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There are many regularization parameter selection methods that can be used when solving inverse problems, but it is not clear which one is best suited for the inverse dispersion problem. The suitability of three different methods for solving the inverse dispersion problem are evaluated here in order to pick a suitable method for these kinds of problems in the future. The regularization parameter selection methods are used to solve the separable non-linear inverse dispersion problem which is adjusted and solved as a linear inverse problem. It is solved with Tikhonov regularization and the model is a time integrated Gaussian puff model. The dispersion problem is used with different settings and is solved with the three methods. The three methods are generalized cross-validation, L-curve method and quasi-optimality criterion. They produce rather different solutions and the results show that generalized cross-validation is the best choice. The other methods are less stable and the errors are sometimes orders of magnitude larger than the errors from generalized cross-validation.

I detta projekt undersöktes hur numeriska lösningar till PDEer beter sig då randvillkor ges med metoder som används i dagens väder- och klimatsimuleringar, så kallat Daviesrelaxering vilket gör att problemet inte är maximalt semibegränsat. Advektionsekvationen och de linjäriserade Eulerekvationerna löstes numeriskt enligt vedertagen matematisk metodik (maximalt semibegränsat) och jämfördes med lösningar då Daviesrelaxering (inte maximalt semibegränsat) användes. Det visade sig att lösningar till advektionsekvationen blir fel, och resultaten tyder på att de linjäriserade Eulerekvationerna blir fel då Daviesrelaxering används.

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We propose three models to study morphology formations in interacting ternary mixtures with the evaporation of one component. Our models involve three distinct length scales: microscopic, mesoscopic, and respectively, macroscopic. The real-world application we have in mind concerns charge transport through the heterogeneous structures arising in the fabrication of organic solar cells. As first model, we propose a microscopic 3-spins lattice dynamics with short-range interactions between the considered species. This microscopic model is approximated numerically via a Monte Carlo Metropolis-based algorithm. We explore the effect of the model parameters (volatility of the solvent, system's temperature, and interaction strengths) on the structure of the formed morphologies. Our second model is built upon the first one, by introducing a new mesoscale corresponding to the size of block spins. The link between these two models as well as between the effects of the model parameters and formed morphologies are studied in detail. These two models offer insight into cross-sections of the modeling box. Our third model encodes a macroscopic view of the evaporating mixture. We investigate its capability to lead to internal coherent structures. We propose a macroscopic system of nonlinearly coupled Cahn-Hilliard equations to capture numerical results for a top view of the modeling box. Effects of effective evaporation rates, effective interaction energy parameters, and degree of polymerization on the wanted morphology formation are explored via the computational platform FEniCS using a FEM approximation of a suitably linearized system. High-performance computing resources and Python-based parallel implementations have been used to facilitate the numerical approximation of the three models.

In this thesis we study the Cauchy problem for elliptic equations. It arises in many areas of application in science and engineering as a problem of reconstruction of solutions to elliptic equations in a domain from boundary measurements taken on a part of the boundary of this domain. The Cauchy problem for elliptic equations is known to be ill-posed. We use an iterative regularization method based on alternatively solving a sequence of well-posed mixed boundary value problems for the same elliptic equation. This method, based on iterations between Dirichlet-Neumann and Neumann-Dirichlet mixed boundary value problems was first proposed by Kozlov and Maz’ya [13] for Laplace equation and Lame’ system but not Helmholtz-type equations. As a result different modifications of this original regularization method have been proposed in literature. We consider the Robin-Dirichlet iterative method proposed by Mpinganzima et.al [3] for the Cauchy problem for the Helmholtz equation in bounded domains. We demonstrate that the Robin-Dirichlet iterative procedure is convergent for second order elliptic equations with variable coefficients provided the parameter in the Robin condition is appropriately chosen. We further investigate the convergence of the Robin-Dirichlet iterative procedure for the Cauchy problem for the Helmholtz equation in a an unbounded domain. We derive and analyse the necessary conditions needed for the convergence of the procedure. In the numerical experiments, the precise behaviour of the procedure for different values of k2 in the Helmholtz equation is investigated and the results show that the speed of convergence depends on the choice of the Robin parameters, μ0 and μ1. In the unbounded domain case, the numerical experiments demonstrate that the procedure is convergent provided that the domain is truncated appropriately and the Robin parameters, μ0 and μ1 are also chosen appropriately. / <p>Funding Agencies: International Science Programme (ISP) and the Eastern Africa Universities Mathematics Programme (EAUMP).</p>

In this thesis, we study the Cauchy problem for an elliptic equation. We use Dirichlet-Robin iterations for solving the Cauchy problem. This allows us to include in our consideration elliptic equations with variable coefficient as well as Helmholtz type equations. The algorithm consists of solving mixed boundary value problems, which include the Dirichlet and Robin boundary conditions. Convergence is achieved by choice of parameters in the Robin conditions. We have also reformulated the Cauchy problem for the Helmholtz equation as an operator equation. We investigate the conditions under which this operator equation is well-defined. Furthermore, we have also discussed possible extensions to the case where the Helmholtz operator is replaced by non-symmetric differential operators by using similar operator equations and model problems which are used for symmetric differential operators. We have observed that the Dirichlet - Robin iterations are equivalent to the classical Landweber iterations. Having formulated the problem in terms of an operator equation is an advantage since it lets us to implement more sophisticated iterative methods based on Krylov subspaces. In particular, we consider the Conjugate gradient method (CG) and the Generalized minimal residual method (GMRES). The numerical results shows that all the methods work well. / <p>Funding agencies: ISP and the Eastern African UniversitiesMathematics Programme (EAUMP)</p>

We have studied examples of random walk methods (RWM) applied to electric problems, solving the Laplace equation. Especially we did calculationswith digital pictures as domains, and evaluated an improvement in calculation speed for the random walk method. The latter was done by varying thestep length depending on the distance to boundaries and obstructions in thedomain. Statistical numerical experiments have been used to see if the improved method had an effect on the accuracy of the solution. The results inthis work have been compared with the finite element method (FEM) usingthe commercial software Comsol. We have looked at four different geometrieswith varying complexity with respect to inner boundary conditions.

AbstractMaxwell’s laws state that the divergence of a magnetic field always remains zero. In thecontinuous setting this is true for the magnetohydrodynamics equations. However, in thediscrete setting, the solution does not always remain divergence-free. The goal of this project isto find two-dimensional finite element meshes for which the magnetic field solution to themagnetohydrodynamics equations remained divergence-free. The report includes the derivationfor a condition of a divergence-free mesh, examples of meshes for which this condition holdstrue, and lastly numerical results to confirm that the solutions using these meshes aredivergence-free.

A finite difference scheme for the incompressible Navier-Stokes equations in 2-dimensionalcurvilinear coordinates is derived. The scheme uses high-order Summation-By-Parts operatorsand boundaries are handled by the Projection method. Using the energy method, the scheme isproven stable and well-posed with Dirichlet boundary conditions for general curvilineartransforms. Numerical tests show convergence of the scheme for high-order operators on theTaylor-Green vortex problem using three different transformations. Furthermore, the Lid-drivencavity problem is tested and it shows that employing curvilinear transforms to create a densergrid near boundaries can achieve greater accuracy when observing boundary layer effects.