173 research outputs found
Regularisation methods for imaging from electrical measurements
In Electrical Impedance Tomography the conductivity of an object is estimated from
boundary measurements. An array of electrodes is attached to the surface of the object
and current stimuli are applied via these electrodes. The resulting voltages are measured.
The process of estimating the conductivity as a function of space inside the object from
voltage measurements at the surface is called reconstruction. Mathematically the ElT
reconstruction is a non linear inverse problem, the stable solution of which requires regularisation
methods. Most common regularisation methods impose that the reconstructed image should
be smooth. Such methods confer stability to the reconstruction process, but limit the
capability of describing sharp variations in the sought parameter.
In this thesis two new methods of regularisation are proposed. The first method, Gallssian
anisotropic regularisation, enhances the reconstruction of sharp conductivity changes
occurring at the interface between a contrasting object and the background. As such
changes are step changes, reconstruction with traditional smoothing regularisation techniques
is unsatisfactory. The Gaussian anisotropic filtering works by incorporating prior
structural information. The approximate knowledge of the shapes of contrasts allows us
to relax the smoothness in the direction normal to the expected boundary. The construction
of Gaussian regularisation filters that express such directional properties on the basis
of the structural information is discussed, and the results of numerical experiments are
analysed. The method gives good results when the actual conductivity distribution is in
accordance with the prior information. When the conductivity distribution violates the
prior information the method is still capable of properly locating the regions of contrast.
The second part of the thesis is concerned with regularisation via the total variation
functional. This functional allows the reconstruction of discontinuous parameters. The
properties of the functional are briefly introduced, and an application in inverse problems
in image denoising is shown. As the functional is non-differentiable, numerical difficulties
are encountered in its use. The aim is therefore to propose an efficient numerical implementation
for application in ElT. Several well known optimisation methods arc analysed,
as possible candidates, by theoretical considerations and by numerical experiments. Such
methods are shown to be inefficient. The application of recent optimisation methods
called primal- dual interior point methods is analysed be theoretical considerations and
by numerical experiments, and an efficient and stable algorithm is developed. Numerical
experiments demonstrate the capability of the algorithm in reconstructing sharp conductivity profiles
Globally convergent techniques in nonlinear Newton-Krylov
Some convergence theory is presented for nonlinear Krylov subspace methods. The basic idea of these methods is to use variants of Newton's iteration in conjunction with a Krylov subspace method for solving the Jacobian linear systems. These methods are variants of inexact Newton methods where the approximate Newton direction is taken from a subspace of small dimensions. The main focus is to analyze these methods when they are combined with global strategies such as linesearch techniques and model trust region algorithms. Most of the convergence results are formulated for projection onto general subspaces rather than just Krylov subspaces
Weak Sharp Minima on Riemannian Manifolds
This is the first paper dealing with the study of weak sharp minima for constrained optimization problems on Riemannian manifolds, which are important in many applications. We consider the notions of local weak sharp minima, boundedly weak sharp minima, and global weak sharp minima for such problems and obtain their complete characterizations in the case of convex problems on finite-dimensional Riemannian manifolds and their Hadamard counterparts. A number of the results obtained in this paper are also new for the case of conventional problems in linear spaces. Our methods involve appropriate tools of variational analysis and generalized differentiation on Riemannian and Hadamard manifolds developed and efficiently implemented in this paper
Existence and computation of a Cournot-Walras equilibrium
In this paper we present a general approach to existence problems in Cournot-Walras (CW) economies, based on mathematical programming theory. We propose a definition of the decision problem of firms which avoids the profit maximization rule as the only rational criterion for the firms and uses the excess demand function instead of the inverse demand function. We prove the existence of a CW equilibrium and we state practical conditions to characterize a CW equilibrium. We also propose efficient algorithms for computing CW equilibria. Finally, we consider some extensions such as externalities, Stackelberg, collusive and Nash equilibrium model
Načrtovanje in planiranje z metodo simulacije
The use of simulation as a tool to design complex stochastic systems is often inhibited by cost. Extensive computer processing is needed to find a design parameter value given a desired target for the performance measure of a given system. The designer simulates the process numerically and obtains an approximation for that same output. The goal is to match the numerical and experimental results as closely as possible by varying the values of input parameters in the numerical simulation. The most obvious difficulty in solving the design problem is that one cannot simply calculate a straightforward solution and be done. Since the output has to be matched by varying the input, an iterative method of solution is implied. This paper proposes a “stochastic approximation” algorithm to estimate the necessary controllable input parameters within a desired accuracy given a target value for the performance function. The proposed solution algorithm is based on Newton’s methods using a single-run simulation approach to estimate the needed derivative. The proposed approach may be viewed as an optimization scheme, where a loss function must be minimized. The solution algorithm properties and the validity of the estimates are examined by applying it to some reliability and queueing systems with known analytical solutions.Uporaba simulacije kot orodja za načrtovanje kompleksnih stohastičnih sistemov je pogosto časovno zahtevna naloga. Potereben je izdaten računalniški čas da se najde vrednost vhodnih parametrov ki ustrezajo željenim performansam sistema. Načrtovalec simulira proces numerično za izbrane vhodne parametre da dobije oceno želene vrednosti izhoda. Cilj je da dobimo kar se da slične vrednosti experimentalnih in simulacijskih rezultatov z variranjem vhodnih parametrov simulacijskega modela. Pproblem je da ne obstaja enostaven način računanja da direkto dobimo zahtevanno rešitev problema. Ker izhod (rešitev) mora odgovarati enoj od možnih vrednosti vhodnih parametrov metoda reševanja je nujno iterativna kar zahteva veliko računalniškega časa. V tem članku predlagava postopek “stohastičnega približka” za oceno potrebnih controlabinih vhodnih parametrov za določitev željene vrednosti sistema v mejah predpisane zanesljivosti. Predlagani algoritam temelji na Newtonovi metodi, kjer spomočjo (enega) simulaciijskega teka ocenimo prvi odvod potreban za optimizacijo kriterijske funkcije. Predlagani postopek lahko razumemo kot optimizacijsko shemo, kjer funkcijo izgube je treba minimizirati. Predlagani postopek je preizkušen in ovrednoten na nekaj primerih zanesljivosti in sistemov strežbe z znanimi analitičnimi rešitvami
Linearly-Constrained Entropy Maximization Problem with Quadratic Costs and Its Applications to Transportation Planning Problems
Many transportation problems can be formulated as a linearly-constrained convex programming problem whose objective function consists of entropy functions and other cost-related terms. In this paper, we propose an unconstrained convex programming dual approach to solving these problems. In particular, we focus on a class of linearly-constrained entropy maximization problem with quadratic cost, study its Lagrangian dual, and provide a globally convergent algorithm with a quadratic rate of convergence. The theory and algorithm can be readily applied to the trip distribution problem with quadratic cost and many other entropy-based formulations, including the conventional trip distribution problem with linear cost, the entropy-based modal split model, and the decomposed problems of the combined problem of trip distribution and assignment. The efficiency and the robustness of this approach are confirmed by our computational experience
Barrier functions and interior-point algorithms for linear programming with zero-, one-, or two-sided bounds on the variables
Includes bibliographical references (p. 37-39).Supported by NSF, AFOSR, and ONR through NSF grant. DMS-8920550 Supported by the Center for Applied Mathematics.Robert M. Freund and Michael J. Todd
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