動態規劃的問題並不一定都存在封閉解(closed form solution)，即使存在，其過程往往也相當繁雜。本研究擬以 Gerrard & Haberman (2004) 的模型為基礎，並使用逼近動態規劃理論解的數值方法來求解，此方法參考自黃迪揚(2009)，其研究探討在有無封閉解的動態規劃下，使用此數值方法求解可以得到 逼近解。本篇嘗試延伸其方法，針對不同類型的限制，做更多不同的變化。Gerrard & Haberman (2004)推導出退休後投資於風險性資產與無風險性資產之最適投資策略封閉解， 本研究欲將模型投資之兩資產衍生至三資產，分別投資在高風險資產、中風險資產與無風險資產，實際市場狀況下禁止買空賣空的情況與風險趨避程度限制資產投資比例所造成的影響。並探討兩資產與三資產下的投資結果，並加入不同的目標函數:使用控制變異數的限制式來降低破產機率、控制帳戶差異部位讓投資更具效率性。雖然加入這些限制式會導致目標函 數過於複雜，但是用此數值方法還是可以得出逼近解。 / Dynamic Programming’s solution is not always a closed form. If it do exist, the solution of progress may be too complicated. Our research is based on the investing model in Gerrard & Haberman (2004), using the numerical solution by Huang (2009) to solve the dynamic programming problem. In his research, he found out that whether dynamic programming problem has the closed form, using the numerical solution to solve the problems, which could get similar result. So in our research, we try to use this solution to solve more complicate problems. Gerrard & Haberman (2004) derived the closed form solution of optimal investing strategy in post retirement investment plan, investing in risky asset and riskless asset. In this research we try to invest in three assets, investing in high risk asset, middle risk asset and riskless asset. Forbidden short buying and short selling, how risk attitude affect investment behavior in risky asset and riskless asset. We also observe the numerical result of 2 asset and 3 asset, using different objective functions : using variance control to avoid ruin risk, consideration the distance between objective account and actual account to improve investment effective. Although using these restricts may increase the complication of objective functions, but we can use this numerical solution to get the approximating solution.

This research investigates wave propagation in an elastic half-space with a quadratic nonlinearity in its stress-strain relationship. Different boundary conditions on the surface are considered that result in both one- and two-dimensional wave propagation problems. The goal of the research is to examine the generation of second-order frequency effects and static effects which may be used to determine the nonlinearity present in the material. This is accomplished by extracting the amplitudes of those effects in the frequency domain and analyzing their dependency on the third-order elastic constants (TOEC). For the one-dimensional problems, both analytical approximate solutions as well as numerical simulations are presented. For the two-dimensional problems, numerical solutions are presented whose dependency on the material's nonlinearity is compared to the one-dimensional problems. The numerical solutions are obtained by first formulating the problem as a hyperbolic system of conservation laws, which is then solved numerically using a semi-discrete central scheme. The numerical method is implemented using the package CentPack. In the one-dimensional cases, it is shown that the analytical and numerical solutions are in good agreement with each other, as well as how different boundary conditions may be used to measure the TOEC. In the two-dimensional cases, it is shown that there exist comparable dependencies of the second-order frequency effects and static effects on the TOEC. Finally, it is analytically and numerically investigated how multiple reflections in a plate can be used to simplify measurements of the material nonlinearity in an experiment.

Les tunnels profonds sont souvent construits dans les roches sédimentaires et métamorphiques stratifiées qui présentent habituellement des propriétés anisotropes en raison de leur structure et des propriétés des constituants. Le présent travail vise à étudier les tunnels profonds dans un massif rocheux anisotrope élastique en portant une attention particulière sur les effets des couplages hydromécaniques par des approches analytiques et numériques. Une solution analytique pour un tunnel creusé dans un massif rocheux anisotrope saturé est développée en tenant compte du couplage hydro-mécanique dans le régime permanent. Cette solution analytique est utilisée pour réaliser une série d’études paramétriques afin d'évaluer les effets des différents paramètres du matériau anisotrope sur le comportement du tunnel. Dans un deuxième temps la solution analytique est élargie pour décrire le comportement du tunnel pendant la phase transitoire hydraulique. Afin de compléter ces études analytiques qui prennent en compte seulement un couplage unilatéral (dans le sens que seul le comportement hydraulique influence le comportement mécanique et pas l’inverse) de l’analyse numérique avec un couplage complet, ont été réalisés. Une application de la solution analytique sur la méthode de convergence-confinement est aussi bien abordée qui peut prendre en compte l'influence du front de taille du tunnel sur le travail du soutènement ainsi que sur le massif. La solution obtenue peut servir comme un outil de dimensionnement rapide des tunnels en milieux poreux en le combinant avec des approches de dimensionnement comme celle de convergence confinement. / Deep tunnels are often built in the sedimentary and metamorphic foliated rocks which exhibits usually the anisotropic properties due to the presence of the discontinuity. The analysis of rock and liner stresses due to tunnel construction with the assumption of homogeneous and isotropic rock would be incorrect. The present thesis aims to deal with the deep tunnel in anisotropic rock with a particular emphasis on the effects of hydraulic phenomenon on the mechanical responses or reciprocal effects of hydraulic and mechanical phenomena by combining analytical and numerical approach. On that point of view, a closed-formed solution for tunnel excavated in saturated anisotropic ground is developed taking into account the hydromechanical coupling in steady-state. Based on the analytical solution obtained, parametric studies are conducted to evaluate the effects of different parameters of the anisotropic material on the tunnel behavior. The thesis considers also to extend the analytical solution with a time-dependent behavior which takes into account the impact of the pore pressure distribution on mechanical response over time, i.e., one way coupling modeling. In addition, some numerical analysis based on fully-coupled modeling, i.e., two ways coupling, are conducted which are considered as the complete solution for the analytical solution. An application of the closed-form solution on convergence-confinement method is as well referred which can take into account the influence of the tunnel face on the work of the support as well as the massif. The obtained solution could be used as a quick tool to calibrate tunnels in porous media by combining with design approaches such as convergence-confinement method.

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This thesis covers detailed analysis of bond pricing function. It focuses on connections between mathematical definitions and financial practice and it points out advantages and drawbacks of currently used function. Well known properties of this function are extended to negative internal rate of return values. This topic is further discussed with internal rate of return polynomial equations solving. Taylor series approximation is also shown regarding duration and convexity of bonds.

Title: Algebraic Error in Matrix Computations in the Context of Numerical Solution of Partial Differential Equations Author: Jan Papež Department: Department of Numerical Mathematics Supervisor: prof. Ing. Zdeněk Strakoš, DrSc., Department of Numerical Mathe- matics Abstract: Solution of algebraic problems is an inseparable and usually the most time-consuming part of numerical solution of PDEs. Algebraic computations are, in general, not exact, and in many cases it is even principally desirable not to perform them to a high accuracy. This has consequences that have to be taken into account in numerical analysis. This thesis investigates in this line some closely related issues. It focuses, in particular, on spatial distribution of the errors of different origin across the solution domain, backward error interpretation of the algebraic error in the context of function approximations, incorporation of algebraic errors to a posteriori error analysis, influence of algebraic errors to adaptivity, and construction of stopping criteria for (preconditioned) iterative algebraic solvers. Progress in these issues requires, in our opinion, understanding the interconnections between the phases of the overall solution process, such as discretization and algebraic computations. Keywords: Numerical solution of partial...

The thesis deals with the back analysis of the instrumentated embankment on the subsoil, improved by prefabricated vertical drains. The first part is devoted to the theory of consolidation calculation. Furthermore, the author deals with a parametric study of the analytical calculation of prefabricated drains and the comparsion of the analytical and numerical solution. The last and the largest part of the work is a back analysis of the embankment which was built as a part of the subsoil improvement near the Suez Canal in Egypt by Keller GmbH.

<p>Fractional calculus is the integration and differentiation to an arbitrary or fractional order. The techniques of fractional calculus are not commonly taught in engineering curricula since physical laws are expressed in integer order notation. Dr. Richard Magin (2006) notes how engineers occasionally encounter dynamic systems in which the integer order methods do not properly model the physical characteristics and lead to numerous mathematical operations. In the following study, the application of fractional order calculus to approximate the angular position of the disk oscillating in a Newtonian fluid was experimentally validated. The proposed experimental study was conducted to model the nonlinear response of an oscillating system using fractional order calculus. The integer and fractional order mathematical models solved the differential equation of motion specific to the experiment. The experimental results were compared to the integer order and the fractional order analytical solutions. The fractional order mathematical model in this study approximated the nonlinear response of the designed system by using the Bagley and Torvik fractional derivative. The analytical results of the experiment indicate that either the integer or fractional order methods can be used to approximate the angular position of the disk oscillating in the homogeneous solution. The following research was in collaboration with Dr. Richard Mark French, Dr. Garcia Bravo, and Rajarshi Choudhuri, and the experimental design was derived from the previous experiments conducted in 2018.</p>

The aim of this thesis is to calculate the stiffness and damping coefficients for radial hydrodynamic bearings. Cylindrical and lemon hydrodynamic bearings are considered. The solution to this problem mainly depends on the hydrodynamic pressure in the bearing. The numerical solution of the Reynolds equation is used to calculate the pressure. The effect of variable viscosity and density of the lubricant due to temperature changes is considered. The static equilibrium position of the journal centre is also solved. The stiffness and damping coefficients are determined using small amplitude journal motions about the equilibrium position. Three methods for determining these coefficients are presented. The outcome of this thesis is an algorithm for the calculation of stiffness and damping coefficients for cylindrical and lemon bearings. Results for lemon bearings are presented and comparison with the commercial software DynRot BR is made. The benefit of this thesis is the creation of an algorithm for the calculation of journal centre equilibrium position, a new way of incorporating the temperature changes in the viscosity and the density of the lubricant, and the modification of a method for calculating stiffness and damping coefficients based on experimental analogy.

科学研究費補助金 研究種目:総合研究(A) 課題番号:04302008 研究代表者:三井 斌友 研究期間:1992-1994年度