# [diss] Perustieteiden korkeakoulu / SCI

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Item Advancing incorporation of expert knowledge into Bayesian networks(Aalto University, 2022) Laitila, Pekka; Virtanen, Kai, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Perustieteiden korkeakoulu; School of Science; Virtanen, Kai, Prof., Aalto University, Department of Mathematics and Systems Analysis, FinlandBayesian networks (BNs) are used in many areas to support risk management and decision-making under uncertainty. A BN represents probabilistic relationships of variables and allows to explore their interaction through various types of analyses. In applications, a lack of suitable data often necessitates that a BN is constructed at least partly based on the knowledge of a domain expert. Then, in order to manage limited time and the cognitive workload on the expert, it is vital to have efficient means to support the construction process. This Dissertation elaborates and improves so-called ranked nodes method (RNM) that is used to quantify expert views on the probabilistic relationships of variables, i.e., nodes, of a BN. RNM is designed for nodes with discrete ordinal scales. With such nodes, the relationship of a descendant node and its direct ancestors is defined in a conditional probability table (CPT) that may consist of dozens or hundreds of conditional probabilities. RNM allows the generation of the CPT based on a small number of parameters elicited from the expert. However, the effective use of RNM can be difficult due to a lack of exact guidelines concerning the parameter elicitation and other user-controlled features. Furthermore, there remains ambiguity regarding the underlying theoretical principle of RNM. In addition, a scarcity of knowledge exists on the general ability of CPTs generated with RNM to portray probabilistic relationships appearing in application areas of BNs. The Dissertation advances RNM with regard to the above shortcomings. The underlying theoretical principle of RNM is clarified and experimental verification is provided on the general practical applicability of the method. The Dissertation also presents novel approaches for the elicitation of RNM parameters. These include separate designs for nodes whose ordinal scales consist of subjective labeled states and for nodes formed by discretizing continuous scales. Two novel approaches are also presented for the discretization of continuous scales of nodes. The first one produces static discretizations that stay intact when a BN is used. The other one involves discretizations updating dynamically during the use of the BN. The theoretical and experimental insight that the Dissertation provides on RNM clears the way for its further development and helps to justify its deployment in applications. In turn, the novel elicitation and discretization approaches offer thorough and well-structured means for easier as well as more flexible and versatile utilization of RNM in applications. Consequently, the Dissertation also facilitates and promotes the effective and diverse use of BNs in various domains.Item Algebraic Aspects of Hidden Variable Models(Aalto University, 2023) Ardiyansyah, Muhammad; Kubjas, Kaie, Assistant Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Kubjas Group; Perustieteiden korkeakoulu; School of Science; Kubjas, Kaie, Assistant Prof., Aalto University, Department of Mathematics and Systems Analysis, FinlandHidden variables are random variables that we cannot observe in reality but they are important for understanding the phenomenon of our interest because they affect the observable variables. Hidden variable models aim to represent the effect of the presence of hidden variables which are theoretically thought to exist but we have no data on them. In this thesis, we focus on two hidden variable models in phylogenetics and statistics. In phylogenetics, we seek answers to two important questions related to modeling evolution. First, we study the embedding problem in the group-based models and the strand symmetric model and its higher order generalizations. In Publication I, we provide some embeddability criteria in the group-based models equipped with certain labeling. In Publication III, we characterize the embeddability in the strand symmetric model. These results allow us to measure approximately the proportion of the set of embeddable Markov matrices within the space of Markov matrices. These results generalize the previously established embeddability results on the Jukes-Cantor and Kimura models. The second question of our interest concerns with the distinguishability of phylogenetic network models which is related to the notion of generic identifiability. In Publication II, we provide some conditions on the network topology that ensure the distinguishability of their associated phylogenetic network models under some group-based models. The last part of this thesis is dedicated to studying the factor analysis model which is a statistical model that seeks to reduce a large number of observable variables into a fewer number of hidden variables. The factor analysis model assumes that the observed variables can be presented as a linear combination of the hidden variables together with some error terms. Moreover, the observed and the hidden variables together with the error terms are assumed to be Gaussian. We generalize the factor analysis model by dropping the Gaussianity assumption and introduce the higher order factor analysis model. In Publication IV, we provide the dimension of the higher order factor analysis model and present some conditions under which the model has positive codimension.Item Algebraic Statistics(Aalto University, 2013) Norén, Patrik; Engström, Alexander, Prof., Aalto University, Finland; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Perustieteiden korkeakoulu; School of Science; Engström, Alexander, Prof., Aalto University, FinlandThis thesis on algebraic statistics contains five papers. In paper I we define ideals of graph homomorphisms. These ideals generalize many of the toric ideals defined in terms of graphs that are important in algebraic statistics and commutative algebra. In paper II we study polytopes from subgraph statistics. Polytopes from subgraph statistics are important for statistical models for large graphs and many problems in extremal graph theory can be stated in terms of them. We find easily described semi-algebraic sets that are contained in these polytopes, and using them we compute dimensions and get volume bounds for the polytopes. In paper III we study the topological Tverberg theorem and its generalizations. We develop a toolbox for complexes from graphs using vertex decomposability to bound the connectivity. In paper IV we prove a conjecture by Haws, Martin del Campo, Takemura and Yoshida. It states that the three-state toric homogenous Markov chain model has Markov degree two. In algebraic terminology this means that a certain class of toric ideals are generated by quadratic binomials. In paper V we produce cellular resolutions for a large class of edge ideals and their powers. Using algebraic discrete Morse theory it is then possible to make many of these resolutions minimal, for example explicit minimal resolutions for powers of edge ideals of paths are constructed this way.Item Analyticity of point measurements in inverse conductivity and scattering problems(Aalto University, 2013) Seiskari, Otto; Hyvönen, Nuutti, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Inverse Problems; Perustieteiden korkeakoulu; School of Science; Hyvönen, Nuutti, Prof., Aalto University, Department of Mathematics and Systems Analysis, FinlandInverse conductivity and Helmholtz scattering problems with distributional boundary values are studied. In the context of electrical impedance tomography (EIT), the considered concepts can be interpreted in terms of measurements involving point-like electrodes. The notion of bisweep data of EIT, analogous to the far-field pattern in scattering theory, is introduced and applied in the theory of inverse conductivity problems. In particular, it is shown that bisweep data are the Schwartz kernel of the relative Neumann-to-Dirichlet map, and this result is employed in proving new partial data results for Calderon's problem. Similar techniques are also applied in the scattering context in order to prove the joint analyticity of the far-field pattern. Another recent concept, sweep data of EIT, analogous to the far-field backscatter data, is studied further, and a numerical method for locating small inhomogeneities from sweep data is presented. It is also demonstrated how bisweep data and conformal maps can be used to reduce certain numerical inverse conductivity problems in piecewise smooth plane domains to equivalent problems in the unit disk.Item Approximations and Surrogates for Computational Inverse Boundary Value Problems(Aalto University, 2017) Mustonen, Lauri; Hyvönen, Nuutti, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Perustieteiden korkeakoulu; School of Science; Hyvönen, Nuutti, Prof., Aalto University, Department of Mathematics and Systems Analysis, FinlandInverse boundary value problems are closely related to imaging techniques where measurements on the surface are used to estimate, or reconstruct, inner properties of the imaged object. In this thesis, improved reconstruction methods and new computational approaches are presented for elliptic and parabolic inverse boundary value problems. Two imaging applications that are addressed are electrical impedance tomography (EIT) and thermal tomography. The inverse problems that are considered are nonlinear and the reconstructions are sought by least squares minimization. Algorithms for such minimization often rely on iterative evaluation of the target function and its partial derivatives. In this thesis, we approximate these target functions, which themselves are solutions to partial differential equations, with polynomial surrogates that are simple to evaluate and differentiate. In the context of EIT, this method is used to estimate the shape of the object in addition to its electrical properties. The method is also shown to be feasible for thermal tomography, including the case of uncertain object shape. We also present a novel logarithmic linearization method for EIT. Transforming the voltage measurements in a certain logarithmic way reduces the nonlinearity in the relationship between the electrical properties and the measurements, allowing a reconstruction with fewer or only one minimization step. Furthermore, we propose a modification to the complete electrode model for EIT. The new model is shown to be compatible with experimental measurement data, while the increased regularity of the predicted electromagnetic potential improves convergence properties of numerical methods.Item Bayesian networks, influence diagrams, and games in simulation metamodeling(Aalto University, 2011) Poropudas, Jirka; Virtanen, Kai, Dr.; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Perustieteiden korkeakoulu; Hämäläinen, Raimo P., Prof.The Dissertation explores novel perspectives related to time and conflict in the context of simulation metamodeling referring to auxiliary models utilized in simulation studies. The techniques innovated in the Dissertation offer new analysis capabilities that are beyond the scope of the existing metamodeling approaches. In the time perspective, dynamic Bayesian networks (DBNs) allow the probabilistic representation of the time evolution of discrete event simulation by describing the probability distribution of the simulation state as a function of time. They enable effective what-if analysis where the state of the simulation at a given time instant is fixed and the conditional probability distributions related to other time instants are updated revealing the conditional time evolution. The utilization of influence diagrams (IDs) as simulation metamodels extends the use of the DBNs into simulation based decision making and optimization. They are used in the comparison of decision alternatives by studying their consequences represented by the conditional time evolution of the simulation. For additional analyses, random variables representing simulation inputs can be included in both the DBNs and the IDs. In the conflict perspective, the Dissertation introduces the game theoretic approach to simulation metamodeling. In this approach, existing metamodeling techniques are applied to the simulation analysis of game settings representing conflict situations where multiple decision makers pursue their own objectives. Game theoretic metamodels are constructed based on simulation data and used to study the interaction between the optimal decisions of the decision makers determining their best responses to each others' decisions and the equilibrium solutions of the game. Therefore, the game theoretic approach extends simulation based decision making and optimization into multilateral settings. In addition to the capabilities related to time and conflict, the techniques introduced in the Dissertation are applicable for most of the other goals of simulation metamodeling, such as validation of simulation models. The utilization of the new techniques is illustrated with examples considering simulation of air combat. However, they can also be applied to simulation studies conducted with any stochastic or discrete event simulation model.Item Bayesian Optimal Experimental Design in Imaging(Aalto University, 2023) Puska, Juha-Pekka; Hyvönen, Nuutti, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finand; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Perustieteiden korkeakoulu; School of Science; Hyvönen, Nuutti, Prof., Aalto University, Department of Mathematics and Systems Analysis, FinandAn inverse problem is defined as a problem that violates one of the classical criteria of a well posed problem: a solution exists, is unique, and is continuous with respect to the data in some reasonable topology. A problem that is not well posed is called illposed, and the development of tools to tackle illposed problems is the goal of the field of inverse problems research. In imaging, illposedness is often an inevitable consequence of the high dimension of the unknown, compared with the measurement data. In an imaging problem, one aims to reconstruct the spatial two- or three-dimensional structure of an object of interest, leading to unknown parameters in the hundreds of thousands or beyond, while the dimension of the measurement data is determined by the number of sensors, and thus limited by physical constraints to values often at least an order of magnitude lower. Another consequence of the high dimensionality of the problem is the computational cost involved in the computations. In imaging problems, there is also usually a cost involved in acquiring data, and thus one would naturally want to minimize the amount of data collection required. One tool for this is optimal experimental design, where one aims to perform the experiment in a way as to maximize the value of the data obtained. The challenge of this however, is that the search for this optimal design usually leads to a computationally challenging problem, whose size is dependent on the dimension of both the data and the unknown. Overcoming this difficulty is the main objective of this thesis. The problem can be tackled by using Gaussian approximations in the formulation of the imaging problem, which leads to practical solution formulas for the quantities of interest. In this thesis, tools are developed to enable the efficient computation of expected utilities for certain measurement designs, particularily in sequential imaging problems and for non-Gaussian prior models. Additionally, these tools are applied to medical imaging and astronomy.Item Best bilinear shell element: flat, twisted or curved?(Teknillinen korkeakoulu, 2009) Niemi, Antti H; Matematiikan ja systeemianalyysin laitosThis thesis concerns the accuracy of finite element models for shell structures. The focus is on low-order approximations of layer and vibration modes in shell deformations with particular reference to problems with concentrated loads. It is shown that parametric error amplification, or numerical locking, arises in these cases when bilinear elements are used and the formulation is based on the so-called degenerated solid approach. Furthermore, an alternative way for designing bilinear shell elements is discussed. The procedure is based on a refined shallow shell model which allows for an effective coupling between the membrane and bending strain in the energy expression.Item Blow-up in reaction-diffusion equations with exponential and power-type nonlinearities(Aalto University, 2011) Pulkkinen, Aappo; Londen, Stig-Olof, Prof.; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Perustieteiden korkeakoulu; Gripenberg, Gustaf, Prof.In this dissertation we study blow-up phenomena in semilinear parabolic equations with both exponential and power-type nonlinearities. We study the behavior of the solutions as the blow-up moment in time and the blow-up point in space are approached. Our focus is on the supercritical case; however, we also give some results on the subcritical case. We prove results concerning the blow-up rate of solutions, and we obtain the blow-up profile for limit L1-solutions both with respect to the similarity variables and at the blow-up moment. We use techniques that are applicable both for the exponential and power nonlinearities. We also consider immediate regularization for minimal L1-solutions and improve on some earlier results. We are also interested in the behavior of selfsimilar solutions and we prove the existence of regular selfsimilar solutions that intersect the singular one arbitrary number of times.Item Boundary shape analysis of electrical impedance tomography with applications(Aalto University, 2014) Staboulis, Stratos; Hyvönen, Nuutti, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Inverse Problems; Inversio-ongelmat; Perustieteiden korkeakoulu; School of Science; Hyvönen, Nuutti, Prof., Aalto University, Department of Mathematics and Systems Analysis, FinlandA typical electrical impedance tomography (EIT) reconstruction is ruined if the shape of theimaged body or the electrode locations are not accurately known. In this dissertation, a newapproach based on (boundary) shape analysis is presented for online adaptation of themeasurement geometry model. It is shown that the forward operator of the complete electrode model (CEM) of EIT is Fréchet differentiable with respect to both the outer boundary shape of the object and the electrodelocations. A dual technique allows feasible computation of the gradients in a Newton-type'output least squares' algorithm for the simultaneous reconstruction of the conductivity andthe measurement geometry. Shape calculus techniques are also applied to optimal experimentdesign for EIT. To be more precise, numerical optimization of the electrode positions is carriedout with respect to posterior covariance related criteria derived from the Bayesian inversionparadigm. Special attention is paid to the Sobolev regularity properties of the CEM essential for the shape analysis. By interpolation of Sobolev spaces it is proven that the CEM is a perturbation of the less regular shunt model of EIT. Consequently, instability in the computation of the numerical shape derivative can be expected if the contact resistances are small.Item Boundedness of maximal operators and oscillation of functions in metric measure spaces(Aalto-yliopiston teknillinen korkeakoulu, 2010) Aalto, Daniel; Kinnunen, Juha, Prof.; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Aalto-yliopiston teknillinen korkeakoulu; Kinnunen, Juha, Prof.In this dissertation the action of maximal operators and the properties of oscillating functions are studied in the context of doubling measure spaces. The work consists of four articles, in which boundedness of maximal operators is studied in several function spaces and different aspects of the oscillation of functions are considered. In particular, new characterizations for the BMO and the weak L∞ are obtained.Item Characterizations and fine properties of functions of bounded variation on metric measure spaces(Aalto University, 2014) Lahti, Panu; Kinnunen, Juha, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Nonlinear PDE research group; Perustieteiden korkeakoulu; School of Science; Kinnunen, Juha, Prof., Aalto University, Department of Mathematics and Systems Analysis, FinlandIn this thesis we study functions of bounded variation, abbreviated as BV functions, on metric measure spaces. We always assume the space to be equipped with a doubling measure, and mostly we also assume it to support a Poincaré inequality. A central topic in the thesis are the various characterizations of BV functions. We show that BV functions can be characterized by a pointwise inequality involving the maximal function of a finite measure. Furthermore, we study the Federer-type characterization of sets of finite perimeter, according to which a set is of finite perimeter if and only if the codimension one Hausdorff measure of the set's measure theoretic boundary is finite. Through the study of socalled strong relative isoperimetric inequalities, we establish a slightly weakened version of this characterization. Moreover, we prove the Federer-type characterization on spaces that support a geometric Semmes family of curves. On such spaces, between every pair of points there is a curve family with certain uniformity properties that resemble the behavior of parallel lines on a Euclidean space. Our proof relies on first proving a characterization of BV functions in terms of curves. We also study functionals of linear growth, which give a generalization of BV functions. We consider an integral representation for such functionals by means of the variation measure, but contrary to the Euclidean case, the functional and the integral representation are only comparable instead of being equal. As a by-product of our analysis, we are able to characterize those BV functions that are in fact Newton-Sobolev functions. As an application of the integral representation, we consider a minimization problem for the functionals of linear growth, and show that the boundary values of such a problem can be expressed as a penalty term in which we integrate over the boundary of the domain. For this, we need to study boundary traces and extensions of BV functions. Our analysis of traces also produces novel pointwise results on the behavior of BV functions in their jump sets.Item Climate impacts of bioenergy from forest harvest residues(Aalto University, 2015) Repo, Anna; Liski, Jari, Research Prof., Finnish Environment Institute SYKE, Finland; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Systems Analysis Laboratory; Perustieteiden korkeakoulu; School of Science; Salo, Ahti, Prof., Aalto University, Department of Mathematics and Systems Analysis, FinlandClimate change mitigation requires substantial cuts in greenhouse gas emissions in the next decades. One option to reduce these emissions is to replace fossil fuels with low-carbon alternatives. Bioenergy from forest harvest residues has been considered as a carbon neutral source of energy, and therefore it has been regarded as an effective means to reduce the emissions. However, an increase in the extraction of forest harvest residues decreases the carbon stock, and the carbon sink capacity of forests. This effect can lessen the greenhouse gas emission savings and undermine the climate change mitigation potential of this bioenergy source. This dissertation examines the climate impacts of bioenergy produced from forest harvest residues. In this dissertation, an approach was developed to quantify the greenhouse gas emissions and the consequent warming climate impact of bioenergy from forest harvest residues. In addition, this dissertation suggests cost-effective ways to compensate for the carbon loss resulting from residue harvesting, and thus improve the climate impacts of this form of bioenergy. The dissertation illustrates the importance of accounting for reductions in the forest carbon stock in order to estimate the efficiency of bioenergy in reducing CO2 emissions reliably. The findings of this dissertation have implications for renewable energy and climate policies, and forest management. The results presented provide guidance on how to choose and plan bioenergy production practices that deliver the largest climate benefits. The approaches presented in this dissertation can be applied in the development of new forest management, which maximizes climate benefits of bioenergy from forest harvest residues with a low cost to the forest owner and the end-user of bioenergy.Item Computational and theoretical models in diffuse imaging(Aalto University, 2023) Kuutela, Topi; Hyvönen, Nuutti, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Perustieteiden korkeakoulu; School of Science; Hyvönen, Nuutti, Prof., Aalto University, Department of Mathematics and Systems Analysis, FinlandIn this thesis, aspects of parameter estimation problems associated with diffuse imaging modalities are studied. Both considered modalities, electrical impedance tomography (EIT) and diffuse optical tomography (DOT), correspond to ill-posed inverse problems governed by elliptic partial differential equations, with the goal of reconstructing a material parameter field inside an examined body from measurements on the surface of the body. The ill-posedness of these problems is particularly evident in their sensitivity to modeling errors. This thesis studies computational and theoretical models with which some of the reconstruction artifacts can be reduced. The logarithmic forward map of EIT has been previously introduced to facilitate the use of linearization in the parameter estimation problem. The first theoretical contribution of this thesis is proving the Fréchet differentiability of the logarithmic forward map, and establishing its regularity. Curiously the Fréchet derivative is found to be more regular than the logarithmic forward map itself. The second theoretical contribution consists of extensions to the series reversion technique in EIT. Numerical methods with higher order convergence rates can be constructed via series reversion using only directional derivatives, which are cheap to compute for the forward map of EIT. In this thesis, the series reversion method is extended to simultaneously cover the domain conductivity and the electrode contact reconstructions and to also allow arbitrary parametrizations for the reconstructed quantities. The error in electrode locations is known to be one of the most significant error sources for EIT reconstruction problems. A computationally simple method to compensate for the electrode location error in the reconstruction process is presented and numerically evaluated in this thesis. The method is based on an extension to the smoothened complete electrode model in which the contact conductivity between the electrodes and the measured domain is not considered a constant but instead a function on the boundary. The viability of this method is demonstrated by two-dimensional reconstructions based on real-world measurements and by three-dimensional numerical experiments as a part of the study on the generalized series reversion. Finally, this thesis includes a careful evaluation of the effect of anatomical variation in frequency-domain DOT. The variation is studied numerically using 166 segmented neonatal head anatomies with the two commonly used computational models for DOT, the diffusion approximation and Markov chain Monte Carlo. Furthermore, a new segmentation method is presented to separate cerebrospinal fluid into two physiologically plausible types, one present in between the skull and the brain while the other found in the sulci of the brain.Item Computational approaches in electrical impedance tomography with applications to head imaging(Aalto University, 2021) Candiani, Valentina; Hyvönen, Nuutti, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Perustieteiden korkeakoulu; School of Science; Hyvönen, Nuutti, Prof., Aalto University, Department of Mathematics and Systems Analysis, FinlandThis thesis considers computational approaches to address the inverse problem arising from electrical impedance tomography (EIT), where the aim is to reconstruct (useful information about) the conductivity distribution inside a physical body from boundary measurements of current and voltages. The problem is nonlinear and highly ill-posed, and it generally presents several theoretical and numerical challenges. In fact, the search for a solution usually requires either carefully selected regularization techniques or simplifying assumptions on the measurement setting. A particular focus is on applying EIT to stroke detection in medical imaging, where measurement and modelling errors considerably deteriorate the available boundary data. To model these uncertainties, a novel computational three-dimensional head model is introduced and utilized to simulate realistic synthetic electrode measurements. According to the studied application, different models for the forward problem are considered, such as the continuum model, the complete electrode model and its smoothened version. The examined solution strategies correspond to different methodologies, ranging from regularized iterative reconstruction algorithms to machine learning techniques. The performance of these methods is assessed via three-dimensional simulated experiments performed in different settings.Item Computational Methods in Conformal and Harmonic Mappings(Aalto University, 2015) Quach, Tri; Rasila, Antti, Senior Lecturer, Aalto University, Department of Mathematics and Systems Analysis, Finland; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Perustieteiden korkeakoulu; School of Science; Eirola, Timo, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland; Nevanlinna, Olavi, Prof., Aalto University, Department of Mathematics and Systems Analysis, FinlandConformal geometry and the theory of harmonic mappings have a vast number of applications in engineering and physics. Conformal mappings have been used classically, e.g., in electrostatics, fluid dynamics, and potential flows, where the governing partial differential equation is Laplacian. These applications rely on the conformal invariance property of the harmonic solution of a Dirichlet problem as well as the Carathéodory boundary extension theorem. Recently conformal mappings have gained more popularity, e.g., in electrical impedance tomography and in computer graphics, where computational modelling is studied in the context of Riemann surfaces, which includes a medical application to the brain imaging of the cortex. Harmonic mappings can be used in studying minimal surfaces, which arise from many interesting phenomena in natural science and engineering, ranging from mathematical models of soap bubble surfaces, to topics in molecular engineering, and tensile structures. In this thesis a new method, conjugate function method, of constructing a conformal mappings from domains of interest onto a rectangle is developed. This algorithm makes use of the harmonic conjugate function as well as properties of modulus of quadrilaterals, and it is suitable for a very general class of domains, which may have curved boundaries and even cusps. The elaborated method is also suitable for multiply connected domains, where connectivity is greater than two. The second part of this thesis deals with the harmonic shearing method of obtaining harmonic mappings, and its application to minimal surfaces. Harmonic shearing involves integration of predetermined analytic function, which is the complex dilatation of the mapping being constructed, and a conformal mapping, which has to be convex in the direction of the real axis. This shearing can be done in numerically as well, thus, in particular, the conformal mappings do not need to be given in a closed form.Item Computational methods in electromagnetic biomedical inverse problems(Teknillinen korkeakoulu, 2008) Pursiainen, Sampsa; Matematiikan ja systeemianalyysin laitosThis work concerns computational methods in electromagnetic biomedical inverse problems. The following biomedical imaging modalities are studied: electro/magnetoencephalography (EEG/MEG), electrical impedance tomography (EIT), and limited-angle computerized tomography (limited-angle CT). The use of a priori information about the unknown feature is necessary for finding an adequate answer to an inverse problem. Both classical regularization techniques and Bayesian methodology are applied to utilize the a priori knowledge. The inverse problems specifically considered in this work include determination of relatively small electric conductivity anomalies in EIT, dipole-like sources in EEG/MEG, and multiscale X-ray absorbing structures in limited-angle CT. Computational methods and techniques applied for solving these have been designed case-by-case. Results concern, among others, appropriate parametrization of inverse problems; two-stage reconstruction processes, in which a region of interest (ROI) is determined in the first stage and the actual reconstruction is found in the second stage; effective forward simulation through h- and p- versions of the finite element method (FEM); localization of dipole-like electric sources through hierarchical Bayesian models; implementation of the Kirsch factorization method for reconstruction of conductivity anomalies; as well as the use of a coarse-to-fine reconstruction strategy in linear inverse problems.Item Computational models for adversarial risk analysis and probabilistic scenario planning(Aalto University, 2023) Roponen, Juho; Salo, Ahti, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Perustieteiden korkeakoulu; School of Science; Salo, Ahti, Prof., Aalto University, Department of Mathematics and Systems Analysis, FinlandPeople need to make decisions under uncertainty. Both in corporate and public governance, in addition to uncertainty, the decisions can have high costs and far-reaching consequences. Thus, choosing a good decision alternative, or at least avoiding the inferior ones, is crucial. Two sources of uncertainty are especially prevalent in these decision problems: human activity and long planning horizons. In this dissertation, methods for addressing uncertainties arising from both these sources are developed. By quantifying these uncertainties as probability distributions and preferences over outcomes as utility functions, a well-defined mathematical decision problem can be constructed and then solved using optimization techniques. First, methods for adversarial risk analysis are developed to model the decision processes of adversarial actors who deliberately try to advance their own interests. The proposed methods facilitate systematic probabilistic analyses with limited knowledge about the adversary's preferences and their available information. This can be especially useful when the exact way the adversary analyzes the situation is difficult to assess or when their goals are deliberately hidden, as is often the case when analyzing military combat or security problems. The dissertation also demonstrates how combat modeling and simulation tools can be applied in adversarial risk analysis. This expands the types of analyses these tools can be used for, making it possible to answer questions such as, how the adversary's actions are impacted by changing circumstances, or how the outcomes of individual battles impact the larger strategic situation. Second, a new probabilistic cross-impact analysis model is developed to quantify uncertainties associated with future scenarios based on information elicited from subject matter experts. Two different computational approaches are presented for analyzing the elicited cross-impact statements. One takes information about upper and lower bounds on probabilities and then calculates upper and lower bounds on system risk or utility. The other takes the best estimates about probabilities of specific uncertainty factors and their interactions and constructs a joint probability distribution and a Bayesian network. These approaches can be useful when probability information based on statistics or simulations is not available, for example when results need to be produced quickly or the uncertainties are associated with relatively far-off future events or human activity.Item Computational Problems in Simulation of Electrical Machines(Aalto University, 2020) Perkkiö, Lauri; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Applied mathematics; Perustieteiden korkeakoulu; School of Science; Hannukainen, Antti, Asst. Prof., Aalto University, Department of Mathematics and Systems Analysis, FinlandThis thesis deals with computational challenges related to simulation of electrical machines. Electromagnetic fields and heat conduction in the machines are modeled by partial differential equations (PDEs), which are treated numerically by using the finite element (FE) method combined with appropriate time integration schemes. The uniting theme of this work is the prediction of iron losses in electrical machines. Simulating energy losses and mechanical torque in an electrical machine involves computation of the energy in the system, and a time integration method may introduce numerical errors in such computations. Additional complications are caused by a moving subdomain in a machine, and the fact that the resulting discretized problem does not lead into a system of ordinary differential equations (ODEs), but to a differential-algebraic equation (DAE). All this has to be taken into account to construct proper time integration schemes, which is the first topic of thesis. A core of an electrical machine often consists of a hysteretic ferromagnetic material. Conventionally, hysteresis is neglected in electromagnetic simulations, as its inclusion is complicated and computationally expensive. In this thesis, we propose and test numerically a method to incorporate the Jiles-Atherton magnetic hysteresis model into a FE simulation. The last article approaches the iron loss prediction from an inverse problem perspective. The iron loss acts as an unknown heat source term in the heat equation, and the source is reconstructed from a limited number of temperature measurements conducted on and inside the machine. A computational framework and temperature sensor placement optimization is proposed and numerically tested in the thesis.Item Conformally invariant scaling limits of random curves and correlations(Aalto University, 2019) Karrila, Alex; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Mathematical physics group; Perustieteiden korkeakoulu; School of Science; Kytölä, Kalle, Assoc. Prof., Department of Mathematics and Systems Analysis, Aalto University, FinlandThis thesis studies scaling limits of critical random models on planar graphs, when a fine-mesh graph approximates a planar domain. Such studies are motivated by quantum field theoretic predictions that suggest the emergence of intricate, conformally invariant structures from simple combinatorial models. We take a mathematical approach, formalizing our results in terms of SLE type conformally invariant random curves or certain expected values, called boundary correlation functions in physics. In the first publication of this thesis, we study two related random models, the uniform spanning tree (UST) and the loop-erased random walk (LERW). We obtain conformally covariant expressions for the scaling limit probabilities of certain UST branch connectivities and of LERW boundary visits. These expressions are solutions to partial differential equations (PDEs) of second and third order, respectively, and such solutions appear in Conformal field theory (CFT) as boundary correlation functions. CFT predicts such PDEs of arbitrarily high order, and this is among the first verifications of higher-than-second order PDEs. The PDE solutions from the first publication can also be interpreted as weights that conjecturally convert the SLE(2) random curve measure, the scaling limit of a UST branch and a LERW, to multiple or boundary-visiting SLE(2). In the second publication, we elaborate this connection by finding an explicit relation between certain multiple SLE(k) weights, called pure partition functions, and certain CFT boundary correlation functions, called conformal blocks. The third and fourth publication concern the weak convergence of the joint law of multiple lattice curves to multiple SLE type random curves. We first provide a result that guarantees at least subsequential convergence to some limiting random curves, given certain standard crossing estimates in the lattice models. Second, we show how such limits can be described by iteratively sampling the curves one by one from the weighted one-curve SLE(k) measures described above. These tools are applied to characterize the scaling limits of multiple curves in various random models, such as multiple UST branches.