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Pfefferer, Johannes [Verfasser], Thomas [Akademischer Betreuer] Apel, and Arnd [Akademischer Betreuer] Rösch. "Numerical analysis for elliptic Neumann boundary control problems on polygonal domains / Johannes Pfefferer. Universität der Bundeswehr München, Fakultät für Bauingenieurwesen und Umweltwissenschaften. Gutachter: Thomas Apel ; Arnd Rösch. Betreuer: Thomas Apel." Neubiberg : Universitätsbibliothek der Universität der Bundeswehr München, 2014. http://d-nb.info/1054706824/34.
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Lu, Xing. "La contrôlabilité frontière exacte et la synchronisation frontière exacte pour un système couplé d’équations des ondes avec des contrôles frontières de Neumann et des contrôles frontières couplés de Robin." Thesis, Strasbourg, 2018. http://www.theses.fr/2018STRAD013/document.
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Dans cette thèse, nous étudions la synchronisation, qui est un phénomène bien répandu dans la nature. Elle a été observée pour la première fois par Huygens en 1665. En se basant sur les résultats de la contrôlabilité frontière exacte, pour un système couplé d’équations des ondes avec des contrôles frontières de Neumann, nous considérons la synchronisation frontière exacte (par groupes), ainsi que la détermination de l’état de synchronisation. Ensuite, nous considérons la contrôlabilité exacte et la synchronisation exacte (par groupes) pour le système couplé avec des contrôles frontières couplés de Robin. A cause du manque de régularité de la solution, nous rencontrons beaucoup plus de difficultés. Afin de surmonter ces difficultés, on obtient un résultat sur la trace de la solution faible du problème de Robin grâce aux résultats de régularité optimale de Lasiecka-Triggiani sur le problème de Neumann. Ceci nous a permis d’établir la contrôlabilité exacte, et, par la méthode de la perturbation compacte, la non-contrôlabilité exacte du système. De plus, nous allons étudier la détermination de l’état de synchronisation, ainsi que la nécessité des conditions de compatibilité des matrices de couplageThis thesis studies the widespread natural phenomenon of synchronization, which was first observed by Huygens en 1665. On the basis of the results on the exact boundary controllability, for a coupled system of wave equations with Neumann boundary controls, we consider its exact boundary synchronization (by groups), as well as the determination of the state of synchronization. Then, we consider the exact boundary controllability and the exact boundary synchronization (by groups) for the coupled system with coupled Robin boundary controls. Due to difficulties from the lack of regularity of the solution, we have to face a bigger challenge. In order to overcome this difficulty, we take advantage of the regularity results for the mixed problem with Neumann boundary conditions (Lasiecka and Triggiani) to discuss the exact boundary controllability, and by the method of compact perturbation, to obtain the non-exact controllability for the system
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Winkler, Max [Verfasser], Thomas [Akademischer Betreuer] Apel, Olaf [Akademischer Betreuer] Steinbach, and Roland [Akademischer Betreuer] Herzog. "Finite Element Error Analysis for Neumann Boundary Control Problems on Polygonal and Polyhedral Domains / Max Winkler. Universität der Bundeswehr München, Fakultät für Bauingenieurwesen und Umweltwissenschaften. Betreuer: Thomas Apel. Gutachter: Thomas Apel ; Olaf Steinbach ; Roland Herzog." Neubiberg : Universitätsbibliothek der Universität der Bundeswehr München, 2015. http://d-nb.info/1077773129/34.
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Alsaedy, Ammar, and Nikolai Tarkhanov. "Normally solvable nonlinear boundary value problems." Universität Potsdam, 2013. http://opus.kobv.de/ubp/volltexte/2013/6507/.
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We study a boundary value problem for an overdetermined elliptic system of nonlinear first order differential equations with linear boundary operators.
Such a problem is solvable for a small set of data, and so we pass to its variational formulation which consists in minimising the discrepancy. The Euler-Lagrange equations for the variational problem are far-reaching analogues of the classical Laplace equation. Within the framework of Euler-Lagrange equations we specify an operator on the boundary whose zero set consists precisely of those boundary data for which the initial problem is solvable. The construction of such operator has much in common with that of the familiar Dirichlet to Neumann operator. In the case of linear problems we establish complete results.
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Yang, Xue. "Neumann problems for second order elliptic operators with singular coefficients." Thesis, University of Manchester, 2012. https://www.research.manchester.ac.uk/portal/en/theses/neumann-problems-for-second-order-elliptic-operators-with-singular-coefficients(2e65b780-df58-4429-89df-6d87777843c8).html.
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In this thesis, we prove the existence and uniqueness of the solution to a Neumann boundary problem for an elliptic differential operator with singular coefficients, and reveal the relationship between the solution to the partial differential equation (PDE in abbreviation) and the solution to a kind of backward stochastic differential equations (BSDE in abbreviation).This study is motivated by the research on the Dirichlet problem for an elliptic operator (\cite{Z}). But it turns out that different methods are needed to deal with the reflecting diffusion on a bounded domain. For example, the integral with respect to the boundary local time, which is a nondecreasing process associated with the reflecting diffusion, needs to be estimated. This leads us to a detailed study of the reflecting diffusion. As a result, two-sided estimates on the heat kernels are established. We introduce a new type of backward differential equations with infinity horizon and prove the existence and uniqueness of both L2 and L1 solutions of the BSDEs. In this thesis, we use the BSDE to solve the semilinear Neumann boundary problem. However, this research on the BSDEs has its independent interest. Under certain conditions on both the "singular" coefficient of the elliptic operator and the "semilinear coefficient" in the deterministic differential equation, we find an explicit probabilistic solution to the Neumann problem, which supplies a L2 solution of a BSDE with infinite horizon. We also show that, less restrictive conditions on the coefficients are needed if the solution to the Neumann boundary problem only provides a L1 solution to the BSDE.
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Orey, Maria de Serpa Salema Reis de. "Factorization of elliptic boundary value problems by invariant embedding and application to overdetermined problems." Doctoral thesis, Faculdade de Ciências e Tecnologia, 2011. http://hdl.handle.net/10362/8677.
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Dissertação para obtenção do Grau de Doutor em
MatemáticaThe purpose of this thesis is the factorization of elliptic boundary value problems defined in cylindrical domains, in a system of decoupled first order initial value problems. We begin with the Poisson equation with mixed boundary conditions, and use the method of invariant embedding: we embed our initial problem in a family of similar problems, defined in sub-domains of the initial domain, with a moving boundary, and an additional condition in the moving boundary. This factorization is inspired by the technique of invariant temporal embedding used in Control Theory when computing the optimal feedback, for, in fact, as we show, our initial problem may be defined as an optimal control problem. The factorization thus obtained may be regarded as a generalized block Gauss LU factorization. From this procedure emerges an operator that can be either the Dirichlet-to-Neumann or the Neumann-to-Dirichlet operator, depending on which boundary data is given on the moving boundary. In any case this operator verifies a Riccati equation that is studied directly by using an Yosida regularization. Then we extend the former results to more general strongly elliptic operators. We also obtain a QR type factorization of the initial problem, where Q is an orthogonal operator and R is an upper triangular operator. This is related to a least mean squares formulation of the boundary value problem. In addition, we obtain the factorization of overdetermined boundary value problems, when we consider an additional Neumann boundary condition: if this data is not compatible with the initial data, then the problem has no solution. In order to solve it, we introduce a perturbation in the original problem and minimize the norm of this perturbation, under the hypothesis of existence of solution. We deduce the normal equations for the overdetermined problem and, as before, we apply the method of invariant embedding to factorize the normal equations in a system of decoupled first order initial value problems.
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López, Ríos Juan Carlos. "Water-wave equations and free boundary problems: inverse problems and control." Tesis, Universidad de Chile, 2015. http://repositorio.uchile.cl/handle/2250/135179.
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Doctor en Ciencias de la Ingeniería, Mención Modelación MatemáticaEn este trabajo se aborda el problema de existencia de algunos tipos de soluciones para las ecuaciones de ondas en el agua así como la relación que existe entre estas soluciones y la forma de un fondo impermeable sobre la que se desliza el fluido. Empezamos por describir las ecuaciones que modelan el fenómeno físico a partir de las leyes de conservación; el modelo general de las ecuaciones de ondas en el agua, escrito para la restricción de la velocidad potencial a la superficie libre, es \begin{equation*} \left\{ \begin{aligned} &\partial_t\zeta-G(\zeta,b)\psi=0, \\ &\partial_t\psi+g\zeta+\frac{1}{2}|\nabla_X\psi|^2-\frac{1}{2(1+|\nabla_X\zeta|^2)}(G(\zeta,b)\psi+\nabla_X\zeta\cdot\nabla_X\psi)^2=0, \end{aligned} \right. \end{equation*} donde $G=G(\zeta,b)\psi$ es el operador Dirichlet-Neumann, el cual contiene la información del fondo $b$, \begin{equation*} G(\zeta,b)\psi:=-\sqrt{1+|\nabla_X\zeta|^2}\partial_n\phi|_{y=\zeta(t,X)}, \end{equation*} y \begin{equation*} \left\{ \begin{array}{rl} & \Delta\phi=0, \quad \R\times(b,\zeta), \\ & \phi|_{y=\zeta}=\psi, \quad \partial_n \phi|_{y=b(X)}=0. \end{array} \right. \end{equation*} Después de describir las condiciones para un teorema de existencia y unicidad de soluciones de las ecuaciones de ondas en el agua, en espacios de Sobolev, nos preguntamos sobre el mínimo de datos necesarios, sobre la superficie libre, para identificar el fondo de manera única. Por la relación que existe entre el operador Dirichlet-Neumann y la velocidad dentro del fluido y utilizando la propiedad de continuación única de las funciones armónicas hemos probado que basta conocer el perfil, la velocidad potencial y la velocidad normal en un instante de tiempo dado y un abierto de $\R$, aún cuando nuestro sistema es de evolución. En la segunda parte se estudia la existencia de soluciones en forma de salto hidráulico para las ecuaciones estacionarias de ondas en el agua, en dimensión dos y su relación con la velocidad aguas arriba, caracterizada por un parámetro adimensional, llamado el número de Froude, $F$, como consecuencia de la existencia de ramas de bifurcación de la solución trivial para el problema \begin{equation*} \mathcal{F}(\eta,F)=\eta+F\widetilde{\psi}_{y^{\prime }}+\frac{\epsilon}{2}(% \widetilde{\psi}_{x^{\prime }}^2+\widetilde{\psi}_{y^{\prime }}^2)-\epsilon^2\eta_x\widetilde{\psi}_{x^{\prime }}\widetilde{\psi}% _{y^{\prime }}+\frac{\epsilon^3}{2}\eta_x^2\widetilde{\psi}_{y^{\prime }}^2; \end{equation*} donde \begin{equation*} \left\{ \begin{aligned} &\Delta\widetilde{\psi}=\epsilon G, && (-L,L)\times(0,1), \\ &\widetilde{\psi}_{x'}=0, && x'=-L,L, \\ &\widetilde{\psi}=0, && y'=0, \\ &\widetilde{\psi}=-F\eta, && y'=1. \end{aligned} \right. \end{equation*}
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PERROTTA, Antea. "Differential Formulation coupled to the Dirichlet-to-Neumann operator for scattering problems." Doctoral thesis, Università degli studi di Cassino, 2020. http://hdl.handle.net/11580/75845.
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This Thesis proposes the use of the Dirichlet-to-Neumann (DtN) operator to improve the accuracy and the efficiency of the numerical solution of an electromagnetic scattering problem, described in terms of a differential formulation. From a general perspective, the DtN operator provides the “connection” (the mapping) between the Dirichlet and the Neumann data onto a proper closed surface. This allows truncation of the computational domain when treating a scattering problem in an unbounded media. Moreover, the DtN operator provides an exact boundary condition, in contrast to other methods such as Perfectly Matching Layer (PML) or Absorbing Boundary Conditions (ABC). In addition, when the surface where the DtN is introduced has a canonical shape, as in the present contribution, the DtN operator can be computed analytically. This thesis is focused on a 2D geometry under TM illumination. The numerical model combines a differential formulation with the DtN operator defined onto a canonical surface where it can be computed analytically. Test cases demonstrate the accuracy and the computational advantage of the proposed technique.
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Kulkarni, Mandar S. "Multi-coefficient Dirichlet Neumann type elliptic inverse problems with application to reflection seismology." Birmingham, Ala. : University of Alabama at Birmingham, 2009. https://www.mhsl.uab.edu/dt/2010r/kulkarni.pdf.
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Thesis (Ph. D.)--University of Alabama at Birmingham, 2009.Title from PDF t.p. (viewed July 21, 2010). Additional advisors: Thomas Jannett, Tsun-Zee Mai, S. S. Ravindran, Günter Stolz, Gilbert Weinstein. Includes bibliographical references (p. 59-64).
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Kamyad, A. V. "Boundary control problems for the multi-dimensional diffusion equation." Thesis, University of Leeds, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.382023.
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Wahbi, Wassim. "Contrôle stochastique sur les réseaux." Thesis, Paris Sciences et Lettres (ComUE), 2018. http://www.theses.fr/2018PSLED072.
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Cette thèse se décompose en trois grandes parties, qui traitent des EDP quasi linéaires paraboliques sur une jonction, des diffusions stochastiques sur une jonction, et du contrôle optimal également sur une jonction, avec contrôle au point de jonction. Nous commençons au premier Chapitre par introduire une nouvelle classe d'EDP non dégénérée et quasi linéaire, satisfaisant une condition de Neumann (ou de Kirchoff) non linéaire et non dynamique au point de jonction. Nous prouvons l'existence d'une solution classique, ainsi que son unicité. L'une des motivations portant sur l'étude de ce type d'EDP, est de faire le lien avec la théorie du contrôle optimale sur les jonctions, et de caractériser la fonction valeur de ce type de problème à l'aide des équations d'Hamilton Jacobi Bellman. Ainsi, au Chapitre suivant, nous formulons une preuve donnant l'existence d'une diffusion sur une jonction. Ce processus admet un temps local, dont l'existence et la variation quadratique dépendent essentiellement de l'hypothèse d'ellipticité des termes du second ordre au point de jonction. Nous formulerons une formule d'Itô pour ce processus. Ainsi, grâce aux résultats de ces deux Chapitres, nous formulerons dont le dernier Chapitre un problème de contrôle stochastique sur les jonctions, avec contrôle au point de jonction. L'espace des contrôles est celui des mesures de Probabilités résolvant un problème martingale. Nous prouvons la compacité de l'espace des contrôles admissibles, ainsi que le principe de la programmation dynamiqueThis thesis consists of three parts which deal with quasi linear parabolic PDE on a junction, stochastic diffusion on a junction and stochastic control on a junction with control at the junction point. We begin in the first Chapter by introducing and studying a new class of non degenerate quasi linear parabolic PDE on a junction, satisfying a Neumann (or Kirchoff) non linear and non dynamical condition at the junction point. We prove the existence and the uniqueness of a classical solution. The main motivation of studying this new mathematical object is the analysis of stochastic control problems with control at the junction point, and the characterization of the value function of the problem in terms of Hamilton Jacobi Bellman equations. For this end, in the second Chapter we give a proof of the existence of a diffusion on a junction. The process is characterized by its local time at the junction point, whose quadratic approximation is centrally related to the ellipticty assumption of the second order terms around the junction point.We then provide an It's formula for this process. Thanks to the previous results, in the last Chapter we study a problem of stochastic control on a junction, with control at the junction point. The set of controls is the set of the probability measures (admissible rules) satisfying a martingale problem. We prove the compactness of the admissible rules and the dynamic programming principle
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Goldberg, H., and F. Tröltzsch. "On a SQP-multigrid technique for nonlinear parabolic boundary control problems." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199801210.
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An optimal control problem governed by the heat equation with nonlinear boundary
conditions is considered. The objective functional consists of a quadratic terminal
part and a quadratic regularization term. It is known, that an SQP method converges
quadratically to the optimal solution of the problem. To handle the quadratic optimal
control subproblems with high precision, very large scale mathematical programming
problems have to be treated. The constrained problem is reduced to an unconstrained
one by a method due to Bertsekas. A multigrid approach developed by Hackbusch is
applied to solve the unconstrained problems. Some numerical examples illustrate the
behaviour of the method.
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Fernando, Chathuri [Verfasser]. "Optimal Control of Free Boundary Value Problems in Thermoelasticity / Chathuri Fernando." München : Verlag Dr. Hut, 2018. http://d-nb.info/1164294075/34.
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Benincasa, Tommaso <1981>. "Analysis and optimal control for the phase-field transition system with non-homogeneous Cauchy-Neumann boundary conditions." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2010. http://amsdottorato.unibo.it/3066/1/benincasa_tommaso_tesi.pdf.
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Benincasa, Tommaso <1981>. "Analysis and optimal control for the phase-field transition system with non-homogeneous Cauchy-Neumann boundary conditions." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2010. http://amsdottorato.unibo.it/3066/.
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Lakhany, Asif. "Finite element recovery techniques in adaptive error control." Thesis, Brunel University, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.262505.
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Brancati, Alessandro. "Boundary element method for fast solution of acoustic problems : active and passive noise control." Thesis, Imperial College London, 2010. http://hdl.handle.net/10044/1/6139.
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This thesis presents boundary element formulations for three-dimensional acoustic problems of active (ANC) and passive (PNC) noise control. A new boundary element strategy, referred to as RABEM (Rapid Acoustic Boundary Element Method), has been formulated and implemented for acoustic problems. The assembly time for both the system matrix and the right hand side vector is accelerated using a Hierarchical-matrix approach based on the Adaptive Cross Approximation (ACA). Two different H-matrix-GMRES solvers (one without preconditioners and one with a block diagonal preconditioner) are developed and tested for low and high frequency problems including noise emanated by aircraft approaching an airport. A new formulation for solving the ANC based on attenuating the unwanted sound in a control volume (CV) rather than cancelling it at a single point is presented. The noise attenuation is obtained by minimising the square modules of two acoustic quantities - the potential and one component of the particle velocity - within the CV. The two formulations presented include a single and a double secondary source, respectively. Several examples are presented to demonstrate the e fficiency of the proposed technique. A new approach, based on sensitivity analysis, for determining the optimum locations of the CV and the optimum location/orientation of the secondary source is presented. The optimisation procedure is based upon a first order method and minimises a suitable cost function by using its gradient. The procedure to calculate the cost function gradients is explained in detail. Finally, a PNC strategy applied to the interior of an aircraft cabin is investigated. A lower noise level is achieved through the introduction of a new textile with a higher noise absorbing coe cient than a conventional textile, especially at low frequencies. The so-called "bubble concept", which consists of adding cap insertions at the sides of the passenger head, is also investigated.
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Chierici, Andrea <1992>. "Mathematical and Numerical Models for Boundary Optimal Control Problems Applied to Fluid-Structure Interaction." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amsdottorato.unibo.it/9856/1/thesis_chierici.pdf.
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The main purpose of this work is to develop mathematical and
numerical methods for the optimal control of fluid-structure interaction simulations. In particular, we focus on the Koiter shell fluid-structure model and on the adjoint formalism for the control problem. Using the Koiter approach, the dimensionality of the solid is reduced to reduce the computational cost of the fluid-structure simulations. In order to couple the fluid and the structure domains, the Koiter shell equations are embedded into the fluid equations as a Robin boundary condition. The coupling fluid-structure conditions are automatically treated in an implicit way, so the stability of the numerical scheme is preserved.
This model has many applications in cases where a fluid interacts with a thin membrane that deforms mainly in the normal direction. Then, an adjoint-based optimal control theory of the presented Koiter fluid-structure model in the steady case is studied. In fact, a boundary optimal control theory is applied to the fluid-structure Koiter model, including the existence of the solution of the fluid-structure problem, the existence of the optimal solution and regularity and differentiability properties.
Moreover, the fractional operators are introduced to be applied to the framework described above, in order to model properly the regularization term in the boundary optimal control problems.
All the numerical simulations presented in this work, with the exception of the fractional simulations, have been simulated with the in-house multigrid finite element based code FEMuS.
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Coco, Armando. "Finite-Difference Ghost-Cell Multigrid Methods for Elliptic problems with Mixed Boundary Conditions and Discontinuous Coefficients." Doctoral thesis, Università di Catania, 2012. http://hdl.handle.net/10761/1107.
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The work of this thesis is devoted to the development of an original and general numerical method for solving the elliptic equation in an arbitrary domain (described by a level-set function) with general boundary conditions (Dirichlet, Neumann, Robin, ...) using Cartesian grids. It can be then considered an immersed boundary method, and the scheme we use is based on a finite-difference ghost-cell technique. The entire problem is solved by an effective multigrid solver, whose components have been suitably constructed in order to be applied to the scheme.
The method is extended to the more challenging case of discontinuous coefficients, and the multigrid is suitable modified in order to attain the optimal convergence factor of the whole iteration procedure.
The development of the multigrid is an important feature of this thesis, since multigrid solvers for discontinuous coefficients maintaining the optimal convergence factor without depending on the jump in the coefficient and on the problem size is recently studied in literature.
The method is second order accurate in the solution and its gradient. A convergence proof for the first order scheme is provided, while second order is confirmed by several numerical tests.
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Zhang, Jindong. "Nonlinear dynamic analysis and optimal control of shallow shells by field-boundary-element approach." Diss., Georgia Institute of Technology, 1987. http://hdl.handle.net/1853/32961.
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Schuhmann, Patrick [Verfasser]. "On some Two-Dimensional Singular Stochastic Control Problems and their Free-Boundary Analysis / Patrick Schuhmann." Bielefeld : Universitätsbibliothek Bielefeld, 2021. http://d-nb.info/1238780865/34.
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O'Donoghue, Padraic Eimear. "Boundary integral equation approach to nonlinear response control of large space structures : alternating technique applied to multiple flaws in three dimensional bodies." Diss., Georgia Institute of Technology, 1985. http://hdl.handle.net/1853/20685.
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John, Christian [Verfasser], and Fredi [Akademischer Betreuer] Tröltzsch. "Optimal Dirichlet boundary control problems of high-lift configurations with control and integral state constraints / Christian John. Betreuer: Fredi Tröltzsch." Berlin : Universitätsbibliothek der Technischen Universität Berlin, 2011. http://d-nb.info/1014971624/34.
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Gao, Guangyue. "Some Controllability and Stabilization Problems of Surface Waves on Water with Surface tension." Diss., Virginia Tech, 2015. http://hdl.handle.net/10919/64377.
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The thesis consists of two parts. The first part discusses the initial value problem of a fifth-order Korteweg-de Vries type of equation
wt + wxxx - wxxxxx - n∑j=1 ajwjwx = 0, w(x, 0) = w0(x)
posed on a periodic domain x ∈ [0, 2π] with boundary conditions wix(0, t) = wix(2π, t), i = 0, 2, 3, 4 and an L2-stabilizing feedback control law wx(2π, t) = αwx(0, t) + (1 - α)wxxx(0; t) where n is a fixed positive integer, aj, j = 1, 2, ... n, α are real constants, and |α| < 1. It is shown that for w0(x) ∈ H1α(0, 2π) with the boundary conditions described above, the problem is locally well-posed for w ∈ C([0, T]; H1α(0, 2π)) with a conserved volume of w, [w] = ∫2π0 w(x, t)dx. Moreover, the solution with small initial condition exists globally and approaches to [w0(x)]/(2π) as t → + ∞. The second part concerns wave motions on water in a rectangular basin with a wave generator mounted on a side wall. The linear governing equations are used and it is assumed that the surface tension on the free surface is not zero. Two types of generators are considered, flexible and rigid. For the flexible case, it is shown that the system is exactly controllable. For the rigid case, the system is not exactly controllable in a finite-time interval. However, it is approximately controllable. The stability problem of the system with the rigid generator controlled by a static feedback is also studied and it is proved that the system is strongly stable for this case.Ph. D.
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Frenander, Hannes. "High-order finite difference approximations for hyperbolic problems : multiple penalties and non-reflecting boundary conditions." Doctoral thesis, Linköpings universitet, Beräkningsmatematik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-134127.
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In this thesis, we use finite difference operators with the Summation-By-Partsproperty (SBP) and a weak boundary treatment, known as SimultaneousApproximation Terms (SAT), to construct high-order accurate numerical schemes.The SBP property and the SAT’s makes the schemes provably stable. The numerical procedure is general, and can be applied to most problems, but we focus on hyperbolic problems such as the shallow water, Euler and wave equations. For a well-posed problem and a stable numerical scheme, data must be available at the boundaries of the domain. However, there are many scenarios where additional information is available inside the computational domain. In termsof well-posedness and stability, the additional information is redundant, but it can still be used to improve the performance of the numerical scheme. As a first contribution, we introduce a procedure for implementing additional data using SAT’s; we call the procedure the Multiple Penalty Technique (MPT). A stable and accurate scheme augmented with the MPT remains stable and accurate. Moreover, the MPT introduces free parameters that can be used to increase the accuracy, construct absorbing boundary layers, increase the rate of convergence and control the error growth in time. To model infinite physical domains, one need transparent artificial boundary conditions, often referred to as Non-Reflecting Boundary Conditions (NRBC). In general, constructing and implementing such boundary conditions is a difficult task that often requires various approximations of the frequency and range of incident angles of the incoming waves. In the second contribution of this thesis,we show how to construct NRBC’s by using SBP operators in time. In the final contribution of this thesis, we investigate long time error bounds for the wave equation on second order form. Upper bounds for the spatial and temporal derivatives of the error can be obtained, but not for the actual error. The theoretical results indicate that the error grows linearly in time. However, the numerical experiments show that the error is in fact bounded, and consequently that the derived error bounds are probably suboptimal.
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Raymond, Jean-Pierre, and Fredi Tröltzsch. "Second Order Sufficient Optimality Conditions for Nonlinear Parabolic Control Problems with State Constraints." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199801014.
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In this paper, optimal control problems for semilinear parabolic equations with
distributed and boundary controls are considered. Pointwise constraints on the control and on
the state are given. Main emphasis is laid on the discussion of second order sufficient optimality
conditions. Sufficiency for local optimality is verified under different assumptions imposed
on the dimension of the domain and on the smoothness of the given data.
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Tröltzsch, F. "Lipschitz stability of solutions to linear-quadratic parabolic control problems with respect to perturbations." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199801229.
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We consider a class of control problems governed by a linear parabolic initial-boundary
value problem with linear-quadratic objective and pointwise constraints on the control.
The control system contains different types of perturbations. They appear in the
linear part of the objective functional, in the right hand side of the equation,
in its boundary condition, and in the initial value. Making use of parabolic regularity
in the whole scale of $L^p$ the known Lipschitz stability in the $L^2$-norm
is improved to the supremum-norm.
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Schmitt, Johann Michael [Verfasser]. "Optimal Control of Initial-Boundary Value Problems for Hyperbolic Balance Laws with Switching Controls and State Constraints / Johann Michael Schmitt." München : Verlag Dr. Hut, 2019. http://d-nb.info/1188516450/34.
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Kasnakoglu, Cosku. "Reduced order modeling, nonlinear analysis and control methods for flow control problems." Columbus, Ohio : Ohio State University, 2007. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1195629380.
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Hegarty-Cremer, Solene G. "Spatial control and cell guidance in evolving biological tissues." Thesis, Queensland University of Technology, 2021. https://eprints.qut.edu.au/207246/1/Solene_Hegarty-Cremer_Thesis.pdf.
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In this thesis, a mathematical model for tissue growth under curvature control and directed cell guidance is derived. The model extends previous mathematical work by adding a tangential cell velocity. A numerical solver is implemented to solve the model and the solutions show that new cases of tissue growth can now be simulated thanks to the extension derived in this thesis. Finally, the model is fit to data on bone pore infilling and is used to examine hypotheses about atypical tissue growth.
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Alves, Michele de Oliveira. "Um problema de extensão relacionado a raiz quadrada do Laplaciano com condição de fronteira de Neumann." Universidade de São Paulo, 2010. http://www.teses.usp.br/teses/disponiveis/45/45132/tde-19012011-231320/.
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Neste trabalho definimos o operador não local, raiz quadrada do Laplaciano com condição de fronteira de Neumann, através do método de extensão harmônica. O estudo foi feito com o auxílio das séries de Fourier em domínios limitados, como sendo o intervalo, o quadrado e a bola. Posteriormente, aplicamos nosso estudo, à problemas elípticos não lineares envolvendo o operador não local raiz quadrada do Laplaciano com condição de fronteira de Neumann.In this work we define the non-local operator, square root of the Laplacian with Neumann boundary condition, using the method of harmonic extension. The study was done with the aid of Fourier series in bounded domains, as the interval, the square and the ball. Subsequently, we apply our study, the nonlinear elliptic problems involving non-local operator square root of the Laplacian with Neumann boundary condition.
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CISTERNINO, MARCO. "A parallel second order Cartesian method for elliptic interface problems and its application to tumor growth model." Doctoral thesis, Politecnico di Torino, 2012. http://hdl.handle.net/11583/2497156.
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This theses deals with a parallel Cartesian method to solve elliptic problems
with complex interfaces and its application to elliptic irregular domain problems in the framework of a tumor growth model. This method is based on a finite differences scheme and is second order accurate in the whole domain.
The originality of the method lies in the use of additional unknowns located on the interface, allowing to express the interface transmission conditions. The method is described and the details of its parallelization, performed with the PETSc library, are provided. Numerical validations of the method follow with comparisons to other related methods in literature. A numerical study of the parallelized method is also given. Then, the method is applied to solve elliptic irregular domain problems appearing in a three-dimensional continuous tumor growth model, the two-species Darcy model. The approach used in this application
is based on the penalization of the interface transmission conditions, in order to impose homogeneous Neumann boundary conditions on the border of an irregular domain. The simulations of model are provided and they show the ability of the method to impose a good approximation of the considered boundary conditions.
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Cisternino, Marco. "A parallel second order Cartesian method for elliptic interface problems and its application to tumor growth model." Phd thesis, Université Sciences et Technologies - Bordeaux I, 2012. http://tel.archives-ouvertes.fr/tel-00690743.
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Cette thèse porte sur une méthode cartésienne parallèle pour résoudre des problèmes elliptiques avec interfaces complexes et sur son application aux problèmes elliptiques en domaine irrégulier dans le cadre d'un modèle de croissance tumorale. La méthode est basée sur un schéma aux différences fi nies et sa précision est d'ordre deux sur tout le domaine. L'originalité de la méthode consiste en l'utilisation d'inconnues additionnelles situées sur l'interface et qui permettent d'exprimer les conditions de transmission à l'interface. La méthode est décrite et les détails sur la parallélisation, réalisée avec la bibliothèque PETSc, sont donnés. La méthode est validée et les résultats sont comparés avec ceux d'autres méthodes du même type disponibles dans la littérature. Une étude numérique de la méthode parallélisée est fournie. La méthode est appliquée aux problèmes elliptiques dans un domaine irrégulier apparaissant dans un modèle continue et tridimensionnel de croissance tumorale, le modèle à deux espèces du type Darcy . L'approche utilisée dans cette application est basée sur la pénalisation des conditions de transmission a l'interface, afin de imposer des conditions de Neumann homogènes sur le bord d'un domaine irrégulier. Les simulations du modèle sont fournies et montrent la capacité de la méthode à imposer une bonne approximation de conditions au bord considérées.
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Rogovs, Sergejs [Verfasser], Thomas [Akademischer Betreuer] Apel, Thomas [Gutachter] Apel, Olaf [Gutachter] Steinbach, and Dmitriy [Gutachter] Leykekhman. "Pointwise Error Estimates for Boundary Control Problems on Polygonal Domains / Sergejs Rogovs ; Gutachter: Thomas Apel, Olaf Steinbach, Dmitriy Leykekhman ; Akademischer Betreuer: Thomas Apel ; Universität der Bundeswehr München, Fakultät für Bauingenieurwesen und Umweltwissenschaften." Neubiberg : Universitätsbibliothek der Universität der Bundeswehr München, 2018. http://d-nb.info/1193497329/34.
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Rogovs, Sergejs [Verfasser], Thomas [Akademischer Betreuer] Apel, Thomas Gutachter] Apel, Olaf [Gutachter] [Steinbach, and Dmitriy [Gutachter] Leykekhman. "Pointwise Error Estimates for Boundary Control Problems on Polygonal Domains / Sergejs Rogovs ; Gutachter: Thomas Apel, Olaf Steinbach, Dmitriy Leykekhman ; Akademischer Betreuer: Thomas Apel ; Universität der Bundeswehr München, Fakultät für Bauingenieurwesen und Umweltwissenschaften." Neubiberg : Universitätsbibliothek der Universität der Bundeswehr München, 2018. http://nbn-resolving.de/urn:nbn:de:bvb:706-6144.
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Cekić, Mihajlo. "The Calderón problem for connections." Thesis, University of Cambridge, 2017. https://www.repository.cam.ac.uk/handle/1810/267829.
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This thesis is concerned with the inverse problem of determining a unitary connection $A$ on a Hermitian vector bundle $E$ of rank $m$ over a compact Riemannian manifold $(M, g)$ from the Dirichlet-to-Neumann (DN) map $\Lambda_A$ of the associated connection Laplacian $d_A^*d_A$. The connection is to be determined up to a unitary gauge equivalence equal to the identity at the boundary. In our first approach to the problem, we restrict our attention to conformally transversally anisotropic (cylindrical) manifolds $M \Subset \mathbb{R}\times M_0$. Our strategy can be described as follows: we construct the special Complex Geometric Optics solutions oscillating in the vertical direction, that concentrate near geodesics and use their density in an integral identity to reduce the problem to a suitable $X$-ray transform on $M_0$. The construction is based on our proof of existence of Gaussian Beams on $M_0$, which are a family of smooth approximate solutions to $d_A^*d_Au = 0$ depending on a parameter $\tau \in \mathbb{R}$, bounded in $L^2$ norm and concentrating in measure along geodesics when $\tau \to \infty$, whereas the small remainder (that makes the solution exact) can be shown to exist by using suitable Carleman estimates. In the case $m = 1$, we prove the recovery of the connection given the injectivity of the $X$-ray transform on $0$ and $1$-forms on $M_0$. For $m > 1$ and $M_0$ simple we reduce the problem to a certain two dimensional $\textit{new non-abelian ray transform}$. In our second approach, we assume that the connection $A$ is a $\textit{Yang-Mills connection}$ and no additional assumption on $M$. We construct a global gauge for $A$ (possibly singular at some points) that ties well with the DN map and in which the Yang-Mills equations become elliptic. By using the unique continuation property for elliptic systems and the fact that the singular set is suitably small, we are able to propagate the gauges globally. For the case $m = 1$ we are able to reconstruct the connection, whereas for $m > 1$ we are forced to make the technical assumption that $(M, g)$ is analytic in order to prove the recovery. Finally, in both approaches we are using the vital fact that is proved in this work: $\Lambda_A$ is a pseudodifferential operator of order $1$ acting on sections of $E|_{\partial M}$, whose full symbol determines the full Taylor expansion of $A$ at the boundary.
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Santos, Simão Pedro Silva. "Calculus of variations of Herglotz type." Doctoral thesis, Universidade de Aveiro, 2017. http://hdl.handle.net/10773/22503.
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Doutoramento em MatemáticaWe consider several problems based on Herglotz’s generalized variational problem. We dedicate two chapters to extensions on Herglotz’s generalized variational problem to higher-order and first-order problems with time delay, using a variational approach. In the last four chapters, we rewrite Herglotz's type problems in the optimal control form and use an optimal control approach. We prove generalized higher- order Euler-Lagrange equations, first without and then with time delay; higher-order natural boundary conditions; Noether's first theorem for the first-order problem of Herglotz with time delay; Noether's first theorem for higher-order problems of Herglotz without and with time delay; and existence of Noether currents as a version of Noether's second theorem of optimal control.
Consideramos vários problemas com base no problema variacional generalizado de Herglotz. Dois capítulos são dedicados à extensão do problema variacional generalizado de Herglotz para ordem superior e para problemas de primeira ordem com atraso no tempo, utilizando uma abordagem variacional. Nos últimos quatro capítulos, reescrevemos os problemas de Herglotz na forma do controlo ótimo e usamos essa abordagem. Demonstramos equações generalizadas de Euler-Lagrange de ordem superior, inicialmente sem e depois com atraso no tempo; condições de fronteira de ordem superior; o primeiro teorema de Noether para o problema de Herglotz de primeira ordem com atraso no tempo; o primeiro teorema de Noether para problemas de ordem superior de Herglotz sem e com atraso no tempo; e a existência de leis de conservação de Noether numa versão do segundo teorema de Noether do controlo ótimo.
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Chowdhury, Sudipto. "Finite Element Analysis of Interior and Boundary Control Problems." Thesis, 2016. http://etd.iisc.ernet.in/2005/3717.
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The primary goal of this thesis is to study finite element based a priori and a posteriori error estimates of optimal control problems of various kinds governed by linear elliptic PDEs (partial differential equations) of second and fourth orders. This thesis studies interior and boundary control (Neumann and Dirichlet) problems.
The initial chapter is introductory in nature. Some preliminary and fundamental results of finite element methods and optimal control problems which play key roles for the subsequent analysis are reviewed in this chapter. This is followed by a brief literature survey of the finite element based numerical analysis of PDE constrained optimal control problems. We conclude the chapter with a discussion on the outline of the thesis.
An abstract framework for the error analysis of discontinuous Galerkin methods for control constrained optimal control problems is developed in the second chapter. The analysis establishes the best approximation result from a priori analysis point of view and delivers a reliable and efficient a posteriori error estimator. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. Subsequently, the applications of p p - interior penalty methods for a boundary control problem as well as a distributed control problem governed by the bi-harmonic equation subject to simply supported boundary conditions are discussed through the abstract analysis.
In the third chapter, an alternative energy space based approach is proposed for the Dirichlet boundary control problem and then a finite element based numerical method is designed and analyzed for its numerical approximation. A priori error estimates of optimal order in the energy norm and the m norm are derived. Moreover, a reliable and efficient a posteriori error estimator is derived with the help an auxiliary problem.
An energy space based Dirichlet boundary control problem governed by bi-harmonic equation is investigated and subsequently a l y - interior penalty method is proposed and analyzed for it in the fourth chapter. An optimal order a priori error estimate is derived under the minimal regularity conditions. The abstract error estimate guarantees optimal order of convergence whenever the solution has minimum regularity. Further an optimal order l l norm error estimate is derived.
The fifth chapter studies a super convergence result for the optimal control of an interior control problem with Dirichlet cost functional and governed by second order linear elliptic PDE. An optimal order a priori error estimate is derived and subsequently a super convergence result for the optimal control is derived. A residual based reliable and efficient error estimators are derived in a posteriori error control for the optimal control.
Numerical experiments illustrate the theoretical results at the end of every chapter. We conclude the thesis stating the possible extensions which can be made of the results presented in the thesis with some more problems of future interest in this direction.
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Sardar, Bidhan Chandra. "Study of Optimal Control Problems in a Domain with Rugose Boundary and Homogenization." Thesis, 2016. http://hdl.handle.net/2005/2883.
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Mathematical theory of partial differential equations (PDEs) is a pretty old classical area with wide range of applications to almost every branch of science and engineering. With the advanced development of functional analysis and operator theory in the last century, it became a topic of analysis. The theory of homogenization of partial differential equations is a relatively new area of research which helps to understand the multi-scale phenomena which has tremendous applications in a variety of physical and engineering models, like in composite materials, porous media, thin structures, rapidly oscillating boundaries and so on. Hence, it has emerged as one of the most interesting and useful subject to study for the last few decades both as a theoretical and applied topic.
In this thesis, we study asymptotic analysis (homogenization) of second-order partial differential equations posed on an oscillating domain. We consider a two dimensional oscillating domain (comb shape type) consisting of a fixed bottom region and an oscillatory (rugose) upper region. We introduce optimal control problems for the Laplace equation. There are mainly two types of optimal control problems; namely distributed control and boundary control. For distributed control problems in the oscillating domain, one can apply control on the oscillating part or on the fixed part and similarly for boundary control problem (control on the oscillating boundary or on the fixed part the boundary). We consider all the four cases, namely distributed and boundary controls both on the oscillating part and away from the oscillating part.
The present thesis consists of 8 chapters. In Chapter 1, a brief introduction to homogenization and optimal control is given with relevant references. In Chapter 2, we introduce the oscillatory domain and define the basic unfolding operators which will be used throughout the thesis. Summary of the thesis is given in Chapter 3 and future plan in Chapter 8. Our main contribution is contained in Chapters 4-7.
In chapters 4 and 5, we study the asymptotic analysis of optimal control problems namely distributed and boundary controls, respectively, where the controls act away from the oscillating part of the domain. We consider both L2 cost functional as well as Dirichlet (gradient type) cost functional. We derive homogenized problem and introduce the limit optimal control problems with appropriate cost functional. Finally, we show convergence of the optimal solution, optimal state and associate adjoint solution. Also convergence of cost-functional.
In Chapter 6, we consider the periodic controls on the oscillatory part together with Neumann condition on the oscillating boundary. One of the main contributions is the characterization of the optimal control using unfolding operator. This characterization is new and also will be used to study the limiting analysis of the optimality system.
Chapter 7 deals with the boundary optimal control problem, where the control is applied through Neumann boundary condition on the oscillating boundary with a suitable scaling parameter. To characterize the optimal control, we introduce boundary unfolding operators which we consider as a novel approach. This characterization is used in the limiting analysis. In the limit, we obtain two limit problems according to the scaling parameters. In one of the limit optimal control problem, we observe that it contains three controls namely; a distributed control, a boundary control and an interface control.
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Jeng, Bor-Wen, and 鄭博文. "Exploiting Symmetries for Semilinear Elliptic Problems with Neumann Boundary Conditions." Thesis, 1997. http://ndltd.ncl.edu.tw/handle/84015655255280839989.
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碩士國立中興大學
應用數學系
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We exploit symmetries in certain semilinear elliptic eigenvalue problems withNeumann boundary conditions for the continuation of solution curves. We showthat symmetry makes the problem decomposable into small ones, and thediscretization matrix obtained via central differences associated to theLaplacian is similar to a symmetric one. Furthermore, the discrete problemspreserve some basic properties on eigenvalues of the continuous problems.Thus the continuation-Lanczos algorithm can be adapted to trace the solutioncurves of the reduced problems. Sample numerical results are reported.
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Lippi, Edoardo Proietti. "Nonlocal Neuman boundary conditions: properties and problems." Doctoral thesis, 2022. http://hdl.handle.net/2158/1270261.
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"Boundary control of quasi-linear hyperbolic initial boundary-value problems." Université catholique de Louvain, 2004. http://edoc.bib.ucl.ac.be:81/ETD-db/collection/available/BelnUcetd-09242004-170922/.
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Ravi, Prakash *. "Homogenization of Optimal Control Problems in a Domain with Oscillating Boundary." Thesis, 2013. http://hdl.handle.net/2005/2807.
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Mathematical theory of homogenization of partial differential equations is relatively a new area of research (30-40 years or so) though the physical and engineering applications were well known. It has tremendous applications in various branches of engineering and science like : material science ,porous media, study of vibrations of thin structures, composite materials to name a few. There are at present various methods to study homogenization problems (basically asymptotic analysis) and there is a vast amount of literature in various directions. Homogenization arise in problems with oscillatory coefficients, domain with large number of perforations, domain with rough boundary and so on. The latter one has applications in fluid flow which is categorized as oscillating boundaries.
In fact ,in this thesis, we consider domains with oscillating boundaries. We plan to study to homogenization of certain optimal control problems with oscillating boundaries. This thesis contains 6 chapters including an introductory Chapter 1 and future proposal Chapter 6. Our main contribution contained in chapters 2-5. The oscillatory domain under consideration is a 3-dimensional cuboid (for simplicity) with a large number of pillars of length O(1) attached on one side, but with a small cross sectional area of order ε2 .As ε0, this gives a geometrical domain with oscillating boundary. We also consider 2-dimensional oscillatory domain which is a cross section of the above 3-dimensional domain.
In chapters 2 and 3, we consider the optimal control problem described by the Δ operator with two types of cost functionals, namely L2-cost functional and Dirichlet cost functional. We consider both distributed and boundary controls. The limit analysis was carried by considering the associated optimality system in which the adjoint states are introduced. But the main contribution in all the different cases(L2 and Dirichlet cost functionals, distributed and boundary controls) is the derivation of error estimates what is known as correctors in homogenization literature. Though there is a basic test function, one need to introduce different test functions to obtain correctors. Introducing correctors in homogenization is an important aspect of study which is indeed useful in the analysis, but important in numerical study as well.
The setup is the same in Chapter 4 as well. But here we consider Stokes’ Problem and study asymptotic analysis as well as corrector results. We obtain corrector results for velocity and pressure terms and also for its adjoint velocity and adjoint pressure. In Chapter 5, we consider a time dependent Kirchhoff-Love equation with the same domain with oscillating boundaries with a distributed control. The state equation is a fourth order hyperbolic type equation with associated L2-cost functional. We do not have corrector results in this chapter, but the limit cost functional is different and new. In the earlier chapters the limit cost functional were of the same type.
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Krumbiegel, Klaus [Verfasser]. "Numerical concepts and error analysis for elliptic Neumann control problems with pointwise state and control constraints / von Klaus Krumbiegel." 2009. http://d-nb.info/994128479/34.
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COKER, ESTELLE MATHILDA. "SEQUENTIAL GRADIENT-RESTORATION ALGORITHM FOR OPTIMAL CONTROL PROBLEMS WITH CONTROL INEQUALITY CONSTRAINTS AND GENERAL BOUNDARY CONDITIONS." Thesis, 1985. http://hdl.handle.net/1911/15889.
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The problem of minimizing a functional I subject to differential constraints, control inequality constraints, and terminal constraints is considered in this thesis. It consists of finding the state x(t), the control u(t), and the parameter (pi) so that the functional I is minimized, while the constraints are satisfied to a predetermined accuracy.
A sequential gradient-restoration algorithm is developed. It involves a sequence of two-phase cycles, the gradient phase and the restoration phase. In the gradient phase, the value of the functional is decreased, while avoiding excessive constraint violation; in the restoration phase, the constraint error is decreased, while avoiding excessive change in the value of the functional.
The variations (DELTA)x(t), (DELTA)u(t), (DELTA)(pi) are generated by requiring the first variation of the augmented functional J to be negative during the gradient phase; and by requiring the first variation of the constraint error P to be negative, while imposing a least-square criterion on the variations of the control, the parameter, and the initial state during the restoration phase. This leads to a linear, two-point boundary-value problem, which is solved via the method of particular solutions.
Various transformation schemes are employed, so as to convert control inequality constraints into control equality constraints. Considerable simplifications are possible if the control inequality constraints have a special form. In this connection, the following cases are studied: (P1) lower bounds on u(t); (P2) upper bounds on u(t); and (P3) upper and lower bounds on u(t).
The algorithmic work per iteration is reduced due to (i) the special structure of problems (P1) through (P3); (ii) the fact that the multiplier (rho) associated with the auxiliary nondifferential constraint can be computed explicitly, bypassing the need of matrix inversion; and (iii) the fact that the auxiliary nondifferential constraint involves only the augmented control and not the state and/or the parameter. Because of this special situation, the number of integrations required to solve the linear, two-point boundary-value problem at each iteration can be reduced from n + p + 1. Here, n, p and b are the dimensions of the state vector, the parameter vector, and the vector of final conditions respectively.
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Yan, Yan. "Smooth and Robust Solutions for Dirichlet Boundary Control of Fluid-Solid Conjugate Heat Transfer Problems." Thesis, 2015. https://doi.org/10.7916/D8H70DNK.
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This work offers new computational methods for the optimal control of the conjugate heat transfer (CHT) problem in thermal science. Conjugate heat transfer has many important industrial applications, such as heat exchange processes in power plants and cooling in electronic packaging industry, and has been a staple of computational methods in thermal science for many years. This work considers the Dirichlet boundary control of fluid-solid CHT problems. The CHT system falls into the category of multi-physics problems. Its domain typically consists of two parts, namely, a solid region subject to thermal heating or cooling and a conjugate fluid region responsible for thermal convection transport. These two different physical systems are strongly coupled through the thermal boundary condition at the fluid-solid interface. The objective in the CHT boundary control problem is to select optimally the fluid inflow profile that minimizes an objective function that involves
the sum of the mismatch between the temperature distribution in the system and a prescribed temperature profile and the cost of the control. This objective is realized by minimizing a nonlinear objective function of the boundary control and the fluid temperature variables, subject to partial differential equations (PDE) constraints governed by the coupled heat diffusion equation in the solid region and mass, momentum and energy conservation equations in the fluid region.
Although CHT has received extensive attention as a forward problem, the optimal Dirichlet
velocity boundary control for the coupled CHT process to our knowledge is only very sparsely studied analytically or computationally in the literature [131]. Therefore, in Part I, we describe the formulation of the optimal control problem and introduce the building blocks for the finite element modeling of the CHT problem, namely, the diffusion equation for the solid temperature, the convection-diffusion equation for the fluid temperature, the incompressible viscous Navier-Stokes equations for the fluid velocity and pressure, and the model verification of CHT simulations.
In Part II, we provide theoretical analysis to explain the nonsmoothness issue which has been observed in this study and in Dirichlet boundary control of Navier-Stokes flows by other scientists. Based on these findings, we use either explicit or implicit numerical smoothing to resolve the nonsmoothness issue. Moreover, we use the numerical continuation on regularization parameters to alleviate the difficulty of locating the global minimum in one shot for highly nonlinear optimization problems even when the initial guess is far from optimal. Two suites of numerical experiments have been provided to demonstrate the feasibility, effectiveness and robustness of the optimization scheme.
In Part III, we demonstrate the strategy of achieving parallel scalable algorithms for CHT models in Simulations of Reactor Thermal Hydraulics. Our motivation originates from the observation that parallel processing is necessary for optimal control problems of very large scale, when the simulation of the underlying physics (or PDE constraints) involves millions or billions of degrees of freedom. To achieve the overall scalability of optimal control problems governed by PDE constraints, scalable components that resolve the PDE constraints and their adjoints are the key. In this Part, first we provide the strategy of designing parallel scalable solvers for each building blocks of the CHT modeling, namely, for the discrete diffusive operator, the discrete convection-diffusion operator, and the discrete Navier-Stokes operator. Second, we demonstrate a pair of effective, robust, parallel, and scalable solvers built with collaborators for simulations of reactor thermal hydraulics. Finally, in the the section of future work, we outline the roadmap of parallel and scalable solutions for Dirichlet boundary control of fluid-solid conjugate heat transfer processes.
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(9838208), David Sturgess. "Using genetic and evolutionary algorithms to solve boundary control problems in soil-water-plant interaction." Thesis, 2002. https://figshare.com/articles/thesis/Using_genetic_and_evolutionary_algorithms_to_solve_boundary_control_problems_in_soil-water-plant_interaction/13429310.
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In this thesis we investigate how modern artificial intelligent techniques, namely the genetic algorithm /evolutionary algorithm can be applied to find irrigation strategies for a cropped soil. We begin by providing an introductory chapter detailing the work that is to be carried out and the results obtained from research. In Chapter 2 we introduce some basic concepts of soil physics in order to give an understanding of the nature of soil composition and the movement of water within a cropped soil. We then summarise background research undertaken by Terry Janz in his Masters Thesis which shows how an irrigation schedule can be obtained using classical methods to solve the Richards' flow equation with realistic parameters and field data. Genetic and evolutionary algorithms are introduced in Chapter 3; their algorithmic structure is defined and contrasted with classic search techniques. In Chapter 4 we apply a genetic algorithm to the problem posed in Chapter 2 to obtain a schedule of irrigation defined as a sequence of irrigation on and irrigation off switches, to control moisture content at specific levels at certain depths within the soil, so that "nutrient uptake" by the root can be maximised. The problem posed is the classical optimal control problem in which the tracking of a desired set of final states is to be achieved. Finally in Chapter 5, we undertake an initial research study into how an evolutionary algorithm can be applied to solve the tracking problem associated with boundary control of a parabolic distributed process. The problem is first transformed into a classical optimal control problem with ordinary differential equations as differential constraints, by using the method of semi-discretisation, or method of lines. Our results are compared with classical techniques commonly used to solve this type of problem, including the finite element method which uses full discretisation of both the state and time variables. It is shown that it is feasible to apply evolutionary learning to problems of boundary control which arise in determining realistic irrigation strategies.
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Ko, Shuh-Hung. "Comparison of gradient-restoration algorithms for optimal control problems with nondifferential constraints and general boundary conditions." Thesis, 1994. http://hdl.handle.net/1911/13854.
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The problem considered here involves a functional I subject to differential constraints, nondifferential constraints, and general boundary conditions. It consists of finding the state x(t), control u(t), and parameter $\pi$ so that the functional I is minimized, while the differential constraints, nondifferential constraints, and boundary conditions are satisfied to a predetermined accuracy. Here, I is a scalar, x an n-vector, u an m-vector, and $\pi$ a p-vector.
Four types of gradient-restoration algorithms are considered, and their relative efficiency in terms of the number of iterations for convergence and CPU time is evaluated. The algorithms considered are as follows: sequential gradient-restoration algorithm, complete restoration (SGRA-CR); sequential gradient-restoration algorithm, incomplete restoration (SGRA-IR); combined gradient-restoration algorithm, no restoration (CGRA-NR); and combined gradient-restoration algorithm, incomplete restoration (CGRA-IR).
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Schwarz, Christian [Verfasser]. "Computation of confluent hypergeometric functions and application to parabolic boundary control problems / vorgelegt von Christian Schwarz." 2004. http://d-nb.info/971747660/34.
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Bai, Xiaoli. "Modified Chebyshev-Picard Iteration Methods for Solution of Initial Value and Boundary Value Problems." Thesis, 2010. http://hdl.handle.net/1969.1/ETD-TAMU-2010-08-8240.
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The solution of initial value problems (IVPs) provides the evolution of dynamic
system state history for given initial conditions. Solving boundary value problems
(BVPs) requires finding the system behavior where elements of the states are defined
at different times. This dissertation presents a unified framework that applies modified
Chebyshev-Picard iteration (MCPI) methods for solving both IVPs and BVPs.
Existing methods for solving IVPs and BVPs have not been very successful in
exploiting parallel computation architectures. One important reason is that most
of the integration methods implemented on parallel machines are only modified versions
of forward integration approaches, which are typically poorly suited for parallel
computation.
The proposed MCPI methods are inherently parallel algorithms. Using Chebyshev
polynomials, it is straightforward to distribute the computation of force functions
and polynomial coefficients to different processors. Combining Chebyshev polynomials
with Picard iteration, MCPI methods iteratively refine estimates of the solutions
until the iteration converges. The developed vector-matrix form makes MCPI methods
computationally efficient.
The power of MCPI methods for solving IVPs is illustrated through a small perturbation
from the sinusoid motion problem and satellite motion propagation problems.
Compared with a Runge-Kutta 4-5 forward integration method implemented in MATLAB, MCPI methods generate solutions with better accuracy as well as orders
of magnitude speedups, prior to parallel implementation. Modifying the algorithm
to do double integration for second order systems, and using orthogonal polynomials
to approximate position states lead to additional speedups. Finally, introducing
perturbation motions relative to a reference motion results in further speedups.
The advantages of using MCPI methods to solve BVPs are demonstrated by
addressing the classical Lambert’s problem and an optimal trajectory design problem.
MCPI methods generate solutions that satisfy both dynamic equation constraints and
boundary conditions with high accuracy. Although the convergence of MCPI methods
in solving BVPs is not guaranteed, using the proposed nonlinear transformations,
linearization approach, or correction control methods enlarge the convergence domain.
Parallel realization of MCPI methods is implemented using a graphics card that
provides a parallel computation architecture. The benefit from the parallel implementation
is demonstrated using several example problems. Larger speedups are achieved
when either force functions become more complicated or higher order polynomials are
used to approximate the solutions.