165 research outputs found
Absorbing boundary conditions for the Westervelt equation
The focus of this work is on the construction of a family of nonlinear
absorbing boundary conditions for the Westervelt equation in one and two space
dimensions. The principal ingredient used in the design of such conditions is
pseudo-differential calculus. This approach enables to develop high order
boundary conditions in a consistent way which are typically more accurate than
their low order analogs. Under the hypothesis of small initial data, we
establish local well-posedness for the Westervelt equation with the absorbing
boundary conditions. The performed numerical experiments illustrate the
efficiency of the proposed boundary conditions for different regimes of wave
propagation
A modified time-of-flight method for precise determination of high speed ratios in molecular beams
Time-of-flight (TOF) is a standard experimental technique for determining, among others, the speed ratio S (velocity spread) of a molecular beam. The speed ratio is a measure for the monochromaticity of the beam and an accurate determination of S is crucial for various applications, for example, for characterising chromatic aberrations in focussing experiments related to helium microscopy or for precise measurements of surface phonons and surface structures in molecular beam scattering experiments. For both of these applications, it is desirable to have as high a speed ratio as possible. Molecular beam TOF measurements are typically performed by chopping the beam using a rotating chopper with one or more slit openings. The TOF spectra are evaluated using a standard deconvolution method. However, for higher speed ratios, this method is very sensitive to errors related to the determination of the slit width and the beam diameter. The exact sensitivity depends on the beam diameter, the number of slits, the chopper radius, and the chopper rotation frequency. We present a modified method suitable for the evaluation of TOF measurements of high speed ratio beams. The modified method is based on a systematic variation of the chopper convolution parameters so that a set of independent measurements that can be fitted with an appropriate function are obtained. We show that with this modified method, it is possible to reduce the error by typically one order of magnitude compared to the standard method
Local Analysis of Inverse Problems: H\"{o}lder Stability and Iterative Reconstruction
We consider a class of inverse problems defined by a nonlinear map from
parameter or model functions to the data. We assume that solutions exist. The
space of model functions is a Banach space which is smooth and uniformly
convex; however, the data space can be an arbitrary Banach space. We study
sequences of parameter functions generated by a nonlinear Landweber iteration
and conditions under which these strongly converge, locally, to the solutions
within an appropriate distance. We express the conditions for convergence in
terms of H\"{o}lder stability of the inverse maps, which ties naturally to the
analysis of inverse problems
The Iteratively Regularized Gau{\ss}-Newton Method with Convex Constraints and Applications in 4Pi-Microscopy
This paper is concerned with the numerical solution of nonlinear ill-posed
operator equations involving convex constraints. We study a Newton-type method
which consists in applying linear Tikhonov regularization with convex
constraints to the Newton equations in each iteration step. Convergence of this
iterative regularization method is analyzed if both the operator and the right
hand side are given with errors and all error levels tend to zero. Our study
has been motivated by the joint estimation of object and phase in 4Pi
microscopy, which leads to a semi-blind deconvolution problem with
nonnegativity constraints. The performance of the proposed algorithm is
illustrated both for simulated and for three-dimensional experimental data
Parameter identification in a semilinear hyperbolic system
We consider the identification of a nonlinear friction law in a
one-dimensional damped wave equation from additional boundary measurements.
Well-posedness of the governing semilinear hyperbolic system is established via
semigroup theory and contraction arguments. We then investigte the inverse
problem of recovering the unknown nonlinear damping law from additional
boundary measurements of the pressure drop along the pipe. This coefficient
inverse problem is shown to be ill-posed and a variational regularization
method is considered for its stable solution. We prove existence of minimizers
for the Tikhonov functional and discuss the convergence of the regularized
solutions under an approximate source condition. The meaning of this condition
and some arguments for its validity are discussed in detail and numerical
results are presented for illustration of the theoretical findings
Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data
We study Newton type methods for inverse problems described by nonlinear
operator equations in Banach spaces where the Newton equations
are regularized variationally using a general
data misfit functional and a convex regularization term. This generalizes the
well-known iteratively regularized Gauss-Newton method (IRGNM). We prove
convergence and convergence rates as the noise level tends to 0 both for an a
priori stopping rule and for a Lepski{\u\i}-type a posteriori stopping rule.
Our analysis includes previous order optimal convergence rate results for the
IRGNM as special cases. The main focus of this paper is on inverse problems
with Poisson data where the natural data misfit functional is given by the
Kullback-Leibler divergence. Two examples of such problems are discussed in
detail: an inverse obstacle scattering problem with amplitude data of the
far-field pattern and a phase retrieval problem. The performence of the
proposed method for these problems is illustrated in numerical examples
Building the impedance model of a real machine
A reliable impedance model of a particle accelerator can be built by combining the beam coupling impedances of all the components. This is a necessary step to be able to evaluate the machine performance limitations, identify the main contributors in case an impedance reduction is required, and study the interaction with other mechanisms such as optics nonlinearities, transverse damper, noise, space charge, electron cloud, beam-beam (in a collider).
The main phases to create a realistic impedance model, and verify it experimentally, will be reviewed, highlighting the main challenges. Some examples will be presented revealing the levels of precision of machine impedance models that have been achieved
Well-posedness for degenerate third order equations with delay and applications to inverse problems
[EN] In this paper, we study well-posedness for the following third-order in time equation with delay <disp-formula idoperators defined on a Banach space X with domains D(A) and D(B) such that t)is the state function taking values in X and u(t): (-, 0] X defined as u(t)() = u(t+) for < 0 belongs to an appropriate phase space where F and G are bounded linear operators. Using operator-valued Fourier multiplier techniques we provide optimal conditions for well-posedness of equation (0.1) in periodic Lebesgue-Bochner spaces Lp(T,X), periodic Besov spaces Bp,qs(T,X) and periodic Triebel-Lizorkin spaces Fp,qs(T,X). A novel application to an inverse problem is given.The first, second and third authors have been supported by MEC, grant MTM2016-75963-P. The second author has been supported by AICO/2016/30. The fourth author has been supported by MEC, grant MTM2015-65825-P.Conejero, JA.; Lizama, C.; Murillo-Arcila, M.; Seoane SepĂșlveda, JB. (2019). 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Joint multi-field T1 quantification for fast field-cycling MRI
Acknowledgment This article is based upon work from COST Action CA15209, supported by COST (European Cooperation in Science and Technology). Oliver Maier is a Recipient of a DOC Fellowship (24966) of the Austrian Academy of Sciences at the Institute of Medical Engineering at TU Graz. The authors would like to acknowledge the NVIDIA Corporation Hardware grant support.Peer reviewedPublisher PD
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