Elliptical pore regularisation of the inverse problem for microstructured optical fibre fabrication

2015 ◽  
Vol 778 ◽  
pp. 5-38 ◽  
Author(s):  
Peter Buchak ◽  
Darren G. Crowdy ◽  
Yvonne M. Stokes ◽  
Heike Ebendorff-Heidepriem
Keyword(s):  
Inverse Problem ◽  
Broad Class ◽  
Outer Boundary ◽  
Coupled System ◽  
Pore Model ◽  
Holey Fibres ◽  
Experimental Parameters ◽  
Axial Variation ◽  
Ill Posed ◽  
Microstructured Optical Fibres

A mathematical model is presented describing the deformation, under the combined effects of surface tension and draw tension, of an array of channels in the drawing of a broad class of slender viscous fibres. The process is relevant to the fabrication of microstructured optical fibres, also known as MOFs or holey fibres, where the pattern of channels in the fibre plays a crucial role in guiding light along it. Our model makes use of two asymptotic approximations, that the fibre is slender and that the cross-section of the fibre is a circular disc with well-separated elliptical channels that are not too close to the outer boundary. The latter assumption allows us to make use of a suitably generalised ‘elliptical pore model (EPM)’ introduced previously by one of the authors (Crowdy, J. Fluid Mech., vol. 501, 2004, pp. 251–277) to quantify the axial variation of the geometry during a steady-state draw. The accuracy of the elliptical pore model as an approximation is tested by comparison with full numerical simulations. Our model provides a fast and accurate reduction of the full free-boundary problem to a coupled system of nonlinear ordinary differential equations. More significantly, it also allows a regularisation of an important ill-posed inverse problem in MOF fabrication: how to find the initial preform geometry and the experimental parameters required to draw MOFs with desired cross-plane geometries.

scholarly journals Bayesian Reconstruction through Adaptive Image Notion

Proceedings ◽  
2019 ◽  
Vol 33 (1) ◽  
pp. 21
Author(s):  
Fabrizia Guglielmetti ◽  
Eric Villard ◽  
Ed Fomalont
Keyword(s):  
Inverse Problem ◽  
Image Plane ◽  
Source Distribution ◽  
Bayesian Probability ◽  
Model Parameters ◽  
Uncertain Information ◽  
Source Detection ◽  
Component Mixture ◽  
Ill Posed ◽  
Image Planes

A stable and unique solution to the ill-posed inverse problem in radio synthesis image analysis is sought employing Bayesian probability theory combined with a probabilistic two-component mixture model. The solution of the ill-posed inverse problem is given by inferring the values of model parameters defined to describe completely the physical system arised by the data. The analysed data are calibrated visibilities, Fourier transformed from the ( u , v ) to image planes. Adaptive splines are explored to model the cumbersome background model corrupted by the largely varying dirty beam in the image plane. The de-convolution process of the dirty image from the dirty beam is tackled in probability space. Probability maps in source detection at several resolution values quantify the acquired knowledge on the celestial source distribution from a given state of information. The information available are data constrains, prior knowledge and uncertain information. The novel algorithm has the aim to provide an alternative imaging task for the use of the Atacama Large Millimeter/Submillimeter Array (ALMA) in support of the widely used Common Astronomy Software Applications (CASA) enhancing the capabilities in source detection.

ON THE WAY TOWARDS INCREASING THE PREDICTIVE POWER OF THE NUCLEAR MEAN FIELD THEORIES: EVALUATION OF TWO-BODY MATRIX ELEMENTS

2012 ◽  
Vol 21 (05) ◽  
pp. 1250037
Author(s):  
HERVÉ MOLIQUE ◽  
JERZY DUDEK
Keyword(s):  
Inverse Problem ◽  
Predictive Power ◽  
Mean Field ◽  
Field Theories ◽  
Matrix Elements ◽  
The Matrix ◽  
Ill Posed ◽  
Technical Issues ◽  
Numerical Precision ◽  
Mean Field Theories

In this paper we collect a number of technical issues that arise when constructing the matrix representation of the most general nuclear mean field Hamiltonian within which "all terms allowed by general symmetries are considered not only in principle but also in practice". Such a general posing of the problem is necessary when investigating the predictive power of the mean field theories by means of the well-posed inverse problem. [J. Dudek et al., Int. J. Mod. Phys. E21 (2012) 1250053]. To our knowledge quite often ill-posed mean field inverse problems arise in practical realizations what makes reliable extrapolations into the unknown areas of nuclei impossible. The conceptual and technical issues related to the inverse problem have been discussed in the above-mentioned topic whereas here we focus on "how to calculate the matrix elements, fast and with high numerical precision when solving the inverse problem" [For space-limitation reasons we illustrate the principal techniques on the example of the central interactions].

scholarly journals Regularization of a sideways problem for a time-fractional diffusion equation with nonlinear source

2020 ◽  
Vol 28 (2) ◽  
pp. 211-235
Author(s):  
Tran Bao Ngoc ◽  
Nguyen Huy Tuan ◽  
Mokhtar Kirane
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Regularization Method ◽  
A Priori ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Nonlinear Source ◽  
Numerical Example ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation

AbstractIn this paper, we consider an inverse problem for a time-fractional diffusion equation with a nonlinear source. We prove that the considered problem is ill-posed, i.e., the solution does not depend continuously on the data. The problem is ill-posed in the sense of Hadamard. Under some weak a priori assumptions on the sought solution, we propose a new regularization method for stabilizing the ill-posed problem. We also provide a numerical example to illustrate our results.

On the identification of a single body immersed in a Navier-Stokes fluid

2007 ◽  
Vol 18 (1) ◽  
pp. 57-80 ◽  
Author(s):  
A. DOUBOVA ◽  
E. FERNÁNDEZ-CARA ◽  
J. H. ORTEGA
Keyword(s):  
Inverse Problem ◽  
Rigid Body ◽  
Stokes Equations ◽  
Outer Boundary ◽  
Unique Continuation ◽  
Uniqueness Result ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Friction Forces ◽  
Regularity Results

In this work we consider the inverse problem of the identification of a single rigid body immersed in a fluid governed by the stationary Navier-Stokes equations. It is assumed that friction forces are known on a part of the outer boundary. We first prove a uniqueness result. Then, we establish a formula for the observed friction forces, at first order, in terms of the deformation of the rigid body. In some particular situations, this provides a strategy that could be used to compute approximations to the solution of the inverse problem. In the proofs we use unique continuation and regularity results for the Navier-Stokes equations and domain variation techniques.

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