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1

Regularising Ill-posed Discrete Optimisation: Quests with P Systems

Nicolescu R., Gimel’farb G., Morris J., Delmas P., Gong R.

Fundamenta Informaticae

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2014

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Vol. 131, nr 3-4

465--483

EN

We propose a novel approach to justify and guide regularisation of an ill-posed one-dimensional global optimisation with multiple solutions using a massively parallel (P system) model of the solution space. Classical optimisation assumes a well-posed problem with a stable unique solution. Most of important practical problems are ill posed due to an unstable or non-unique global optimum and are regularised to get a unique best-suited solution. Whilst regularisation theory exists largely for unstable unique solutions, its recommendations are often routinely applied to inverse optical problems with essentially non-unique solutions, e.g. computer stereo vision or image segmentation, typically formulated in terms of global energy minimisation. In these cases the recommended regularisation becomes purely heuristic and does not guarantee a unique solution. As a result, classical optimisation algorithms: dynamic programming (DP) and belief propagation (BP) – meet with difficulties. Our recent concurrent propagation (CP), leaning upon the P systems paradigm, extends DP and BP to always detect whether the problem is ill posed or not and store in the ill-posed case an entire space of solutions that yield the same global optimum. This suggests a radically new path to proper regularisation: select the best-suited unique solution by exploring statistical and structural features of this space. We propose a P systems based implementation of CP and set out as a case study an application of CP to the image matching problem in stereo vision.

2

Regularization parameter selection in discrete ill-posed problems—The use of the U-curve

Krawczyk-Stańdo D., Rudnicki M.

International Journal of Applied Mathematics and Computer Science

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2007

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Vol. 17, no 2

157-164

EN

To obtain smooth solutions to ill-posed problems, the standard Tikhonov regularization method is most often used. For the practical choice of the regularization parameter \alfa we can then employ the well-known L-curve criterion, based on the L-curve which is a plot of the norm of the regularized solution versus the norm of the corresponding residual for all valid regularization parameters. This paper proposes a new criterion for choosing the regularization parameter \alfa, based on the so-called U-curve. A comparison of the two methods made on numerical examples is additionally included.

3

Revisiting least squares: A discussion on the leading estimation principle in geodesy

Kotsakis C., Sideris M. G.

Geodezja i Kartografia

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2006

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Vol. 55, no 1

3-22

EN

Least squares (LS) estimation is one of the most important tools in geodetic data analysis, However, its prevailing use is not often complemented by an objective view of its rudiments, Within the standard formalism of LS estimation theory there are actually several paradoxical and curious issues which are seldom explicitly formulated. The aim of this expository paper is to present some of these issues and to discuss their implications for geodetic data analysis and parameter estimation problems, In the first part of the paper, an alternative view of the statistical principles that are traditionally linked to LS estimation is given. Particularly, we show that the property of unbiasedness for the ordinary LS estimators can be replaced with a different, yet equivalent, constraint which implies that the numerical range of the unknown parameters is boundless. In the second part of the paper, the shortcomings of the LS method are exposed from a purely algebraic perspective, without employing any concepts from the probabilistic/statistical framework of estimation theory, In particular, it is explained that what is 'least' in least squares is certainly not the errors in the estimated model parameters, and that in every LS-based inversion of a linear model there exists a critical trade-off between the Euclidean norms of the parameter estimation errors and the adjusted residuals

PL

Estymacja metodą najmniejszych kwadratów (LS) jest jednym z najważniejszych narzędzi w analizowaniu danych geodezyjnych. Jednakże powszechne korzystanie z tej metody nie zawsze idzie w parze z pelnym uświadomieniem sobie jej podstaw. W standardowym formalizmie teorii estymacji LS w rzeczywistości istnieje kilka paradoksalnych i osobliwych zagadnień rzadko formułowanych wprost. Celem niniejszej pracy jest przedstawienie niektórych z tych zagadnień i przedyskutowanie ich konsekwencji w analizie danych geodezyjnych oraz problematyce estymacji parametrów. W pierwszej części pracy przedstawiony jest alternatywny pogląd na podstawy statystyczne, które są tradycyjnie łączone z estymacją LS. W szczególności pokazano, że właściwość nieobciążoności dla zwykłych estymatorów LS może być zastąpiona przez inne, równoważne jej uwarunkowanie, które powoduje, że zakres numeryczny nieznanych parametrów jest nieograniczony. W drugiej części pracy przedstawiono wady metody LS z czysto algebraicznego punktu widzenia, bez uwzględnienia pojęć z zakresu probabilistycznego/statystycznego teorii estymacji. W szczególności wyjaśnione zostało, do czego odnosi się 'najmniejszy' (Ieast) w metodzie najmniej szych kwadratów. Z pewnością nie odnosi się do błędów wyznaczanych parametrów modelu. Ponadto stwierdzono, że w kazdej inwersji modelu liniowego opartej na metodzie LS istnieje krytyczna zamiana pomiędzy normami euklidesowymi blędów wyznaczanych parametrów i wyrównanych residuów.

4

On the Choice of Subspace for Iterative Methods for Linear Discrete Ill-Posed Problems

Calvetti D., Lewis B., Reichel L.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 5

1069-1092

EN

Many iterative methods for the solution of linear discrete ill-posed problems with a large matrix require the computed approximate solutions to be orthogonal to the null space of the matrix. We show that when the desired solution is not smooth, it may be possible to determine meaningful approximate solutions with less computational work by not imposing this orthogonality condition.