[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems.

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Inverse problems: A Bayesian perspective

Finite element methods for Maxwell's equations.

As companies strive to develop artefacts intended for services instead of traditional sell-off, new challenges in the product development process arise to promote continuous improvement and increasing market profits. This creates a focus on product life-cycle components as companies then make life-cycle commitments, where they are responsible for the function availability during the extent of the life-cycle, i.e. functional products. One of these life-cycle components is manufacturing; therefore, companies search for new approaches of success during manufacturability evaluation already in engineering design. Efforts have been done to support early engineering design, as this phase sets constraints and opportunities for manufacturing. These efforts have turned into design for manufacturing methods and guidelines. A further step to improve the life-cycle focus during early engineering design is to reuse results and use experience from earlier projects. However, because results and experiences created during project work are often not documented for reuse, only remembered by some people, there is a need for design support. Knowledge engineering (KE) is a methodology for creating knowledge-based systems, e.g. systems that enable reuse of earlier results and make available both explicit and tacit corporate knowledge, enabling the automated generation and evaluation of new engineering design solutions during early product development. There are a variety of KE-approaches, such as knowledge-based engineering, case-based reasoning and programming, which have been used in research to develop design for manufacturing methods and applications. There are, however, opportunities for research where several approaches and their interdependencies, to create a transparent picture of how KE can be used to support engineering design, are investigated. The aim of the research presented in this thesis is to create new methods for design for manufacturing, by using several approaches of KE, and find the beneficial and less beneficial aspects of these methods in comparison to each other and earlier research. This thesis presents methods and applications for design for manufacturing using KE. KE has been employed in several ways, namely rule-based, rule-, programmingand finite element analysis (FEA)-based, and ruleand plan-based, which are tested and compared with each other. Results show that KE can be used to generate information about manufacturing in several ways. The rule-based way is suitable for supporting life-cycle commitments, as engineering design and manufacturing can be integrated with maintenance and performance predictions during early engineering design, though limited to the firing of production rules. The rule-, programmingand FEA-based way can be used to integrate computer-aided design tools and virtual manufacturing for non-linear stress and displacement analysis. This way may also bridge the gap between engineering designers and computational experts, even though this way requires a larger effort to program than the rule-based. The ruleand planbased way can enable design for manufacturing in two fashions – based on earlier manufacturing plans and based on rules. Because earlier manufacturing plans, together with programming algorithms, can handle knowledge that may be more intricate to capture as rules, as opposed to the time demanding routine work that is often automated by means of rules, several opportunities for designing for manufacturing exist.

Methods and Applications

Special Functions and Their Applications. By N. N. Lebedev, translated by R. A. Silverman. Pp. xii, 308. 96s. 1965. (Prentice-Hall)

Cet article a pour objet de faire une synthese des approches realisees dans le cadre de la modelisation fonctionnelle et structurale des arbres (FSM). Ces modeles resultent du couplage entre la modelisation du fonctionnement ecophysiologique d'arbres, d'une part, et la modelisation des processus morphologiques, d'autre part. Apres une breve presentation des approches existantes, nous introduisons la notion «d'unite elementaire ideale» (IEU) qui peut etre consideree comme la composante fondamentale des FSM au regard de la souplesse qu'elle confere dans l'articulation des processus. La distribution des metabolites et la croissance sont ensuite abordees comme etant les processus a resoudre de facon prioritaire dans le developpement des FSM, et les differentes approches pouvant etre mises a contribution dans la construction de ces modeles sont discutees. Enfin nous analysons les besoins en programmation des FSM, discutons des avancees necessaires et evaluons leur adequation a la resolution d'objectifs divers.

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Components of functional-structural tree models

The factorization method is a tool for recovering inclusions inside a body when the Neumann-to-Dirichlet operator, which maps applied currents to measured voltages, is known. In practice this information is never at hand due to the discreteness and physical properties of the measurement devices. The complete electrode model of impedance tomography includes these physical characteristics but leads to a finite-dimensional data set, called the resistivity matrix. The main result of this work is an approximation link relating the resistivity matrix to the Neumann-to-Dirichlet operator in the $L^2$-operator norm. This result allows us to extend the factorization method to the framework of real-life electrode measurements using a regularized series criterion which is easy to implement in practice. The truncation index of the sequence criterion, which represents the stopping index of the regularization scheme, can be computed solely from the measured, perturbed, and finite-dimensional data. The functionality of ...

The factorization method applied to the complete electrode model of impedance tomography

Bayesian modeling and analysis of the magnetoencephalography and electroencephalography modalities provide a flexible framework for introducing prior information complementary to the measured data. This prior information is often qualitative in nature, making the translation of the available information into a computational model a challenging task. We propose a generalized gamma family of hyperpriors which allows the impressed currents to be focal and we advocate a fast and efficient iterative algorithm, the iterative alternating sequential algorithm for computing maximum a posteriori (MAP) estimates. Furthermore, we show that for particular choices of the scalar parameters specifying the hyperprior, the algorithm effectively approximates popular regularization strategies such as the minimum current estimate and the minimum support estimate. The connection between priorconditioning and adaptive regularization methods is also pointed out. The posterior densities are explored by means of a Markov chain Monte Carlo strategy suitable for this family of hypermodels. The computed experiments suggest that the known preference of regularization methods for superficial sources over deep sources is a property of the MAP estimators only, and that estimation of the posterior mean in the hierarchical model is better adapted for localizing deep sources.

Conditionally Gaussian Hypermodels for Cerebral Source Localization

Scale resolution, locking, and high-order finite element modelling of shells

Moduli of rings and quadrilaterals are frequently applied in geometric function theory; see, e.g., the handbook by Kuhnau [Handbook of Complex Analysis: Geometric Function Theory, Vols. 1 and 2, North-Holland, Amsterdam, 2005]. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation. We present here a new $hp$-FEM algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the Schwarz-Christoffel Toolbox of Driscoll and Trefethen. We also demonstrate that the $hp$-FEM algorithm applies to the case of nonpolygonal boundary and report results with concrete error bounds.

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Harri Hakula

Papers

Components of functional-structural tree models

The factorization method applied to the complete electrode model of impedance tomography

Conditionally Gaussian Hypermodels for Cerebral Source Localization

Scale resolution, locking, and high-order finite element modelling of shells

On Moduli of Rings and Quadrilaterals: Algorithms and Experiments