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A new version of an open form localisation algorithm, called regularised location estimator (RLE), for time difference of arrival (TDOA) multilateration systems is proposed. This algorithm is based on Tikhonov regularisation and on maximum-likelihood estimation. It is aimed to solve the localisation problem for both well-conditioned and ill-conditioned situations. It provides solutions that are statistically and numerically efficient, in contrast to the algorithms in the literature providing solutions that are only efficient in one sense and that do not solve the ill-conditioned situations, which can be found in some operational conditions. A recursive formulation for estimating the regularisation parameter is provided and some mathematical analyses about the proposed estimator are also presented. A comparison with a large number of open and closed form algorithms in the literature is performed with both simulated and real data. The results show that the novel TDOA-RLE provides superior performance even in the case of well-conditioned situations.

An investigation of the applicability of neural network-based methods in predicting the values of multiple parameters, given the value of a single parameter within a particular problem domain is presented. In this context, the input parameter may be an important source of variation that is related with a complex mapping function to the remaining sources of variation within a multivariate distribution. The definition of the relationship between the variables of a multivariate distribution and a single source of variation allows the estimation of the values of multiple variables given the value of the single variable, addressing in that way an ill-conditioned one-to-many mapping problem. As part of our investigation, two problem domains are considered: predicting the values of individual stock shares, given the value of the general index, and predicting the grades received by high school pupils, given the grade for a single course or the average grade. With our work, the performance of standard neural network-based methods and in particular multilayer perceptrons (MLPs), radial basis functions (RBFs), mixture density networks (MDNs) and a latent variable method, the general topographic mapping (GTM), is compared. According to the results, MLPs and RBFs outperform MDNs and the GTM for these one-to-many mapping problems. © 2010 Springer-Verlag London Limited. 20 6 775 785 Cited By :5

Relative entropy has been employed, as an alternative to other regularization methods, in solving ill-conditioned linear inverse problems. Damage detection when treated as structural modification imparted by the damage leads to a linear inverse problem involving frequency response functions. This problem is amenable to ill-conditioning issues that could arise from the low frequency response values and noisy experiments. This article formulates and solves using the minimum relative entropy method the damage detection and localization problem on a simulated cantilever beam. © 2009 Society for Experimental Mechanics Inc.

We describe two algorithms of approximate GCD (greatest common divisor) for polynomials with coefficients of floating-point numbers. One is for univariate polynomials and the other for multivariate polynomials. The algorithms are careful extensions of the conventional Euclidean algorithm, and they are applied to solving ill-conditioned algebraic equations. Ill-conditioned univariate algebraic equations have multiple or close roots, and we can ill-conditioned system of algebraic equations which have an approprimate common divisor. Conventional numeric root-finding methods such as Newton's method give no satisfactory result for such ill-conditioned systems. By using an approximate GCD algorithm for multivariate polynomials, the system is transformed into a well-conditioned problems by a combination of algebraic and numeric methods; we call such algorithms hybrid algorithms. This paper shows the importance and possible fruitfulness of the hybrid algorithm.

Let $\mathbb{F}$ be a floating-point number system with basis $\beta \geqq 2$ and an exponent range consisting of at least the exponents 1 and 2. A class of arbitrarily ill conditioned matrices is described, the coefficients of which are elements of $\mathbb{F}$. Due to the very rapidly increasing sensitivity of those matrices, they might be regarded as “almost” ill posed problems.The condition of those matrices and their sensitivity with respect to inversion is given by means of a closed formula. The condition is rapidly increasing with the dimension. For example, in the double precision of the IEEE 754 floating-point standard (base 2, 53 bits in the mantissa including implicit 1), matrices with $2n$ rows and columns are given with a condition number of approximately $4\cdot10^{32n} $.

Date of Conference: 11-16 Nov. 2007 Conference name: 2nd European Conference on Antennas and Propagation (EuCAP 2007) We consider accurate solution of scattering problems involving complicated metamaterial (MM) structures consisting of thin wires and split-ring resonators. The scattering problems are formulated by the electric-field integral equation (EFIE) discretized with the Rao-Wilton- Glisson basis functions defined on planar triangles. The resulting dense matrix equations are solved iteratively, where the matrix-vector multiplications that are required by the iterative solvers are accelerated with the multilevel fast multipole algorithm (MLFMA). Since EFIE usually produces matrix equations that are ill-conditioned and difficult to solve iteratively, we employ nested preconditioners to achieve rapid convergence of the iterative solutions. To further accelerate the simulations, we parallelize our algorithm and perform the solutions on a cluster of personal computers. This way, we are able to solve problems of MMs involving thousands of unit cells.

In this article we discuss ill-posed inverse problems, with an emphasis on hierarchical variable order regularization. Traditionally, smoothness penalties in Tikhonov regularization assume a fixed degree of regularity of the unknown over the whole domain. Using a Bayesian framework with hierarchical priors, we derive a prior model, formally represented as a convex combination of autoregressive (AR) models, in which the parameter controlling the mixture of the AR models can dynamically change over the domain of the signal. Moreover, the mixture parameter itself is an unknown and is to be estimated using the data. Also, the variance of the innovation processes in the AR model is a free parameter, which leads to conditionally Gaussian priors that have been previously shown to be much more flexible than the traditional Gaussian priors, capable, e.g., to deal with sparsity type prior information. The suggested method, the Weighted Variable Order Autoregressive model (WVO-AR) is tested with a computed example. Fil: Calvetti, Daniela. Case Western Reserve University; Estados Unidos Fil: Somersalo, Erkki. Case Western Reserve University; Estados Unidos Fil: Spies, Ruben Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Santa Fe. Instituto de Matemática Aplicada "Litoral"; Argentina