Example Of Ill-posed Problem Research Articles

Ill-posed problems in mechanics

The notion of ill-posed initial and boundary value problems for partial differential equations was introduced by Hadamard, who also presented the first example of an ill-posed problem for a specific partial differential equation. At the same time, there are numerous examples of ill-posed problems in any field of mechanics.

Software for retrieval of aerosol particle size distribution from multiwavelength lidar signals

Software to retrieve profiles of aerosol particle size distribution (APSD) from multiwavelength lidar signals is presented. The approach consists in direct fit of artificial signal generated using predefined distribution to the experimental signals. Combination of two lognormal functions with a few free parameters is applied for the predefined APSD. The minimization technique allows finding lognormal function parameters which provide the best fit. The approach was tested on the experimental signals registered at 1064, 532 and 355 nm. The software is designated for processing on PCs. The computation time was about several minutes. Program summaryProgram title: APSD_SoftwareCatalogue identifier: AEXV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEXV_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Open LicenceNo. of lines in distributed program, including test data, etc.: 12813No. of bytes in distributed program, including test data, etc.: 878099Distribution format: tar.gzProgramming language: Delphi 2010.Computer: PC.Operating system: Windows XP, Vista, 7, 8, 8.1.RAM: 37MBClassification: 13.Nature of problem:Aerosol Particle Size Distribution (APSD) is a very significant function for evaluation of atmospheric optical properties. Remote determination of APSD might be performed with multiwavelength lidar. Retrieval of profiles of APSD from multiwavelength lidar signals is an example of ill-posed problem in the atmospheric physics (in the sense of Jacques Hadamard).Solution method:The approach consists in direct fit of artificial signals to the experimental signals. The artificial signals are generated using predefined aerosol particle distribution, Combination of two lognormal functions with a few free parameters is applied for the predefined APSD. The minimization technique used in the software allows finding lognormal function parameters which provide the best fit.Additional comments:The input signal should be located into the SIGNALS directory which is automatically created by the software. The results are presented in the main form of the software. They are also saved into the same directory from which it was loaded input file. They are saved as a BitMap as well as the ASCII files. The software returns two main sets of files: first one “APSD.bmp” and “APSD_Iet.txt”, and the second one “EffectiveRadius.bmp” and “EffectiveRadius.txt”.Running time:195 s for an input file consisting 12 measurement points (altitudes).

Fredholm Integral Equations of the First Kind and Topological Information Theory

The Fredholm integral equations of the first kind are a classical example of ill-posed problem in the sense of Hadamard. If the integral operator is self-adjoint and admits a set of eigenfunctions, then a formal solution can be written in terms of eigenfunction expansions. One of the possible methods of regularization consists in truncating this formal expansion after restricting the class of admissible solutions through a-priori global bounds. In this paper we reconsider various possible methods of truncation from the viewpoint of the $\varepsilon$-coverings of compact sets.

Hybrid neuro-genetic based method for solving ill-posed inverse problem occurring in synthesis of electromagnetic fields

Fredholm Integral Equation of the First Kind (FIEFK) is an example of ill-posed problems. Solving this type of equation using conventional methods of discretization often leads to an ill-conditioned system of linear equations. This paper deals with the numerical solution for the FIEFK occurring in the synthesis of the electromagnetic fields. To tackle this problem, we propose a hybrid method based on Genetic Algorithms (GAs) and Artificial Neural Networks. The method consists of two major steps. The first step is to find an initial solution by utilizing a GA, and the second is to refine the solution using a regularized neural network. Experimental results prove the efficiency of our proposed method in comparison with a previous work.

Regularization of the backward-in-time Kuramoto–Sivashinsky equation

We are interested in solution techniques for backward-in-time evolutionary PDE problems arising in fluid mechanics. In addition to their intrinsic interest, such techniques have applications in the recently proposed retrograde data assimilation. As our model system we consider the terminal value problem for the Kuramoto–Sivashinsky equation in a 1D periodic domain. Such backward problems are typical examples of ill-posed problems, where any disturbances are amplified exponentially during the backward march. Hence, regularization is required in order to solve such a problem efficiently in practice. We consider regularization approaches in which the original ill-posed problem is approximated with a less ill-posed problem obtained by adding a regularization term to the original equation. While such techniques are relatively well understood for simple linear problems, in this work we investigate them carefully in the nonlinear setting and report on some interesting universal behavior. In addition to considering regularization terms with fixed magnitudes, we also mention briefly a novel approach in which these magnitudes are adapted dynamically using simple concepts from the Control Theory.

On the uniqueness of optical images and solutions of inverse lithographical problems

We analyze the uniqueness of the solutions of inverse lithographical problems, stated as optimization problems, and optical images. By considering a band-limitedness argument, examples of ill-posed problems are constructed with multiple solutions in the domain of real non-negative passive masks. We conclude that the necessary condition for the solution m to be unique is to touch (or to pass infinitely close to) the boundary of the constraint 0m1. In the domain of binary masks, we propose a procedure to construct two nonunique solutions from a possibly unique one. We prove the existence of unique binary solutions in some class of optical systems and suggest a thresholding procedure to generate a unique solution from a possibly nonunique one.

Lattice-free finite difference method for numerical solution of inverse heat conduction problem

Inverse heat conduction problem consists of finding an initial temperature distribution from the knowledge of a distribution of the temperature at the present time. Here, we assume that the associated boundary conditions are known. The heat conduction problem backward in time is a typical example of ill-posed problems in the sense that the solution exists only for regular functions of some kind describing the present temperature distribution and also the solution is unstable for the present temperature distribution function. Conventional numerical methods often suffer from instability of the problem itself when high accuracy is intended in the approximation. Our aim is to create a meshless method which is applicable to the ill-posed inverse heat conduction problem. We construct a high order finite difference method in which quadrature points do not need to have a lattice structure. In order to develop our new method we show a tool in using exponential functions in Taylor's expansion. From numerical experiments we confirmed that our method is effective for solving two-dimensional inverse heat conduction problem numerically subject to mixed boundary conditions.

Lattice-free Finite Difference Method ForBackward Heat Conduction Problems

We construct a high order finite difference method in which quadrature points do not need to have a lattice structure. In order to develop our method we show two tools using Fourier transform and Taylor expansion, respectively. On the other hand, the backward heat conduction problem is a typical example of ill-posed problems in the sense that the solution is unstable for errors in data. Our aim is to create a measles method which can be applied to the ill-posed problem. From numerical experiments we confirmed that our method is effective in order to solve the two-dimensional backward heat conduction equation subject to mixed boundary conditions.

Ill-posed inverse problems of the limb remote sounding of the atmosphere

We consider inverse problems of limb refractometry and microwave sounding. Such problems are usually reduced to Abel-type integral equations and solved by virtue of the known inverse transform. However, this method does not yield a solution for some formulations of the problem. In particular, this concerns cases where the measured value is known at only a fraction of the impact height range in which the solution is searched for. In refractometry, this situation is inevitable if a wavelike stratification of the atmosphere takes place. Other examples of ill-posed problems are the inverse problem of limb microwave sounding with account of the finite width of the antenna pattern and the problem of reconstruction of a gas component profile along a fixed direction from the thermal radiation spectrum. We suggest methods of solution for such problems based on the Tikhonov principle of generalized residual. Numerical simulations of height profile reconstruction are presented for the refraction index and ozone content in inverse problems of refractometry and microwave sounding, respectively.

Inversion of Fredholm integral equations of the first kind with fully connected neural networks

Fredholm integral equations of the first kind are examples of ill-posed problems in the Hadamard sense. Conventional methods of discretization often result in an ill-conditioned system of equations. The traditional approach is to solve this system using a variety of regularization schemes. An alternative is to solve the system using connectionist paradigms. The robustness of this approach is illustrated by using a fully connected network that resembles a Hopfield network.

Some non-standard regularizing algorithms and their numerical realization

1. Considerable experience has noi been accumulated on the numerical solution of operator equations of the 1st kind. which represent typical examples of ill-posed problems. The main advances have been achieved b>, using regularizing algorithms (r.a.). A detailed theory of the construction of r.a. has be-n developed, and the properties of wide classes of r.a. have been studied theoretically. In the numerical analysis of ill-posed problems, however, the r.a. based sn the Tikhonov parametric functlonal has? until recently, mainly been used, and the only essential difference between concrete r.a.s is in the method of fixing the parameter (by a quasi-optimal method, or on the basis of a discrepant!,. or a priori, etc.) [ 1 ] , The success achieved by practical application of these r.a.s is well knoa-n. On the other hand. nea’ mathematical models, leading to ill-posed problems. are continualI> being introduced into research. and the demands for reliable solutions are al\~a~~s increasing with the result thai ne r.a.s. differing from the standard Tikhonov type. are beginning to look promising for practical use [ 11.