This thesis focusses on the development and testing of optimization methods for parameter identification in cardiac electrophysiology models. Cardiac electrophysiology models are systems of differential equations representing the evolution of the trans-membrane potential of cardiac cells. The Mitchell-Schaeffer model is chosen for this thesis. The parameters included in the Mitchell-Schaeffer model are optimally adjusted so that the solution of the model has desired properties. Two optimization problems are formulated using least-square functions to identify parameters that match phase durations and parameters that fit entire potential recordings of swine heart tissue acquired via optical imaging techniques at different stimulation frequencies. The non-differentiable optimization methods (Compass Search and three other variants) are applied to solving both optimization problems for two reasons; First, the methods are studied to evaluate performance and second, the optimization process is evaluated to confirm its ability to identify parameters for the Mitchell-Schaeffer model.

A detailed literature review on the development and applications of several numerical methods in solid mechanics and electromagnetic field analysis is presented in the thesis. Despite the great achievements in this research area, there are always the needs to develop new numerical methods or to explore alternative techniques for the purpose of solving the complicate problems and improve the efficiency and accuracy of the existing or new numerical methods. This thesis presents the development of a novel numerical method, the moving least square Ritz (MLS-Ritz) method, and its applications for solving science and engineering problems. The MLS-Ritz method is based on the moving least square (MLS) data interpolation technique and the Ritz minimization principle. The MLS technique is utilized to establish the Ritz trial functions for two-dimensional (2-D) and three-dimensional (3-D) cases. A point substitution approach is developed to enforce boundary conditions. The proposed MLS-Ritz method has the ability to expand the applicability of the conventional Ritz method and meshless method for analysing problems with complex geometries and multiple mediums. The applications of the MLS-Ritz method are also extended to the analysis of the electromagnetic field problems. Three cases including electrical potential problems in a uniform trough and with dielectric medium and a waveguide eigenvalue problem are analysed and compared with solutions obtained by other methods. Comparison studies show that excellent agreement is achieved for the three cases when comparing with existing results in the open literature. The future directions in the development of the MLS-Ritz method for science and engineering computations are discussed. / Doctor of Philosophy (PhD)

<p>Continuum mechanics is used to model the mechanical behaviour of concrete structures subjected to high rates of loading in defence applications. Large deformation theory is used and an isotropic elastic-plastic constitutive equation with isotropic hardening, damage and strain rate dependent loading surface. The hydrostatic pressure is governed by an equation of state. Numerical analysis is performed using the finite element method and the central difference method for the time integration.</p><p>Projectile penetration is studied and it is concluded that it is not suitable to use material description of the motion of both the target and the projectile together with an erosion criterion. Instead, the material description should be used only for the projectile and the spatial description for the target. In this way the need for an erosion criterion is eliminated. Also, in the constitutive model used it is necessary to introduce a scaling of the softening phase in relation to the finite element size, in order to avoid strain localization.</p><p>Drop weight testing of reinforced concrete beams are analysed, where a regularisation is introduced that renders mesh objectivity regarding fracture energy release. The resulting model can accurately reproduce results from material testing but the regularisation is not sufficient to avoid strain localization when applied to an impact loaded structure. It is finally proposed that a non-local measure of deformation could be a solution to attain convergence.</p><p>The third study presents the behaviour of a concrete constitutive model in a splitting test and a simplified non-local theory applied in a tensile test. The splitting test model exhibits mesh dependency due to a singularity. In the tensile test the non-local theory is shown to give a convergent solution. The report https://www.diva-portal.org/liu/webform/form.jsp#paper0is concluded with a discussion on how to better model concrete materials.</p>

<p>In this thesis, we present algorithms for local and global minimization of some <i>Procrustes</i> type problems. Typically, these problems are about rotating and scaling a known set of data to fit another set with applications related to determination of rigid body movements, factor analysis and multidimensional scaling. The known sets of data are usually represented as matrices, and the rotation to be determined is commonly a matrix <i>Q</i> with orthonormal columns.</p><p>The algorithms presented use Newton and Gauss-Newton search directions with optimal step lengths, which in most cases result in a fast computation of a solution.</p><p>Some of these problems are known to have several minima, e.g., the weighted orthogonal Procrustes problem (WOPP). A study on the maximal amount of minima has been done for this problem. Theoretical results and empirical observations gives strong indications that there are not more than 2ⁿ minimizers, where <i>n</i> is the number of columns in <i>Q</i>. A global optimization method to compute all 2ⁿ minima is presented.</p><p>Also considered in this thesis is a cubically convergent iteration method for solving nonlinear equations. The iteration method presented uses second order information (derivatives) when computing a search direction. Normally this is a computational heavy task, but if the second order derivatives are constant, which is the case for quadratic equations, a performance gain can be obtained. This is confirmed by a small numerical study.</p><p>Finally, regularization of ill-posed nonlinear least squares problems is considered. The quite well known L-curve for linear least squares problems is put in context for nonlinear problems.</p>

Cette thèse est construite autour de l'approximation et la représentation des fonctions sur la sphère avec des applications pour des problèmes inverses issues de la géodésie et de l'imagerie médicale. Le plan de la thèse est structuré de la façon suivante. Dans le premier chapitre, on donne le cadre général d'un problème inverse ainsi que la description du problème de la géophysique et de la M/EEG. L'idée d'un problème inverse est de retrouver une densité à l'intérieur d'un domaine (la boule unité modélisant la terre ou le cerveau humain), à partir des données des mesures d'un certain potentiel à la surface du domaine. On continue par donner les principales définitions et théorèmes qu'on utilisera tout au long de la thèse. De plus, la résolution du problème inverse consiste dans la résolution de deux problèmes : transmission de données et localisation de sources à l'intérieur de la boule. En pratique, les données mesurées sont disponibles que sur des parties de la sphère : calottes sphériques, hémisphère nord de la tête (M/EEG), continents (géodésie). Pour représenter ce type de données, on construit la base de Slepian qui a des bonnes propriétés sur les régions étudiées. Dans le Chapitre 4 on s'intéresse au problème d'estimation de données sur la sphère entière (leur développement sous la base des harmoniques sphériques) à partir des mesures partielles bruitées. Une fois qu'on connait ce développement, on applique la méthode du meilleur approximant rationnel sur des sections planes de la sphère (Chapitre 5). Ce chapitre traite trois types de densité : monopolaire, dipolaire et inclusions pour la modélisation des problèmes, ainsi que des propriétés de la densité et du potentiel associé, quantités mises en relation par un certain opérateur. Dans le Chapitre 6 on regarde les Chapitres 3, 4 et 5 du point de vue numérique. On présente des tests numériques pour la localisation de sources dans la géodésie et la M/EEG lorsqu'on dispose des données partielles sur la sphère.

This thesis consists of five papers and it mainly addresses the theory and schemes to approximate the quadrature domains, QDs. The first deals with the uniqueness and some qualitative properties of the two QDs. The concept of two phase QDs, is more complicated than its one counterpart and consequently introduces significant and interesting open. We present two numerical schemes to approach the one phase QDs in the paper. The first method is based on the properties of the free boundary the level set techniques. We use shape optimization analysis to construct second method. We illustrate the efficiency of the schemes on a variety of experiments. In the third paper we design two finite difference methods for the approximation of the multi phase QDs. We prove that the second method enjoys monotonicity, consistency and stability and consequently it is a convergent scheme by Barles-Souganidis theorem. We also present various numerical simulations in the case of Dirac measures. We introduce the QDs in a sub domain of and Rn study the existence and uniqueness along with a numerical scheme based on the level set method in the fourth paper. In the last paper we study the tangential touch for a semi-linear problem. We prove that there is just one phase free boundary points on the flat part of the fixed boundary and it is also shown that the free boundary is a uniform C1-graph up to that part. / Denna avhandling består av fem artiklar och behandlar främst teori och numeriska metoder för att approximera "quadrature domians", QDs. Den första artikeln behandlar entydighet och allmänna egenskaper hos tvåfas QDs. Begreppet tvåfas QDs, är mer komplicerat än enafasmotsvarigheten och introducerar därmed intressanta öppna problem. Vi presenterar två numeriska metoder för att approximera enfas QDs i andra artikeln. Den första metoden är baserad på egenskaperna hos den fria randen och nivå mängdmetoden. Vi använder forsoptimeringmanalys för att konstruera den andra metoden. Båda metoderna är testade i olika numeriska simuleringar. I det tredje artikeln vi approximera flerafas QDs med konstruktionen tvåmetoder finita differens. Vi visar att den andra metoden har monotonicitat, konsistens och stabilitet och följaktligen är metoden konvergent tack vare Barles-Souganidis sats. Vi presenterar också olika numeriska simuleringar i fallet med Diracmåt. Vi introducerar QDs i en delmängd av Rn och studerar existens och entydighet jämte en numerisk metod baserad på nivå mängdmetoden i fjärde pappret. I det sista pappret studerar vi den tangentiella touchen för ett semilinjärt problem. Vi visar att det enbart är enafasrandpunkter på den platta delen av den fixerade randen. Vi visar också att den fria randen är en likformig C1-graf upp till den delen av den fixerade randen.

Injuries to the neck, or cervical region, are very importantsince there is a potential risk of damage to the spinal cord.Any neck injury can have devastating if not life threateningconsequences. High-speed transportation as well as leisure-timeadventures have increased the number of serious neck injuriesand made us increasingly aware of its consequences.Surveillance systems and epidemiological studies are importantprerequisites in defining the scope of the problem. Thedevelopment of mechanical and clinical tools is important forprimary prevention of neck injuries. Thus, the main objectives of the present doctoral thesisare:- To illustrate the dimension of cervical injuries inSweden,- To develop a Finite Element (FE) model of the uppercervical spine, and- To study spinal stability for cervical injuries. The incidence studies were undertaken with data from theinjury surveillance program at the Swedish National Board ofHealth and Welfare. All in-patient data from Swedish hospitals,ranging over thirteen years from 1987 to 1999, were analyzed.During this period 14,310 nonfatal and 782 fatal cervicalinjuries occurred. The lower cervical spine is the mostfrequent location for spinal trauma, although, this changeswith age so that the upper cervical spine is the most frequentlocation for the population over 65 years of age. The incidencefor cervical fractures for the Swedish population decreased forall age groups, except for those older than 65 years of age.The male population, in all age groups, has a higher incidencefor neck fractures than females. Transportation relatedcervical fractures have dropped since 1991, leaving fallaccidents as the sole largest cause of cervical trauma. An anatomically detailed FE model of the human uppercervical spine was developed. The model was validated to ensurerealistic motions of the joints, with significant correlationfor flexion, extension, lateral bending, axial rotation, andtension. It was shown that an FE-model could simulate thecomplex anatomy and mechanism of the upper cervical spine withgood correlation to experimental data. Three studies wereconducted with the FE model. Firstly, the model of the uppercervical spine was combined with an FE model of the lowercervical spine and a head model. The complete model was used toinvestigate a new car roof structure. Secondly, the FE modelwas used for a parameter study of the ligament materialcharacteristics. The kinematics of the upper cervical spine iscontrolled by the ligamentous structures. The ligaments have tomaintain spinal stability while enabling for large rotations ofthe joints. Thirdly, the FE-model was used to study spinalinjuries and their effect on cervical spinal stability inflexion, extension, and lateral bending. To do this, the intactupper cervical spine FE model was modified to implementruptures of the various spinal ligaments. Transection of theposterior atlantooccipital membrane, the ligametum flavum andthe capsular ligament had the most impact on flexion, while theanterior longitudinal ligament and the apical ligamentinfluenced extension. It is concluded that neck injuries in Sweden is a problemthat needs to be address with new preventive strategies. It isespecially important that results from the research on fallaccidents among the elderly are implemented in preventiveprograms. Secondly, it is concluded that an FE model of thecervical region is a powerful tool for development andevaluation of preventive systems. Such models will be importantin defining preventive strategies for the future. Lastly, it isconcluded that the FE model of the cervical spine can increasethe biomechanical understanding of the spine and contribute inanalyses of spinal stability.

In this thesis, we present algorithms for local and global minimization of some Procrustes type problems. Typically, these problems are about rotating and scaling a known set of data to fit another set with applications related to determination of rigid body movements, factor analysis and multidimensional scaling. The known sets of data are usually represented as matrices, and the rotation to be determined is commonly a matrix Q with orthonormal columns. The algorithms presented use Newton and Gauss-Newton search directions with optimal step lengths, which in most cases result in a fast computation of a solution. Some of these problems are known to have several minima, e.g., the weighted orthogonal Procrustes problem (WOPP). A study on the maximal amount of minima has been done for this problem. Theoretical results and empirical observations gives strong indications that there are not more than 2n minimizers, where n is the number of columns in Q. A global optimization method to compute all 2n minima is presented. Also considered in this thesis is a cubically convergent iteration method for solving nonlinear equations. The iteration method presented uses second order information (derivatives) when computing a search direction. Normally this is a computational heavy task, but if the second order derivatives are constant, which is the case for quadratic equations, a performance gain can be obtained. This is confirmed by a small numerical study. Finally, regularization of ill-posed nonlinear least squares problems is considered. The quite well known L-curve for linear least squares problems is put in context for nonlinear problems.

The presence of wave motion is the defining feature in many fields of application,such as electro-magnetics, seismics, acoustics, aerodynamics,oceanography and optics. In these fields, accurate numerical simulation of wave phenomena is important for the enhanced understanding of basic phenomenon, but also in design and development of various engineering applications. In general, numerical simulations must be confined to truncated domains, much smaller than the physical space were the wave phenomena takes place. To truncate the physical space, artificial boundaries, and corresponding boundary conditions, are introduced. There are four main classes of methods that can be used to truncate problems on unbounded or large domains: boundary integral methods, infinite element methods, non-reflecting boundary condition methods and absorbing layer methods. In this thesis, we consider different aspects of non-reflecting boundary conditions and absorbing layers. In paper I, we construct discretely non-reflecting boundary conditions for a high order centered finite difference scheme. This is done by separating the numerical solution into spurious and physical waves, using the discrete dispersion relation. In paper II-IV, we focus on the perfectly matched layer method, which is a particular absorbing layer method. An open issue is whether stable perfectly matched layers can be constructed for a general hyperbolic system. In paper II, we present a stable perfectly matched layer formulation for 2 x 2 symmetric hyperbolic systems in (2 + 1) dimensions. We also show how to choose the layer parameters as functions of the coefficient matrices to guarantee stability. In paper III, we construct a new perfectly matched layer for the simulation of elastic waves in an anisotropic media. We present theoretical and numerical results, showing that the stability properties of the present layer are better than previously suggested layers. In paper IV, we develop general tools for constructing PMLs for first order hyperbolic systems. We present a model with many parameters which is applicable to all hyperbolic systems, and which we prove is well-posed and perfectly matched. We also use an automatic method, derived in paper V, for analyzing the stability of the model and establishing energy inequalities. We illustrate our techniques with applications to Maxwell s equations, the linearized Euler equations, as well as arbitrary 2 x 2 systems in (2 + 1) dimensions. In paper V, we use the method of Sturm sequences for bounding the real parts of roots of polynomials, to construct an automatic method for checking Petrowsky well-posedness of a general Cauchy problem. We prove that this method can be adapted to automatically symmetrize any well-posed problem, producing an energy estimate involving only local quantities. / QC 20100830

Partially Recursive Acceptance Rejection (PRAR) and bounding chains used in conjunction with coupling from the past (CFTP) are two perfect simulation protocols which can be used to sample from a variety of unnormalized target distributions. This paper first examines and then implements these two protocols to sample from the hardcore gas process. We empirically determine the subset of the hardcore process's parameters for which these two algorithms run in polynomial time. Comparing the efficiency of these two algorithms, we find that PRAR runs much faster for small values of the hardcore process's parameter whereas the bounding chain approach is vastly superior for large values of the process's parameter.