noncompact multiplication operator M has only a
restricted influence on this degree of ill-posedness even if m has essential zeros of
various order.
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New results on the degree of ill-posedness for integration operators with weightsHofmann, Bernd, von Wolfersdorf, Lothar 16 May 2008 (has links) (PDF)
We extend our results on the degree of ill-posedness for linear integration opera-
tors A with weights mapping in the Hilbert space L^2(0,1), which were published in
the journal 'Inverse Problems' in 2005 ([5]). Now we can prove that the degree one
also holds for a family of exponential weight functions. In this context, we empha-
size that for integration operators with outer weights the use of the operator AA^*
is more appropriate for the analysis of eigenvalue problems and the corresponding
asymptotics of singular values than the former use of A^*A.
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About an autoconvolution problem arising in ultrashort laser pulse characterizationBürger, Steven 03 November 2014 (has links) (PDF)
We are investigating a kernel-based autoconvolution problem, which has its origin in the physics of ultra short laser pulses. The task in this problem is to reconstruct a complex-valued function $x$ on a finite interval from measurements of its absolute value and a kernel-based autoconvolution of the form [[F(x)](s)=int k(s,t)x(s-t)x(t)de t.]
This problem has not been studied in the literature. One reason might be that one has more information than in the classical autoconvolution case, where only the right hand side is available. Nevertheless we show that ill posedness phenomena may occur. We also propose an algorithm to solve the problem numerically and demonstrate its performance with artificial data. Since the algorithm fails to produce good results with real data and we suspect that the data for $|F(x)|$ are not dependable we also consider the whole problem with only $arg(F(x))$ given instead of $F(x)$.
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Ill-posedness of parameter estimation in jump diffusion processesDüvelmeyer, Dana, Hofmann, Bernd 25 August 2004 (has links)
In this paper, we consider as an inverse problem the simultaneous estimation
of the five parameters of a jump diffusion process from return observations of a
price trajectory. We show that there occur some ill-posedness phenomena in the
parameter estimation problem, because the forward operator fails to be injective
and small perturbations in the data may lead to large changes in the solution. We
illustrate the instability effect by a numerical case study. To overcome the difficulty
coming from ill-posedness we use a multi-parameter regularization approach that
finds a trade-off between a least-squares approach based on empircal densities and
a fitting of semi-invariants. In this context, a fixed point iteration is proposed that
provides good results for the example under consideration in the case study.
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On Ill-Posedness and Local Ill-Posedness of Operator Equations in Hilbert SpacesHofmann, B. 30 October 1998 (has links) (PDF)
In this paper, we study ill-posedness concepts of nonlinear and linear inverse problems
in a Hilbert space setting. We define local ill-posedness of a nonlinear operator
equation $F(x) = y_0$ in a solution point $x_0$ and the interplay between the nonlinear
problem and its linearization using the Frechet derivative $F\acent(x_0)$ . To find an
appropriate ill-posedness concept for the linarized equation we define intrinsic
ill-posedness for linear operator equations $Ax = y$ and compare this approach with
the ill-posedness definitions due to Hadamard and Nashed.
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