Numerical approximation of the one-dimensional inverse Cauchy–Stefan problem using a method of fundamental solutions

We investigate an application of the method of fundamental solutions (MFS) to the one-dimensional parabolic inverse Cauchy–Stefan problem, where boundary data and the initial condition are to be determined from the Cauchy data prescribed on a given moving interface. In [B.T. Johansson, D. Lesnic, and T. Reeve, A method of fundamental solutions for the one-dimensional inverse Stefan Problem, Appl. Math Model. 35 (2011), pp. 4367–4378], the inverse Stefan problem was considered, where only the boundary data is to be reconstructed on the fixed boundary. We extend the MFS proposed in Johansson et al. (2011) and show that the initial condition can also be simultaneously recovered, i.e. the MFS is appropriate for the inverse Cauchy-Stefan problem. Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate results can be efficiently obtained with small computational cost.

Bayesian analysis of the scatterometer wind retrieval inverse problem:Some new approaches

The retrieval of wind vectors from satellite scatterometer observations is a non-linear inverse problem. A common approach to solving inverse problems is to adopt a Bayesian framework and to infer the posterior distribution of the parameters of interest given the observations by using a likelihood model relating the observations to the parameters, and a prior distribution over the parameters. We show how Gaussian process priors can be used efficiently with a variety of likelihood models, using local forward (observation) models and direct inverse models for the scatterometer. We present an enhanced Markov chain Monte Carlo method to sample from the resulting multimodal posterior distribution. We go on to show how the computational complexity of the inference can be controlled by using a sparse, sequential Bayes algorithm for estimation with Gaussian processes. This helps to overcome the most serious barrier to the use of probabilistic, Gaussian process methods in remote sensing inverse problems, which is the prohibitively large size of the data sets. We contrast the sampling results with the approximations that are found by using the sparse, sequential Bayes algorithm.

The application of knowledge based expert systems to mechanical engineering design

Investigation of the different approaches used by Expert Systems researchers to solve problems in the domain of Mechanical Design and Expert Systems was carried out. The techniques used for conventional formal logic programming were compared with those used when applying Expert Systems concepts. A literature survey of design processes was also conducted with a view to adopting a suitable model of the design process. A model, comprising a variation on two established ones, was developed and applied to a problem within what are described as class 3 design tasks. The research explored the application of these concepts to Mechanical Engineering Design problems and their implementation on a microcomputer using an Expert System building tool. It was necessary to explore the use of Expert Systems in this manner so as to bridge the gap between their use as a control structure and for detailed analytical design. The former application is well researched into and this thesis discusses the latter. Some Expert System building tools available to the author at the beginning of his work were evaluated specifically for their suitability for Mechanical Engineering design problems. Microsynics was found to be the most suitable on which to implement a design problem because of its simple but powerful Semantic Net Knowledge Representation structure and the ability to use other types of representation schemes. Two major implementations were carried out. The first involved a design program for a Helical compression spring and the second a gearpair system design. Two concepts were proposed in the thesis for the modelling and implementation of design systems involving many equations. The method proposed enables equation manipulation and analysis using a combination of frames, semantic nets and production rules. The use of semantic nets for purposes other than for psychology and natural language interpretation, is quite new and represents one of the major contributions to knowledge by the author. The development of a purpose built shell program for this type of design problems was recommended as an extension of the research. Microsynics may usefully be used as a platform for this development.

Fast reconstruction of harmonic functions from Cauchy data using the Dirichlet-to-Neumann map and integral equations

We propose and investigate a method for the stable determination of a harmonic function from knowledge of its value and its normal derivative on a part of the boundary of the (bounded) solution domain (Cauchy problem). We reformulate the Cauchy problem as an operator equation on the boundary using the Dirichlet-to-Neumann map. To discretize the obtained operator, we modify and employ a method denoted as Classic II given in [J. Helsing, Faster convergence and higher accuracy for the Dirichlet–Neumann map, J. Comput. Phys. 228 (2009), pp. 2578–2576, Section 3], which is based on Fredholm integral equations and Nyström discretization schemes. Then, for stability reasons, to solve the discretized integral equation we use the method of smoothing projection introduced in [J. Helsing and B.T. Johansson, Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques, Inverse Probl. Sci. Eng. 18 (2010), pp. 381–399, Section 7], which makes it possible to solve the discretized operator equation in a stable way with minor computational cost and high accuracy. With this approach, for sufficiently smooth Cauchy data, the normal derivative can also be accurately computed on the part of the boundary where no data is initially given.

Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques

We consider the problem of stable determination of a harmonic function from knowledge of the solution and its normal derivative on a part of the boundary of the (bounded) solution domain. The alternating method is a procedure to generate an approximation to the harmonic function from such Cauchy data and we investigate a numerical implementation of this procedure based on Fredholm integral equations and Nyström discretization schemes, which makes it possible to perform a large number of iterations (millions) with minor computational cost (seconds) and high accuracy. Moreover, the original problem is rewritten as a fixed point equation on the boundary, and various other direct regularization techniques are discussed to solve that equation. We also discuss how knowledge of the smoothness of the data can be used to further improve the accuracy. Numerical examples are presented showing that accurate approximations of both the solution and its normal derivative can be obtained with much less computational time than in previous works.

An alternating boundary integral based method for a Cauchy problem for the Laplace equation in a quadrant

We study the Cauchy problem for the Laplace equation in a quadrant (quarter-plane) containing a bounded inclusion. Given the values of the solution and its derivative on the edges of the quadrant the solution is reconstructed on the boundary of the inclusion. This is achieved using an alternating iterative method where at each iteration step mixed boundary value problems are being solved. A numerical method is also proposed and investigated for the direct mixed problems reducing these to integral equations over the inclusion. Numerical examples verify the efficiency of the proposed scheme.

On the numerical solution of a Cauchy problem in an elastostatic half-plane with a bounded inclusion

We propose an iterative procedure for the inverse problem of determining the displacement vector on the boundary of a bounded planar inclusion given the displacement and stress fields on an infinite (planar) line-segment. At each iteration step mixed boundary value problems in an elastostatic half-plane containing the bounded inclusion are solved. For efficient numerical implementation of the procedure these mixed problems are reduced to integral equations over the bounded inclusion. Well-posedness and numerical solution of these boundary integral equations are presented, and a proof of convergence of the procedure for the inverse problem to the original solution is given. Numerical investigations are presented both for the direct and inverse problems, and these results show in particular that the displacement vector on the boundary of the inclusion can be found in an accurate and stable way with small computational cost.

High stress monitoring of prestressing tendons in nuclear concrete vessels using fibre-optic sensors

Maintaining the structural health of prestressed concrete nuclear containments is a key element in ensuring nuclear reactors are capable of meeting their safety requirements. This paper discusses the attachment, fabrication and characterisation of optical fibre strain sensors suitable for the prestress monitoring of irradiated steel prestressing tendons. The all-metal fabrication and welding process allowed the instrumented strand to simultaneously monitor and apply stresses up to 1300 MPa (80% of steel's ultimate tensile strength). There were no adverse effects to the strand's mechanical properties or integrity. After sensor relaxation through cyclic stress treatment, strain transfer between the optical fibre sensors and the strand remained at 69%. The fibre strain sensors could also withstand the non-axial forces induced as the strand was deflected around a 4.5 m bend radius. Further development of this technology has the potential to augment current prestress monitoring practices, allowing distributed measurements of short- and long-term prestress losses in nuclear prestressed-concrete vessels. © 2014 Elsevier B.V.

Numerical solution of a Cauchy problem for Laplace equation in 3-dimensional domains by integral equations

A numerical method based on integral equations is proposed and investigated for the Cauchy problem for the Laplace equation in 3-dimensional smooth bounded doubly connected domains. To numerically reconstruct a harmonic function from knowledge of the function and its normal derivative on the outer of two closed boundary surfaces, the harmonic function is represented as a single-layer potential. Matching this representation against the given data, a system of boundary integral equations is obtained to be solved for two unknown densities. This system is rewritten over the unit sphere under the assumption that each of the two boundary surfaces can be mapped smoothly and one-to-one to the unit sphere. For the discretization of this system, Weinert’s method (PhD, Göttingen, 1990) is employed, which generates a Galerkin type procedure for the numerical solution, and the densities in the system of integral equations are expressed in terms of spherical harmonics. Tikhonov regularization is incorporated, and numerical results are included showing the efficiency of the proposed procedure.

Mixture density networks

Minimization of a sum-of-squares or cross-entropy error function leads to network outputs which approximate the conditional averages of the target data, conditioned on the input vector. For classifications problems, with a suitably chosen target coding scheme, these averages represent the posterior probabilities of class membership, and so can be regarded as optimal. For problems involving the prediction of continuous variables, however, the conditional averages provide only a very limited description of the properties of the target variables. This is particularly true for problems in which the mapping to be learned is multi-valued, as often arises in the solution of inverse problems, since the average of several correct target values is not necessarily itself a correct value. In order to obtain a complete description of the data, for the purposes of predicting the outputs corresponding to new input vectors, we must model the conditional probability distribution of the target data, again conditioned on the input vector. In this paper we introduce a new class of network models obtained by combining a conventional neural network with a mixture density model. The complete system is called a Mixture Density Network, and can in principle represent arbitrary conditional probability distributions in the same way that a conventional neural network can represent arbitrary functions. We demonstrate the effectiveness of Mixture Density Networks using both a toy problem and a problem involving robot inverse kinematics.

Theoretical foundations of neural networks

Neural networks have often been motivated by superficial analogy with biological nervous systems. Recently, however, it has become widely recognised that the effective application of neural networks requires instead a deeper understanding of the theoretical foundations of these models. Insight into neural networks comes from a number of fields including statistical pattern recognition, computational learning theory, statistics, information geometry and statistical mechanics. As an illustration of the importance of understanding the theoretical basis for neural network models, we consider their application to the solution of multi-valued inverse problems. We show how a naive application of the standard least-squares approach can lead to very poor results, and how an appreciation of the underlying statistical goals of the modelling process allows the development of a more general and more powerful formalism which can tackle the problem of multi-modality.

Hydrodynamic modeling of mineral wool fiber suspensions in a two-dimensional flow

A consequence of a loss of coolant accident is that the local insulation material is damaged and maybe transported to the containment sump where it can penetrate and/or block the sump strainers. An experimental and theoretical study, which examines the transport of mineral wool fibers via single and multi-effect experiments is being performed. This paper focuses on the experiments and simulations performed for validation of numerical models of sedimentation and resuspension of mineral wool fiber agglomerates in a racetrack type channel. Three velocity conditions are used to test the response of two dispersed phase fiber agglomerates to two drag correlations and to two turbulent dispersion coefficients. The Eulerian multiphase flow model is applied with either one or two dispersed phases.

CFD-modeling of insulation debris transport phenomena in water flow

The investigation of insulation debris generation, transport and sedimentation becomes important with regard to reactor safety research for PWR and BWR, when considering the long-term behavior of emergency core cooling systems during all types of loss of coolant accidents (LOCA). The insulation debris released near the break during a LOCA incident consists of a mixture of disparate particle population that varies with size, shape, consistency and other properties. Some fractions of the released insulation debris can be transported into the reactor sump, where it may perturb/impinge on the emergency core cooling systems. Open questions of generic interest are the sedimentation of the insulation debris in a water pool, its possible re-suspension and transport in the sump water flow and the particle load on strainers and corresponding pressure drop. A joint research project on such questions is being performed in cooperation between the University of Applied Sciences Zittau/Görlitz and the Forschungszentrum Dresden-Rossendorf. The project deals with the experimental investigation of particle transport phenomena in coolant flow and the development of CFD models for its description. While the experiments are performed at the University at Zittau/Görlitz, the theoretical modeling efforts are concentrated at Forschungszentrum Dresden-Rossendorf. In the current paper the basic concepts for CFD-modeling are described and feasibility studies including the conceptual design of the experiments are presented. © 2009 Elsevier B.V. All rights reserved.