Deep neural networks (DNNs) have successfully been applied across various data intensive applications ranging from computer vision, language modeling, bioinformatics and search engines. Hyper-parameters of a DNN are defined as parameters that remain fixed during model training and heavily influence the DNN performance. Hence, regardless of application, the design-phase of constructing a DNN model becomes critical. Framing the selection and tuning of hyper-parameters as an expensive black-box optimization (BBO) problem, obstacles encountered in manual by-hand tuning could be addressed by taking instead an automated algorithmic approach. In this work, the following BBO algorithms: Nelder-Mead Algorithm (NM), ParticleSwarm Optmization (PSO), Bayesian Optimization with Gaussian Processes (BO-GP) and Tree-structured Parzen Estimator (TPE), are evaluated side-by-side for two hyper-parameter optimization problem instances. These instances are: Problem 1, incorporating a convolutionalneural network and Problem 2, incorporating a recurrent neural network. A simple Random Search (RS) algorithm acting as a baseline for performance comparison is also included in the experiments. Results in this work show that the TPE algorithm achieves the overall highest performance with respect to mean solution quality, speed ofimprovement and with a comparatively low trial-to-trial variability for both Problem 1 and Problem 2. The NM, PSO and BO-GP algorithms are shown capable of outperforming the RS baseline for Problem 1, but fails to do so in Problem 2.

Compact Riemann surfaces of genus greater than 1 can be realized as quotient spaces of the hyperbolic plane by the action of Fuchsian groups. The Teichmüller space is the set of all complex structures of Riemann surfaces and the moduli space the set of conformal equivalence classes of Riemann surfaces. For genus greater than two the branch locus of the covering of the moduli space by the Teichmüller space can be identified wi the set of Riemann surfaces admitting non-trivial automorphisms. Here we give the orbifold structure of the branch locus of surfaces of genus 5 by studying the equisymmetric stratification of the branch locus. This gives the orbifold structure of the moduli space. We also show that the strata corresponding to surfaces with automorphisms of order 2 and 3 belong to the same connected component for every genus. Further we show that the branch locus is connected with the exception of one isolated point for genera 5 and 6, it is connected for genus 7 and it is connected with the exception of two isolated points for genus 8.

This thesis deals with fast and efficient methods for electrostatic calculations with application in molecular dynamics simulations. The electrostatic calculations are often the most expensive part of MD simulations of charged particles. Therefore, fast and efficient algorithms are required to accelerate these calculations. In this thesis, two types of methods have been considered: FFT-based methods and fast multipole methods (FMM). The major part of this thesis deals with fast N.log(N) and spectrally accurate methods for accelerating the computation of pairwise interactions with arbitrary periodicity. These methods are based on the Ewald decomposition and have been previously introduced for triply and doubly periodic problems under the name of Spectral Ewald (SE) method. We extend the method for problems with singly periodic boundary conditions, in which one of three dimensions is periodic. By introducing an adaptive fast Fourier transform, we reduce the cost of upsampling in the non periodic directions and show that the total cost of computation is comparable with the triply periodic counterpart. Using an FFT-based technique for solving free-space harmonic problems, we are able to unify the treatment of zero and nonzero Fourier modes for the doubly and singly periodic problems. Applying the same technique, we extend the SE method for cases with free-space boundary conditions, i.e. without any periodicity. This thesis is also concerned with the fast multipole method (FMM) for electrostatic calculations. The FMM is very efficient for parallel processing but it introduces irregularities in the electrostatic potential and force, which can cause an energy drift in MD simulations. In this part of the thesis we introduce a regularized version of the FMM, useful for MD simulations, which approximately conserves energy over a long time period and even for low accuracy requirements. The method introduces a smooth transition over the boundary of boxes in the FMM tree and therefore it removes the discontinuity at the error level inherent in the FMM. / <p>QC 20171213</p>

District heating is today, in Sweden, the most common method used for heating buildings in cities. More than half of all the buildings, both commercial and residential, are heated using district heating. The load on the district heating networks are affected by, among other things, the time of the day and different external conditions, such as temperature differences. One has to be able to simulate the heat and pressure losses in the network in order to deliver the amount of heat demanded by the customers. Expansions of district heating networks and disrupted pipes also demand good simulations of the networks. To cope with this, energy companies use simulation software. These software need to contain numerical methods that provide accurate and stable results and at the same time be fast and efficient. At the moment there are available software packages that works but these have some limitations. Among other things you may need to divide the whole network into smaller loops or try to guess how the distribution of pressure and flow in the network looks like. The development in recent years makes it possible to use better and more efficient algorithms for these types of problems. The purpose of this report is therefore to introduce a better and more efficient method than that used in the current situation. This work is the first step in order to replace a current method used in a simulation software provided by Vitec energy. Therefore, we will in this report, stick to computing pressure and flow in the network. The method we will introduce in this report is called the gradient method and it is based on the Newton Raphson method. Unlike with older methods like Hardy Cross which is a relaxation method, you do not have to divide the network into loops. Instead you create a matrix representation of the network that is used in the computations. The idea is also that you should not need to make good initial guesses to get the method to converge quickly. We performed a number of test simulations in order to examine how the method performs. We tested how different initial guesses and how different sizes of the networks affected the number of iterations. The results shows that the model is capable of solving large networks within a reasonable number of iterations. The results also show that the initial guesses have little impact on the number of iterations. Changing the initial guess on the pressure does not affect the number at all but it turns out that changing the initial guess on the flow can affect the number of iterations a little, but not much.

This thesis contains five papers and an introduction. The first four of the included papers are related to financial mathematics and the fifth paper studies a case of mean field game equations. The introduction thus provides background in financial mathematics relevant to the first four papers, and an introduction to mean field game equations related to the fifth paper. In Paper I, we use theory from optimal control to calibrate the so called local volatility process given market data on options. Optimality conditions are in this case given by the solution to a Hamiltonian system of differential equations. Regularization is added by mollifying the Hamiltonian in this system and we solve the resulting equation using a trust region Newton method. We find that our resulting algorithm for the calibration problem is both accurate and robust. In Paper II, we solve the local volatility calibration problem using a technique that is related to - but also different from - the Hamiltonian framework in Paper I. We formulate the optimization problem by means of a Lagrangian multiplier and add a Tikhonov type regularization directly on the parameter we are trying to estimate. The resulting equations are solved with the same trust region Newton method as in Paper II, and again we obtain an accurate and robust algorithm for the calibration problem. Paper III formulates the problem of calibrating a local volatility process to option prices in a way that differs entirely from what is done in the first two papers. We exploit the linearity of the Dupire equation governing the prices to write the optimization problem as a quadratic programming problem. We illustrate by a numerical example that method can indeed be used to find a local volatility that gives good match between model prices and observed market prices on options. Paper IV deals with the hedging problem in finance. We investigate if so called quadratic hedging strategies formulated for a stochastic volatility model can generate smaller hedging errors than obtained when hedging with the standard Black-Scholes framework. We thus apply the quadratic hedging technique as well as the Black-Scholes hedging to observed option prices written on an equity index and calculate the empirical errors in the two cases. Our results indicate that smaller errors can be obtained with quadratic hedging in the models used than with hedging in the Black-Scholes framework. Paper V describes a model of an electricity market consisting of households that try to minimize their electricity cost by dynamic battery usage. We assume that the price process of electricity depends on the aggregated momentaneous electricity consumption. With this assumption, the cost minimization problem of each household is governed by a system of mean field game equations. We also provide an existence and uniqueness result for these mean field game equations. The equations are regularized and the approximate equations are solved numerically. We illustrate how the battery usage affects the electricity price. / Den här avhandlingen innehåller fyra artiklar och en introduktion. De första fyra av de inkluderade artiklarna är relaterade till finansmatematik och den femte artikeln studerar ett fall av medelfältsekvationer. Introduktionen ger bakgrund i finansmatematik som har relevans för de fyra första artiklarna och en introduktion till medelfältsekvationer relaterad till den femte artikeln. I Artikel I använder vi teori från optimal styrning för att kalibrera den så kallade lokala volatilitetsprocessen givet marknadsdata för optionspriser. Optimalitetsvillkor ges i det här fallet av lösningen till ett Hamiltonskt system av differentialekvationer. Vi regulariserar problemet genom att släta ut systemets Hamiltonian och vi löser den resulterande ekvationen med en trust region Newtonmetod. Den resulterande algoritmen är både noggrann och robust i att lösa kalibreringsproblemet. I Artikel II löser vi kalibreringsproblemet för lokal volatilitet med en teknik som är besläktad med - men också skiljer sig från - det Hamiltonska ramverket i Artikel I. Vi formulerar optimeringsproblemet med en Lagrangemultiplikator och använder en Tikhonovregularisering direkt på den parameter vi försöker uppskatta. De resulterande ekvationerna löses med samma trust region Newtonmetod som i Artikel II. Även i detta fall erhåller vi en noggrann och robust algoritm för kalibreringsproblemet. Artikel III formulerar problemet att kalibrera en lokal volatilitet till optionspriser på att sätt som skiljer sig helt från vad som görs i de två första artiklarna. Vi utnyttjar linjäriteten hos Dupires ekvation som ger optionspriserna och kan skriva optimieringsproblemet som ett kvadratiskt programmeringsproblem. Vi illusterar genom ett numeriskt exempel att metoden kan användas för att hitta en lokal volatilitet som ger en bra anpassning av modellpriser till observerade marknadspriser på optioner. Artikel IV behandlar hedgingproblemet i finans. Vi undersöker om så kallad kvadratiska hedgingstrategier formulerade för en stokastisk volatilitetsmodell kan generera mindre hedgingfel än vad som erhålls med hedging i den standardmässiga Black-Scholes modellen. Vi tillämpar således teorin för kvadratisk hedging så väl som hedging med Black-Scholes modell på observerade priser för optioner skrivna på ett aktieindex, och beräknar de empiriska felen i båda fallen. Våra resultat indikerar att mindre fel kan erhållas med kvadratisk hedging med de använda modellerna än med hedging genom Black-Scholes modell. Artikel V beskriver en modell av en elmarknad som består av hushåll som försöker minimera sin elkostnad genom dynamisk batterianvändning. Vi antar att prisprocessen för el beror på den aggregerade momentana elkonsumtionen. Med detta antagande kommer kostnadsminimeringen för varje hushåll att styras av ett system av medelfältsekvationer. Vi ger också ett existens- och entydighetsresultat för dessa medelfältsekvationer. Ekvationerna regulariseras och de approximerade ekvationerna löses numeriskt. Vi illustrerar hur batterianvändningen påverkar elpriset. / <p>QC 20170911</p>

In this thesis, we study numerically the one-dimensional Vlasov equation for a plasma consisting of electrons and infinitely heavy ions. This partial differential equation describes the evolution of the distribution function of particles in the two-dimensional phase space (x,v). The Vlasov equation describes, in statistical mechanics terms, the collective dynamics of particles interacting with long-range forces, but neglects the short-range "collisional" forces. A space plasma consists of electrically charged particles, and therefore the most important long-range forces acting on a plasma are the Lorentz forces created by electromagnetic fields. What makes the numerical solution of the Vlasov equation to a challenging task is firstly that the fully three-dimensional problem leads to a partial differential equation in the six-dimensional phase space, plus time, making it even hard to store a discretized solution in the computer's memory. Secondly, the Vlasov equation has a tendency of structuring in velocity space (due to free streaming terms), in which steep gradients are created and problems of calculating the v (velocity) derivative of the function accurately increase with time. The method used in this thesis is based on the technique of Fourier transforming the Vlasov equation in velocity space and then solving the resulting equation. We have developed a method where the small-scale information in velocity space is removed through an outgoing wave boundary condition in the Fourier transformed velocity space. The position of the boundary in the Fourier transformed variable determines the amount of small-scale information saved in velocity space. The numerical method is used to investigate a phenomenon of tunnelling of information through an ionospheric layer, discovered in experiments, and to assess the accuracy of approximate analytic formulæ describing plasma wave dispersion. The numerical results are compared with theoretical predictions, and further physical experiments are proposed.

The solution of the acoustic wave equation in one space dimension is studied. The PDE is discretized using finite element approximation. A cubic piecewise Lagrange polynomial is used as basis. Consistent finite element and lumped mass schemes are obtained. These schemes are interpreted as finite difference schemes. Error analysis is given for these finite differences (only for constant coefficients).

We have developed an MM–PO hybrid solver designed to deliver reasonable accuracy inexpensively in terms of both CPU-time and memory demands. The solver is based on an iterative block Gauss–Seidel process to avoid unnecessary storage and matrix computations, and can be used to solve the radiation and scattering problems for both disjunct and connected regions. It supports thin wires and dielectrica in the MM domain and has been implemented both as a serial and parallel solver. Numerical experiments have been performed on simple objects to demonstrate certain keyfeatures of the solver, and validate the positive and negative aspects of the MM/PO hybrid. Experiments have also been conducted on more complex objects such as a model aircraft, to demonstrate that the good results from the simpler objects are transferrable to the real life situation. The complex geometries have been used to conduct tests to investigate how well parallelised the code is, and the results are satisfactory.

A second order space and time adaptive method for parallel solution of hyperbolic PDEs on structured grids is presented. The grid is adapted to the underlying solution by successive refinement in blocks. Therefore, there may be jumps in the cell size at the block faces. Special attention is needed at the block boundaries to maintain second order accuracy and stability. The stability of the method is proven for a model problem. The step sizes in space and time are selected based on estimates of the local truncation errors and an error tolerance provided by the user. The global error in the solution is also computed by solving an error equation similar to the original problem on a coarser grid. The performance of the method depends on the number of blocks used in the domain. A method of predicting the optimal number of blocks is presented. The cells are distributed in blocks over the processor set using a number of different partitioning schemes. The method is used to successfully solve a number of test problems where the method selects the appropriate space and time steps according to the error tolerance.

Wave propagation is described by the wave equation, or in the time-periodic case, by the Helmholtz equation. For problems with small wavelengths, high order discretizations must be used to resolve the solution. Two different techniques for finding compact finite difference schemes of high order are studied and compared. The first approach is Numerov's idea of using the equation to transfer higher derivatives to lower order ones for the Helmholtz equation, or, for the wave equation, from time to space. The second principle is the method of deferred correction, where a lower order approximation is used for error correction. For the time-independent Helmholtz problem, sharp estimates for the error are derived, in order to compare the arithmetic complexity for both approaches with a non-compact scheme. The characteristics of the errors for fourth order as well as sixth order accuracy are demonstrated and the advantages and disadvantages of the methods are discussed. A time compact, Numerov-type, fourth order method and a fourth order method using deferred correction in time are studied for the wave equation. Schemes are derived for both the second order formulation of the equation, and for the system in first order form. Stability properties are analyzed and numerical experiments have been performed, for both constant and variable coefficients in the equations. For the first order formulation, a staggered grid is used.