Diss. (sammanfattning) Uppsala : Univ., 2003. / Härtill 5 uppsatser.

Diss. Linköping : Linköpings universitet, 2008.

<p>Multi-scale wave propagation problems are computationally costly to    solve by traditional techniques because the smallest scales must be    represented over a domain determined by the largest scales of the    problem.  We have developed new numerical methods for multi-scale wave    propagation in the framework of heterogeneous multi-scale methods.  The    numerical methods couples simulations on macro and micro scales with    data exchange between models of different scales.  With the new method    we are able to consider a general class of problems including some    problems where a homogenized equation is unknown.  We show that the    complexity of the new method is significantly lower than that of    traditional techniques.  Numerical results are presented from problems    in one, two and three dimensional and for finite and long time.  We also    analyze the method, in one and several dimensions and for finite time,    using Fourier analysis.</p>

<p>In this thesis a new method for the option pricing will be introduced with</p><p>the help of the Mellin transforms. Firstly, the Mellin transform techniques for</p><p>options on a single underlying stock is presented. After that basket options</p><p>will be considered. Finally, an improvement of existing numerical results</p><p>applied to Mellin transforms for 1-basket and 2-basket American Put Option</p><p>will be discussed concisely. Our approach does not require either variable</p><p>transformations or solving diusion equations.</p>

This thesis concerns numerical techniques for two phase flowsimulations; the two phases are immiscible and incompressible fluids. Strategies for accurate simulations are suggested. In particular, accurate approximations of the weakly discontinuousvelocity field, the discontinuous pressure, and the surface tension force and a new model for simulations of contact line dynamics are proposed. In two phase flow problems discontinuities arise in the pressure and the gradient of the velocity field due to surface tension forces and differences in the fluids' viscosity. In this thesis, a new finite element method which allows for discontinuities along an interface that can be arbitrarily located with respect to the mesh is presented. Using standard linear finite elements, the method is for an elliptic PDE proven to have optimal convergence order and a system matrix with condition number bounded independently of the position of the interface.The new finite element method is extended to the incompressible Stokes equations for two fluid systemsand enables accurate approximations of the weakly discontinuous velocity field and the discontinuous pressure. An alternative way to handle discontinuities is regularization. In this thesis, consistent regularizations of Dirac delta functions with support on interfaces are proposed. These regularized delta functions make it easy to approximate surface tension forces in level set methods. A new model for simulating contact line dynamics is also proposed. Capillary dominated flows are considered and it is assumed that contact line movement is driven by the deviation of the contact angle from its static value. This idea is used together with the conservative level set method. The need for fluid slip at the boundary is eliminated by providing a diffusive mechanism for contact line movement. Numerical experiments in two space dimensions show that the method is able to qualitatively correctly capture contact line dynamics. / QC 20110503

This thesis concerns the analysis and numerical simulation of wave propagation problems described by systems of linear hyperbolic partial differential equations. A major challenge in wave propagation problems is numerical simulation of high frequency waves. When the wavelength is very small compared to the overall size of the computational domain, we encounter a multiscale problem. Examples include the forward and the inverse seismic wave propagation, radiation and scattering problems in computational electromagnetics and underwater acoustics. In direct numerical simulations, the accuracy of the approximate solution is determined by the number of grid points or elements per wavelength. The computational cost to maintain constant accuracy grows algebraically with the frequency, and for sufficiently high frequency, direct numerical simulations are no longer feasible. Other numerical methods are therefore needed. Asymptotic methods, for instance, are good approximations for very high frequency waves. They are based on constructing asymptotic expansions of the solution. The accuracy increases with increasing frequency for a fixed computational cost. Most asymptotic techniques rely on geometrical optics equations with frequency independent unknowns. There are however two deficiencies in the geometrical optics solution. First, it does not include diffraction effects. Secondly, it breaks down at caustics. Geometrical theory of diffraction provides a technique for adding diffraction effects to the geometrical optics approximation by introducing diffracted rays. In papers 1 and 2 we present a numerical algorithm for computing an important type of diffracted rays known as creeping rays. Another asymptotic model which is valid also at caustics is based on Gaussian beams. In papers 3 and 4, we present an error analysis of Gaussian beams approximation and develop a new numerical algorithm for computing Gaussian beams, respectively. Another challenge in computation of wave propagation problems arises when the system of equations consists of second order hyperbolic equations involving mixed space-time derivatives. Examples include the harmonic formulation of Einstein’s equations and wave equations governing elasticity and acoustics. The classic computational treatment of such second order hyperbolic systems has been based on reducing the systems to first order differential forms. This treatment has however the disadvantage of introducing auxiliary variables with their associated constraints and boundary conditions. In paper 5, we treat the problem in the second order differential form, which has advantages for both computational efficiency and accuracy over the first order formulation. Finally, paper 6 concerns the concept of well-posedness for a class of linear hyperbolic initial boundary value problems which are not boundary stable. The well-posedness is well established for boundary stable hyperbolic systems for which we can obtain sharp estimates of the solution including estimates at boundaries. There are, however, problems which are not boundary stable but are well-posed in a weaker sense, i.e., the problems for which an energy estimate can be obtained in the interior of the domain but not on the boundaries. We analyze a model problem of this type. Possible applications arise in elastic wave equations and Maxwell’s equations describing glancing and surface waves. / QC 20100830

his work is concentrated on efforts to efficiently compute properties of systems, modelled by differential equations, involving multiple scales. Goal oriented adaptivity is the common approach to all the treated problems. Here the goal of a numerical computation is to approximate a functional of the solution to the differential equation and the numerical method is adapted to this task. The thesis consists of four papers. The first three papers concern the convergence of adaptive algorithms for numerical solution of differential equations; based on a posteriori expansions of global errors in the sought functional, the discretisations used in a numerical solution of the differential equiation are adaptively refined. The fourth paper uses expansion of the adaptive modelling error to compute a stochastic differential equation for a phase-field by coarse-graining molecular dynamics. An adaptive algorithm aims to minimise the number of degrees of freedom to make the error in the functional less than a given tolerance. The number of degrees of freedom provides the convergence rate of the adaptive algorithm as the tolerance tends to zero. Provided that the computational work is proportional to the degrees of freedom this gives an estimate of the efficiency of the algorithm. The first paper treats approximation of functionals of solutions to second order elliptic partial differential equations in bounded domains of ℝd, using isoparametric $d$-linear quadrilateral finite elements. For an adaptive algorithm, an error expansion with computable leading order term is derived %. and used in a computable error density, which is proved to converge uniformly as the mesh size tends to zero. For each element an error indicator is defined by the computed error density multiplying the local mesh size to the power of 2+d. The adaptive algorithm is based on successive subdivisions of elements, where it uses the error indicators. It is proved, using the uniform convergence of the error density, that the algorithm either reduces the maximal error indicator with a factor or stops; if it stops, then the error is asymptotically bounded by the tolerance using the optimal number of elements for an adaptive isotropic mesh, up to a problem independent factor. Here the optimal number of elements is proportional to the d/2 power of the Ldd+2 quasi-norm of the error density, whereas a uniform mesh requires a number of elements proportional to the d/2 power of the larger L1 norm of the same error density to obtain the same accuracy. For problems with multiple scales, in particular, these convergence rates may differ much, even though the convergence order may be the same. The second paper presents an adaptive algorithm for Monte Carlo Euler approximation of the expected value E[g(X(τ),\τ)] of a given function g depending on the solution X of an \Ito\ stochastic differential equation and on the first exit time τ from a given domain. An error expansion with computable leading order term for the approximation of E[g(X(T))] with a fixed final time T&gt;0 was given in~[Szepessy, Tempone, and Zouraris, Comm. Pure and Appl. Math., 54, 1169-1214, 2001]. This error expansion is now extended to the case with stopped diffusion. In the extension conditional probabilities are used to estimate the first exit time error, and difference quotients are used to approximate the initial data of the dual solutions. For the stopped diffusion problem the time discretisation error is of order N-1/2 for a method with N uniform time steps. Numerical results show that the adaptive algorithm improves the time discretisation error to the order N-1, with N adaptive time steps. The third paper gives an overview of the application of the adaptive algorithm in the first two papers to ordinary, stochastic, and partial differential equation. The fourth paper investigates the possibility of computing some of the model functions in an Allen--Cahn type phase-field equation from a microscale model, where the material is described by stochastic, Smoluchowski, molecular dynamics. A local average of contributions to the potential energy in the micro model is used to determine the local phase, and a stochastic phase-field model is computed by coarse-graining the molecular dynamics. Molecular dynamics simulations on a two phase system at the melting point are used to compute a double-well reaction term in the Allen--Cahn equation and a diffusion matrix describing the noise in the coarse-grained phase-field. / QC 20100823

In this thesis a new method for the option pricing will be introduced with the help of the Mellin transforms. Firstly, the Mellin transform techniques for options on a single underlying stock is presented. After that basket options will be considered. Finally, an improvement of existing numerical results applied to Mellin transforms for 1-basket and 2-basket American Put Option will be discussed concisely. Our approach does not require either variable transformations or solving diusion equations.

In this thesis the connections between the boundary conditions and the vari- ance of the solution to a stochastic partial differential equation (PDE) are investigated. In particular a hyperbolical system of PDE’s with stochastic initial and boundary data are considered. The problem is shown to be well- posed on a class of boundary conditions through the energy method. Stability is shown by using summation-by-part operators coupled with simultaneous- approximation-terms. By using the energy estimates, the relative variance of the solutions for different boundary conditions are analyzed. It is concluded that some types of boundary conditions yields a lower variance than others. This is verified by numerical computations.

<p>A finite element based numerical method for the two-dimensional elliptic interface problems is presented. Due to presence of these interfaces the problem will contain discontinuities in the coefficients and singularities in the right hand side that are represented by delta functionals along the interface. As a result, the solution to the interface problem and its derivatives may have jump discontinuities. The introduced method is specifically designed to handle this features of the solution using non-body fitted grids, i.e. the grids are not aligned with the interfaces.</p><p>The main idea is to modify the standard basis function in the vicinity of the interface such that the jump conditions are well approximated. The resulting finite element space is, in general, non-conforming. The interface itself is represented by a set of Lagrangian markers together with a parametric description connecting them. To illustrate the abilities of the method, numerical tests are presented. For all the considered test problems, the introduced method has been shown to have super-linear or second order of convergence. Our approach is also compared with the standard finite element method.</p><p>Finally, the method is applied to the interface Stokes problem, where the interface represents an elastic stretched band immersed in fluid. Since we assume the fluid to be homogeneous, the Stokes equations are reduced to a sequence of three Poisson problems that are solved with our method. The numerical results agree well with those found in the literature.</p>