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.

We have defined and analyzed a semi-Toeplitz preconditioner for time-dependent and steady-state convection-diffusion problems. Analytic expressions for the eigenvalues of the preconditioned systems are obtained. An asymptotic analysis shows that the eigenvalue spectrum of the time-dependent problem is reduced to two eigenvalues when the number of grid points go to infinity. The numerical experiments sustain the results of the theoretical analysis, and the preconditioner exhibits a robust behavior for stretched grids. A semi-Toeplitz preconditioner for the linearized Navier-Stokes equations for compressible flow is proposed and tested. The preconditioner is applied to the linear system of equations to be solved in each time step of an implicit method. The equations are solved with flat plate boundary conditions and are linearized around the Blasius solution. The grids are stretched in the normal direction to the plate and the quotient between the time step and the space step is varied. The preconditioner works well in all tested cases and outperforms the method without preconditioning both in number of iterations and execution time.

New preconditioning techniques for the iterative solution of systems of equations arising from discretizations of partial differential equations are considered. Fundamental solutions, both of differential and difference operators, are used as kernels in discrete, truncated convolution operators. The intention is to approximate inverses of difference operators that arise when discretizing the differential equations. The approximate inverses are used as preconditioners. The technique using fundamental solutions of differential operators is applied to scalar problems in two dimensions, and grid independent convergence is obtained for a first order differential equation. The problem of computing fundamental solutions of difference operators is considered, and we propose a new algorithm. It can be used also when the symbol of the difference operator is not invertible everywhere, and it is applicable in two or more dimensions. Fundamental solutions of difference operators are used to construct preconditioners for non-linear systems of difference equations in two dimensions. Grid independent convergence is observed for two standard finite difference discretizations of the Euler equations in a non-axisymmetric duct.

Two unstructured finite volume solvers for the Maxwell equations in 2D and 3D are introduced. The solvers are a generalization of FD–TD to unstructured grids and they use a third-order staggered Adams–Bashforth scheme for time discretization. Analysis and experiments of this time integrator reveal that we achieve a long term stable solution on general triangular grids. A Fourier analysis shows that the 2D solver has excellent dispersion characteristics on uniform triangular grids. In 3D a spatial filter of Laplace type is introduced to enable long simulations without suffering from late time instability. The recursive convolution method proposed by Luebbers et al. to extend FD–TD to permit frequency dispersive materials is here generalized to the 3D solver. A better modelling of materials which have a strong frequency dependence in their constitutive parameters is obtained through the use of a general material model. The finite volume solvers are not intended to be stand-alone solvers but one part in two hybrid solvers with FD–TD. The numerical examples in 2D and 3D demonstrate that the hybrid solvers are superior to stand-alone FD–TD in terms of accuracy and efficiency.

We have developed an iterative solver for the Moment Method. It computes a matrix–vector product with the multilevel Fast Multipole Method, which makes the method scale with the number of unknowns. The iterative solver is of Block Quasi-Minimum Residual type and can handle several right-hand sides at once. The linear system is preconditioned with a Sparse Approximate Inverse, which is modified to handle dense matrices. The solver is parallelized on shared memory machines using OpenMP. To verify the method some tests are conducted on varying geometries. We use simple geometries to show that the method works. We show that the method scales on several processors of a shared memory machine. To prove that the method works for real life problems, we do some tests on large scale aircrafts. The largest test is a one million unknown simulation on a full scale model of a fighter aircraft.