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- Kieri, Emil, 1987-
(författare)
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Numerical Methods for Wave Propagation : Analysis and Applications in Quantum Dynamics
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We study numerical methods for time-dependent partial differential equations describing wave propagation, primarily applied to problems in quantum dynamics governed by the time-dependent Schrödinger equation (TDSE). We consider both methods for spatial approximation and for time stepping. In most settings, numerical solution of the TDSE is more challenging than solving a hyperbolic wave equation. This is mainly because the dispersion relation of the TDSE makes it very sensitive to dispersion error, and infers a stringent time step restriction for standard explicit time stepping schemes. The TDSE is also often posed in high dimensions, where standard methods are intractable.The sensitivity to dispersion error makes spectral methods advantageous for the TDSE. We use spectral or pseudospectral methods in all except one of the included papers. In Paper III we improve and analyse the accuracy of the Fourier pseudospectral method applied to a problem with limited regularity, and in Paper V we construct a matrix-free spectral method for problems with non-trivial boundary conditions. Due to its stiffness, the TDSE is most often solved using exponential time integration. In this thesis we use exponential operator splitting and Krylov subspace methods. We rigorously prove convergence for force-gradient operator splitting methods in Paper IV. One way of making high-dimensional problems computationally tractable is low-rank approximation. In Paper VI we prove that a splitting method for dynamical low-rank approximation is robust to singular values in the approximation approaching zero, a situation which is difficult to handle since it implies strong curvature of the approximation space.
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- Mahjani, Behrang, 1981-
(författare)
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Methods from Statistical Computing for Genetic Analysis of Complex Traits
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The goal of this thesis is to explore, improve and implement some advanced modern computational methods in statistics, focusing on applications in genetics. The thesis has three major directions.First, we study likelihoods for genetics analysis of experimental populations. Here, the maximum likelihood can be viewed as a computational global optimization problem. We introduce a faster optimization algorithm called PruneDIRECT, and explain how it can be parallelized for permutation testing using the Map-Reduce framework. We have implemented PruneDIRECT as an open source R package, and also Software as a Service for cloud infrastructures (QTLaaS).The second part of the thesis focusses on using sparse matrix methods for solving linear mixed models with large correlation matrices. For populations with known pedigrees, we show that the inverse of covariance matrix is sparse. We describe how to use this sparsity to develop a new method to maximize the likelihood and calculate the variance components.In the final part of the thesis we study computational challenges of psychiatric genetics, using only pedigree information. The aim is to investigate existence of maternal effects in obsessive compulsive behavior. We add the maternal effects to the linear mixed model, used in the second part of this thesis, and we describe the computational challenges of working with binary traits.
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