>Magister Scientiae - MSc / We present the Black-Scholes Merton partial differential equation (BSMPDE) and its analytical solution. We present the Black-Scholes option pricing model and list some limitations of this model. We also present a nonlinear model (the Frey-Patie model) that may improve on one of these limitations. We apply various numerical methods on the BSMPDE and run simulations to compare which method performs best in approximating the value of a European put option based on the maximum errors each method produces when we vary some parameters like the interest rate and the volatility. We re-apply the same finite difference methods on the nonlinear model. / 2025

A two-dimensional discontinuous Galerkin finite element method algorithm in the time domain was developed for calculation of the radar cross-section of an arbitrary object. The algorithm was formed using local nodal basis functions in each element and coupling them via numerical upwind flux. Both transverse electric and transverse magnetic polarization, as well as three different dispersive material models, were handled. The computational domain was effectively truncated with low reflections using the uniaxial perfectly matched layer method. Two different time stepping methods were used, low-storage explicit Runge-Kutta and Leap-Frog, to allow for flexibility in the time step and application of a stabilization method. The algorithm was verified with geometries, which have analytical expressions, and an existing validated code. The algorithm was also compared to an existing algorithm, which utilized the continuous finite element method with implicit time stepping, and showed outstanding performance regarding computation time and memory allocation. Since the developed algorithm had explicit time stepping could no general conclusions favoring any of the methods beyond these specific algorithms be made. The results still encouraged continued development of the DGFEM algorithm, where the expansion into three dimensions and optimizations could be explored further.

La résolution numérique directe des problèmes de transport de photons en interaction avec un milieu matériel est très coûteuse en mémoire et temps CPU. Pour pallier ce problème, une méthode consiste à construire des modèles réduits dont la résolution est moins coûteuse. La littérature abonde de ce genre de modèles : modèles probabilistes (Monte-Carlo), modèles aux moments (M₁, PN), modèles aux ordonnées discrètes (SN), modèles de diffusion... Dans cette thèse, nous nous intéressons aux modèles PN dans lesquels l'opérateur de transport est approché par projections sur une base tronquée d'harmoniques sphériques. Ces modèles ont l'avantage d'être arbitrairement précis sur la dimension angulaire et ne présentent pas les défauts connus des autres méthodes (bruit stochastique, "effets de raies") pouvant briser les éventuelles symétries du problème. Ce dernier point est capital pour la simulation d'expériences de fusion par confinement inertiel (FCI) où la symétrie sphérique joue un rôle important dans la précision des résultats. Nous étudions donc dans cette thèse la structure mathématique des modèles PN ainsi que leur discrétisation dans le cas d'une géométrie 1D sphérique.Nous commençons par le cas du transport linéaire dans le vide. Même dans ce cas simple, les équations du modèle PN contiennent des termes sources d'origine géométrique dont la discrétisation s'avère délicate. Jusqu'à présent, les différents schémas utilisés étaient insatisfaisants pour les raisons suivantes : (1) mauvais comportement au voisinage de r = 0 (phénomène de "flux-dip"), (2) non préservation des équilibres stationnaires, (3) pas de preuve formelle de stabilité. À la lumière de récents travaux, nous proposons une nouvelle discrétisation qui capture exactement les états d'équilibres. Nous démontrons en particulier la stabilité en norme L² du schéma. Nous étendons par la suite ce schéma au cas du transport de photons dans un milieu matériel figé et nous nous intéressons au comportement du schéma en limite diffusion (propriété "asymptotic-preserving").Dans un second temps, nous nous intéressons au couplage entre rayonnement et hydrodynamique. Devant l'absence de consensus sur les modèles "transport" d'hydrodynamique radiative issus de la littérature, nous établissons une étude comparative de ceux-ci basée sur leurs propriétés mathématiques. Nous nous intéressons particulièrement aux propriétés suivantes : (1) conservation de l'énergie et de l'impulsion, (2) précision des effets comobiles, (3) existence d'une entropie mathématiques compatible et (4) restitution de la limite diffusion. Notre étude se réduit aux modèles dits "mixed-frame" et une attention particulière est toujours portée sur l'approximation "PN" de l'opérateur de transport. Nous identifions des défauts (conservation ou entropie) sur des modèles existants et proposons une correction entropique conduisant à un modèle PN satisfaisant toutes les propriétés mathématiques listées ci-dessus. / Computational costs for direct numerical simulations of photon transport problemsare very high in terms of CPU time and memory. One way to tackle this issue is todevelop reduced models that a cheaper to solve numerically. There exists number of these models : moments models, discrete ordinates models (SN), diffusion-like models... In this thesis, we focus on PN models in which the transport operator is approached by mean of a truncated development on the spherical harmonics basis. These models are arbitrary accurate in the angular dimension and are rotationnaly invariants (in multiple space dimensions). The latter point is fundamental when one wants to simulate inertial confinment fusion (ICF) experiments where the spherical symmetry plays an important part in the accuracy of the numerical solutions. We study the mathematical structure of the PN models and construct a new numerical method in the special case of a one dimensionnal space dimension with spherical symmetry photon transport problems. We first focus on a linear transport problem in the vacuum. Even in this simple case, it appears in the PN equations geometrical source terms that are stiff in the neighborhood of r = 0 and thus hard to discretise. Existing numerical methods are not satisfactory for multiple reasons : (1) unaccuracy in the neighborhood of r = 0 ("flux-dip"), (2) do not capture steady states (well-balanced scheme), (3) no stability proof. Following recent works, we develop a new well-balanced scheme for which we show the L² stability. We then extend the scheme for photon transport problems within a no moving media, the linear Boltzmann equation, and interest ourselves on its behavior in the diffusion limit (asymptotic-preserving property). In a second part, we consider radiation hydrodynamics problems. Since modelisation of these problems is still under discussion in the litterature, we compare a set of existing models by mean of mathematical analysis and establish a hierarchy. For each model, we focus on the following mathematical properties : (1) energy and impulsion conservation, (2) accuracy of the comobile effects, (3) existence of a mathematical entropy and (4) behavior in the diffusion limit. Our study reduces to « laboratory frame » models and we are still interested in the PN approximation of the transport operator. We identify defects in entropy structure of existing models and propose an entroy correction which leads to PN-based radiation hydrodynamics models which satisfy all the properties listed above.

No description available.

Low frequency acoustic room behaviour can be approximated using numerical methods. Traditionally, music studio control rooms are built with complex geometries, making their eigenmodes difficult to predict mathematically. Hence, a summation-by-parts method with simultaneous-approximation-terms is derived to approximate the time dependent acoustic wave equation in three dimensions. The derived model is limited to rectangular prismatic rooms but planned to be expanded to handle complex geometries in the future. Semi-reflecting boundary conditions are used, corresponding to tabulated reflection and absorption properties of real. walls. Two speakers are modeled as omnidirectional point sources placed on a boundary, to mimic common studio setups. Through tests and examination of eigenvalues of the matrices in the method, conditions for stability and reflection coefficients are derived. Simulations of sound pressure distribution produced by the model correlate well to room mode theory, suggesting the model to be accurate in the application of predicting low frequency acoustic room behaviour. However, the convergence rate of the model turns out to be lower than expected when point sources are introduced. Future steps towards applying the model to real music studio control rooms include modeling the walls as changes in density and wave speed rather than boundaries of the domain. This would potentially allow more complex geometries to be modeled within a larger, rectangular domain.

Calcium signaling has been identified with triggering of gene transcriptions associated with learning and neuroprotection in neurons. Studies indicate that dysregulation of calcium signaling is correlated with severe Alzheimer Disease pathologies. A stable calcium wave or signal arising from triggers in dendritic synapses needs to reach soma with constant amplitude for proper functioning of neurons. In this study, we introduce "CalciumSim", a calcium dynamics simulator which works on dimensionally reduced model. Numerical analysis is conducted to obtain the best configuration of neuron geometry to make the code efficient and fast. Alongside, biologically important insights are derived by modulating and changing parameters of the simulation. The ability of "CalciumSim" to work with real neuron geometries allows user to study calcium signalling in a realistic model. / Mathematics

No description available.

The accuracy of moment method solutions to electromagnetic scattering problems has been studied by many researchers. Error bounds for the moment method have been obtained in terms of Sobolev norms of the current solution. Motivated by the historical origins of Sobolev spaces as energy spaces, it is shown that the Sobolev norm used in these bounds is equivalent to the forward scattering amplitude, for the case of 2D scattering from a PEC circular cylinder. A slightly weaker relationship is obtained for 3D scattering from a PEC sphere. These results provide a physical meaning for abstract solution error bounds in terms of the power radiated by the error in the current solution. It is further shown that bounds on the Sobolev norm of the current error imply a bound on the error in the computed backscattering amplitude. Since Sobolev-based error bounds do not provide the actual error in a solution nor identify its source, the error in typical moment method scattering solutions for smooth cylindrical geometries is analyzed. To quantify the impact of mesh element size, approximate integration of moment matrix elements, and geometrical discretization error on the accuracy of computed surface currents and scattering amplitudes, error estimates are derived analytically for the circular cylinder. These results for the circular cylinder are empirically compared to computed error values for other smooth scatterer geometries, with consistent results obtained. It is observed that moment method solutions to the magnetic field integral equation are often less accurate for a given grid than corresponding solutions to the electric field integral equation. Building from the error analysis, the cause of this observation is proposed to be the identity operator in the magnetic formulation. A regularization of the identity operator is then derived that increases the convergence rate of the discretized 2D magnetic field integral equation by three orders.

We consider the problem of optimizing the choice of interpolation nodes such that the interpolation error is minimized, given the constraint that none of the nodes may be placed inside a forbidden region. Restricting the problem to using one-dimensional polynomial interpolants, we explore different ways of quantifying the interpolation error; such as the integral of the absolute/squared difference between the interpolated function and the interpolant, or the Lebesgue constant, which compares the interpolant with the best possible approximating polynomial of a given degree. The interpolation error then serves as a cost function that we intend to minimize using gradient-based optimization algorithms. The results are compared with existing theory about the optimal choice of interpolation nodes in the absence of a forbidden region (mainly due to Chebyshev) and indicate that the Chebyshev points of the second kind are near-optimal as interpolation nodes for optimizing the Lebesgue constant, whereas placing the points as close as possible to the forbidden region seems optimal for minimizing the integral of the difference between the interpolated function and the interpolant. We conclude that the Chebyshev points of the second kind serve as a great choice of interpolation nodes, even with the constraint on the placement of the nodes explored in this paper, and that the interpolation nodes should be placed as close as possible to the forbidden region in order to minimize the interpolation error.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24168号 / 理博第4859号 / 新制||理||1695(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)准教授 笹倉 直樹, 准教授 髙山 史宏, 教授 橋本 幸士 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM