| 1. |
- Hultgren, Jakob, 1986
(författare)
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Real and complex Monge-Ampère equations, statistical mechanics and canonical metrics
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Recent decades has seen a strong trend in complex geometry to study canonical metrics and the way they relate to geometric analysis, algebraic geometry and probability theory. This thesis consists of four papers each contributing to this field. The first paper sets up a probabilistic framework for real Monge-Ampère equations on tori. We show that solutions to a large class of real Monge-Ampère equations arise as the many particle limit of certain permanental point processes. The framework can be seen as a real, compact analog of the probabilistic framework for Kähler-Einstein metrics on Kähler manifolds. The second paper introduces a variational approach in terms of optimal transport to real Monge-Ampère equations on compact Hessian manifolds. This is applied to prove existence and uniqueness results for various types of canonical Hessian metrics. The results can, on one hand, be seen as a first step towards a probabilistic approach to canonical metrics on Hessian manifolds and, on the other hand, as a remark on the Gross-Wilson and Kontsevich-Soibelmann conjectures in Mirror symmetry. The third paper introduces a new type of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, that generalises Kähler-Einstein metrics. Existence and uniqueness theorems are given as well as a proof of one direction of a generalised Yau-Tian-Donaldson conjecture, establishing a connection between this new notion of canonical metrics and stability in algebraic geometry. The fourth paper gives a necessary and sufficient condition for existence of coupled Kähler-Einstein metrics on toric manifolds in terms of a collection of associated polytopes, proving this generalised Yau-Tian-Donaldson conjecture in the toric setting.
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| 2. |
- Leander, Jacob, 1987
(författare)
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Mixed Effects Modeling of Deterministic and Stochastic Dynamical Systems - Methods and Applications in Drug Development
- 2021
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Mathematical models based on ordinary differential equations (ODEs) are commonly used for describing the evolution of a system over time. In drug development, pharmacokinetic (PK) and pharmacodynamic (PD) models are used to characterize the exposure and effect of drugs. When developing mathematical models, an important step is to infer model parameters from experimental data. This can be a challenging problem, and the methods used need to be efficient and robust for the modeling to be successful. This thesis presents the development of a set of novel methods for mathematical modeling of dynamical systems and their application to PK-PD modeling in drug development. A method for regularizing the parameter estimation problem for dynamical systems is presented. The method is based on an extension of ODEs to stochastic differential equations (SDEs), which allows for stochasticity in the system dynamics, and is shown to lead to a parameter estimation problem that is easier to solve. The combination of parameter variability and SDEs are investigated, allowing for an additional source of variability compared to the standard nonlinear mixed effects (NLME) model. For NLME models with dynamics described using either ODEs or SDEs, a novel parameter estimation algorithm is presented. The method is a gradient-based optimization method where the exact gradient of the likelihood function is calculated using sensitivity equations, which is shown to give a substantial improvement in computational speed compared to existing methods. The methods developed have been integrated into NLMEModeling, a freely available software package for mixed effects modeling in Wolfram Mathematica. The package allows for general model specifications and offers a user-friendly environment for NLME modeling of dynamical systems. The SDE-NLME framework is used in two applied modeling problems in drug development. First, a previously published PK model of nicotinic acid is extended to incorporate SDEs. By extending the ODE model to an SDE model, it is shown that an additional source of variability can be quantified. Second, the SDE-NLME framework is applied in a model-based analysis of peak expiratory flow (PEF) diary data from two Phase III studies in asthma. The established PEF model can describe several aspects of the PEF dynamics, including long-term fluctuations. The association to exacerbation risk is investigated using a repeated time-to-event model, and several characteristics of the PEF dynamics are shown to be associated with exacerbation risk. The research presented in this doctoral thesis demonstrates the development of a set of methods and applications of mathematical modeling of dynamical systems. In this work, the methods were primarily applied in the field of PK-PD modeling, but are also applicable in other scientific fields.
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| 3. |
- Ryeznik, Yevgen, 1979-
(författare)
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Optimal adaptive designs and adaptive randomization techniques for clinical trials
- 2019
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this Ph.D. thesis, we investigate how to optimize the design of clinical trials by constructing optimal adaptive designs, and how to implement the design by adaptive randomization. The results of the thesis are summarized by four research papers preceded by three chapters: an introduction, a short summary of the results obtained, and possible topics for future work.In Paper I, we investigate the structure of a D-optimal design for dose-finding studies with censored time-to-event outcomes. We show that the D-optimal design can be much more efficient than uniform allocation design for the parameter estimation. The D-optimal design obtained depends on true parameters of the dose-response model, so it is a locally D-optimal design. We construct two-stage and multi-stage adaptive designs as approximations of the D-optimal design when prior information about model parameters is not available. Adaptive designs provide very good approximations to the locally D-optimal design, and can potentially reduce total sample size in a study with a pre-specified stopping criterion.In Paper II, we investigate statistical properties of several restricted randomization procedures which target unequal allocation proportions in a multi-arm trial. We compare procedures in terms of their operational characteristics such as balance, randomness, type I error/power, and allocation ratio preserving (ARP) property. We conclude that there is no single “best” randomization procedure for all the target allocation proportions, but the choice of randomization can be done through computer-intensive simulations for a particular target allocation.In Paper III, we combine the results from the papers I and II to implement optimal designs in practice when the sample size is small. The simulation study done in the paper shows that the choice of randomization procedure has an impact on the quality of dose-response estimation. An adaptive design with a small cohort size should be implemented with a procedure that ensures a “well-balanced” allocation according to the D-optimal design at each stage.In Paper IV, we obtain an optimal design for a comparative study with unequal treatment costs and investigate its properties. We demonstrate that unequal allocation may decrease the total study cost while having the same power as traditional equal allocation. However, a larger sample size may be required. We suggest a strategy on how to choose a suitable randomization procedure which provides a good trade-off between balance and randomness to implement optimal allocation. If there is a strong linear trend in observations, then the ARP property is important to maintain the type I error and power at a certain level. Otherwise, a randomization-based inference can be a good alternative for non-ARP procedures.
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| 4. |
- Schoug, Lukas
(författare)
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On two-dimensional conformal geometry related to the Schramm-Loewner evolution
- 2019
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis contains three papers, one introductory chapter and one chapter with overviews of the papers and some additional results. The topic of this thesis is the geometry of models related to the Schramm-Loewner evolution.In Paper I, we derive a multifractal boundary spectrum for SLEκ(ρ) processes with κ<4 and ρ chosen so that the curves hit the boundary. That is, we study the sets of points where the curves hit the boundary with a prescribed ``angle'', and compute the Hausdorff dimension of those sets. We study the moments of the spatial derivatives of the conformal maps gt, use Girsanov's theorem to change to an appropriate measure, and use the imaginary geometry coupling to derive a correlation estimate.In Paper II, we study the two-valued sets of the Gaussian free field, that is, the local sets such the associated harmonic function only takes two values. It turns out that the real part of the imaginary chaos is large close to these sets. We use this to derive a correlation estimate which lets us compute the Hausdorff dimensions of the two-valued sets.Paper III is dedicated to studying quasislits, that is, images of the segment [0,i] under quasiconformal maps of the upper half-plane into itself, fixing ∞, generated by driving the Loewner equation with a Lip-1/2 function. We improve estimates on the cones containing the curves, and hence on the Hölder regularity of the curves, in terms of the Lip-1/2 seminorm of the driving function.
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| 5. |
- Kirchner, Kristin, 1987
(författare)
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Numerical Approximation of Solutions to Stochastic Partial Differential Equations and Their Moments
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The first part of this thesis focusses on the numerical approximation of the first two moments of solutions to parabolic stochastic partial differential equations (SPDEs) with additive or multiplicative noise. More precisely, in Paper I an earlier result (A. Lang, S. Larsson, and Ch. Schwab, Covariance structure of parabolic stochastic partial differential equations, Stoch. PDE: Anal. Comp., 1(2013), pp. 351–364), which shows that the second moment of the solution to a parabolic SPDE driven by additive Wiener noise solves a well-posed deterministic space-time variational problem, is extended to the class of SPDEs with multiplicative Lévy noise. In contrast to the additive case, this variational formulation is not posed on Hilbert tensor product spaces as trial–test spaces, but on projective–injective tensor product spaces, i.e., on non-reflexive Banach spaces. Well-posedness of this variational problem is derived for the case when the multiplicative noise term is sufficiently small. This result is improved in Paper II by disposing of the smallness assumption. Furthermore, the deterministic equations in variational form are used to derive numerical methods for approximating the first and the second moment of solutions to stochastic ordinary and partial differential equations without Monte Carlo sampling. Petrov–Galerkin discretizations are proposed and their stability and convergence are analyzed. In the second part the numerical solution of fractional order elliptic SPDEs with spatial white noise is considered. Such equations are particularly interesting for applications in statistics, as they can be used to approximate Gaussian Matérn fields. Specifically, in Paper III a numerical scheme is proposed, which is based on a finite element discretization in space and a quadrature for an integral representation of the fractional inverse involving only non-fractional inverses. For the resulting approximation, an explicit rate of convergence to the true solution in the strong mean-square sense is derived. Subsequently, in Paper IV weak convergence of this approximation is established. Finally, in Paper V a similar method, which exploits a rational approximation of the fractional power operator instead of the quadrature, is introduced and its performance with respect to accuracy and computing time is compared to the quadrature approach from Paper III and to existing methods for inference in spatial statistics.
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| 6. |
- Muhumuza, Asaph Keikara, 1975-
(författare)
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Extreme points of the Vandermonde determinant in numerical approximation, random matrix theory and financial mathematics
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis discusses the extreme points of the Vandermonde determinant on various surfaces, their applications in numerical approximation, random matrix theory and financial mathematics. Some mathematical models that employ these extreme points such as curve fitting, data smoothing, experimental design, electrostatics, risk control in finance and method for finding the extreme points on certain surfaces are demonstrated.The first chapter introduces the theoretical background necessary for later chapters. We review the historical background of the Vandermonde matrix and its determinant, some of its properties that make it more applicable to symmetric polynomials, classical orthogonal polynomials and random matrices.The second chapter discusses the construction of the generalized Vandermonde interpolation polynomial based on divided differences. We explore further, the concept of weighted Fekete points and their connection to zeros of the classical orthogonal polynomials as stable interpolation points.The third chapter discusses some extended results on optimizing the Vandermonde determinant on a few different surfaces defined by univariate polynomials. The coordinates of the extreme points are shown to be given as roots of univariate polynomials.The fourth chapter describes the symmetric group properties of the extreme points of Vandermonde and Schur polynomials as well as application of these extreme points in curve fitting.The fifth chapter discusses the extreme points of Vandermonde determinant to number of mathematical models in random matrix theory where the joint eigenvalue probability density distribution of a Wishart matrix when optimized over surfaces implicitly defined by univariate polynomials.The sixth chapter examines some properties of the extreme points of the joint eigenvalue probability density distribution of the Wishart matrix and application of such in computation of the condition numbers of the Vandermonde and Wishart matrices. The seventh chapter establishes a connection between the extreme points of Vandermonde determinants and minimizing risk measures in financial mathematics. We illustrate this with an application to optimal portfolio selection.The eighth chapter discusses the extension of the Wishart probability distributions in higher dimension based on the symmetric cones in Jordan algebras. The symmetric cones form a basis for the construction of the degenerate and non-degenerate Wishart distributions.The ninth chapter demonstrates the connection between the extreme points of the Vandermonde determinant and Wishart joint eigenvalue probability distributions in higher dimension based on the boundary points of the symmetric cones in Jordan algebras that occur in both the discrete and continuous part of the Gindikin set.
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| 7. |
- Petersson, Andreas, 1990
(författare)
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Approximating Stochastic Partial Differential Equations with Finite Elements: Computation and Analysis
- 2019
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Stochastic partial differential equations (SPDE) must be approximated in space and time to allow for the simulation of their solutions. In this thesis fully discrete approximations of such equations are considered, with an emphasis on finite element methods combined with rational semigroup approximations. A quantity of interest for SPDE simulations often takes the form of an expected value of a functional applied to the solution. This is the major theme of this thesis, which divides into two minor themes. The first is how to analyze the error resulting from the fully discrete approximation of an SPDE with respect to a given functional, which is referred to as the weak error of the approximation. The second is how to efficiently compute the quantity of interest as well as the weak error itself. The Monte Carlo (MC) and multilevel Monte Carlo (MLMC) methods are common approaches for this. The thesis consists of five papers. In the first paper the additional error caused by MC and MLMC methods in simulations of the weak error is analyzed. Upper and lower bounds are derived for the different methods and simulations illustrate the results. The second paper sets up a framework for the analysis of the asymptotic mean square stability, the stability as measured in a quadratic functional, of a general stochastic recursion scheme, which is applied to several discretizations of an SPDE. In the third paper, a novel technique for efficiently generating samples of SPDE approximations is introduced, based on the computation of discrete covariance operators. The computational complexities of the resulting MC and MLMC methods are analyzed. The fourth paper considers the analysis of the weak error for the approximation of the semilinear stochastic wave equation. In the fifth paper, a Lyapunov equation is derived, which allows for the deterministic approximation of the expected value of a quadratic functional applied to the solution of an SPDE. The paper also includes an error analysis of an approximation of this equation and an analysis of the weak error, with respect to the quadratic functional, of an approximation of the considered SPDE.
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| 8. |
- Aurell, Alexander, 1989-
(författare)
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Topics in the mean-field type approach to pedestrian crowd modeling and conventions
- 2019
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of five appended papers, primarily addressingtopics in pedestrian crowd modeling and the formation of conventions.The first paper generalizes a pedestrian crowd model for competingsubcrowds to include nonlocal interactions and an arbitrary (butfinite) number of subcrowds. Each pedestrian is granted a ’personalspace’ and is effected by the presence of other pedestrians within it.The interaction strength may depend on subcrowd affinity. The paperinvestigates the mean-field type game between subcrowds and derivesconditions for the reduction of the game to an optimization problem.The second paper suggest a model for pedestrians with a predeterminedtarget they have to reach. The fixed and non-negotiablefinal target leads us to formulate a model with backward stochasticdifferential equations of mean-field type. Equilibrium in the game betweenthe tagged pedestrians and a surrounding crowd is characterizedwith the stochastic maximum principle. The model is illustrated by anumber of numerical examples.The third paper introduces sticky reflected stochastic differentialequations with boundary diffusion as a means to include walls andobstacles in the mean-field approach to pedestrian crowd modeling.The proposed dynamics allow the pedestrians to move and interactwhile spending time on the boundary. The model only admits a weaksolution, leading to the formulation of a weak optimal control problem.The fourth paper treats two-player finite-horizon mean-field typegames between players whose state trajectories are given by backwardstochastic differential equations of mean-field type. The paper validatesthe stochastic maximum principle for such games. Numericalexperiments illustrate equilibrium behavior and the price of anarchy.The fifth paper treats the formation of conventions in a large populationof agents that repeatedly play a finite two-player game. Theplayers access a history of previously used action profiles and form beliefson how the opposing player will act. A dynamical model wheremore recent interactions are considered to be more important in thebelief-forming process is proposed. Convergence of the history to acollection of minimal CURB blocks and, for a certain class of games,to Nash equilibria is proven.
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| 9. |
- Burghart, Fabian, 1996-
(författare)
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Building and Destroying Urns, Graphs, and Trees
- 2023
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis, consisting of an introduction and four papers, different models in the mathematical area of combinatorial probability are investigated.In Paper I, two operations for combining generalised Pólya urns, called disjoint union and product, are defined. This is then shown to turn the set of isomorphism classes of Pólya urns into a semiring, and we find that assigning to an urn its intensity matrix is a semiring homomorphism.In paper II, a modification and generalisation of the random cutting model is introduced. For a finite graph with given source and target vertices, we remove vertices at random and discard all resulting components without a source node. The results concern the number of cuts needed to remove all target vertices and the size of the remaining graph, and suggest that this model interpolates between the traditional cutting model and site percolation.In paper III, we define several polynomial invariants for rooted trees based on the modified cutting model in Paper II.We find recursive identities for these invariants and, using an approach via irreducibility of polynomials, prove that two specific invariants are complete, that is, they distinguish rooted trees up to isomorphism.In paper IV, joint with Paul Thévenin, we consider an operation of concatenating t random perfect matchings on 2n vertices. Our analysis of the resulting random graph as t tends to infinity shows that there is a giant component if and only if n is odd, and that the size of this giant component as well as the number of components is asymptotically normally distributed.
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| 10. |
- Johansson, Fredrik, 1988
(författare)
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Learning with Geometric Embeddings of Graphs
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Graphs are natural representations of problems and data in many fields. For example, in computational biology, interaction networks model the functional relationships between genes in living organisms; in the social sciences, graphs are used to represent friendships and business relations among people; in chemoinformatics, graphs represent atoms and molecular bonds. Fields like these are often rich in data, to the extent that manual analysis is not feasible and machine learning algorithms are necessary to exploit the wealth of available information. Unfortunately, in machine learning research, there is a huge bias in favor of algorithms operating only on continuous vector valued data, algorithms that are not suitable for the combinatorial structure of graphs. In this thesis, we show how to leverage both the expressive power of graphs and the strength of established machine learning tools by introducing methods that combine geometric embeddings of graphs with standard learning algorithms. We demonstrate the generality of this idea by developing embedding algorithms for both simple and weighted graphs and applying them in both supervised and unsupervised learning problems such as classification and clustering. Our results provide both theoretical support for the usefulness of graph embeddings in machine learning and empirical evidence showing that this framework is often more flexible and better performing than competing machine learning algorithms for graphs.
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