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1
Park, Elinor Jane. "Regularizations of first order partial differential equations by generators of semigroups." Thesis, Swansea University, 2005. https://cronfa.swan.ac.uk/Record/cronfa42982.
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This thesis investigates the limiting behaviour of solutions to certain partial and pseudo differential equations. Included is a study of the notion of generalised solutions and particular examples, with emphasis on hyperbolic conservation laws. A probabilistic interpretation of some results is also presented.
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Aziz, Waleed. "Analytic and algebraic aspects of integrability for first order partial differential equations." Thesis, University of Plymouth, 2013. http://hdl.handle.net/10026.1/1468.
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This work is devoted to investigating the algebraic and analytic integrability of first order polynomial partial differential equations via an understanding of the well-developed area of local and global integrability of polynomial vector fields. In the view of characteristics method, the search of first integrals of the first order partial differential equations P(x,y,z)∂z(x,y) ∂x +Q(x,y,z)∂z(x,y) ∂y = R(x,y,z), (1) is equivalent to the search of first integrals of the system of the ordinary differential equations dx/dt= P(x,y,z), dy/dt= Q(x,y,z), dz/dt= R(x,y,z). (2) The trajectories of (2) will be found by representing these trajectories as the intersection of level surfaces of first integrals of (1). We would like to investigate the integrability of the partial differential equation (1) around a singularity. This is a case where understanding of ordinary differential equations will help understanding of partial differential equations. Clearly, first integrals of the partial differential equation (1), are first integrals of the ordinary differential equations (2). So, if (2) has two first integrals φ1(x,y,z) =C1and φ2(x,y,z) =C2, where C1and C2 are constants, then the general solution of (1) is F(φ1,φ2) = 0, where F is an arbitrary function of φ1and φ2. We choose for our investigation a system with quadratic nonlinearities and such that the axes planes are invariant for the characteristics: this gives three dimensional Lotka– Volterra systems x' =dx/dt= P = x(λ +ax+by+cz), y' =dy/dt= Q = y(µ +dx+ey+ fz), z' =dz/dt= R = z(ν +gx+hy+kz), where λ,µ,ν 6= 0. v Several problems have been investigated in this work such as the study of local integrability and linearizability of three dimensional Lotka–Volterra equations with (λ:µ:ν)–resonance. More precisely, we give a complete set of necessary and sufficient conditions for both integrability and linearizability for three dimensional Lotka-Volterra systems for (1:−1:1), (2:−1:1) and (1:−2:1)–resonance. To prove their sufficiency, we mainly use the method of Darboux with the existence of inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable. Also, more general three dimensional system have been investigated and necessary and sufficient conditions are obtained. In another approach, we also consider the applicability of an entirely different method which based on the monodromy method to prove the sufficiency of integrability of these systems. These investigations, in fact, mean that we generalized the classical centre-focus problem in two dimensional vector fields to three dimensional vector fields. In three dimensions, the possible mechanisms underling integrability are more difficult and computationally much harder. We also give a generalization of Singer’s theorem about the existence of Liouvillian first integrals in codimension 1 foliations in Cnas well as to three dimensional vector fields. Finally, we characterize the centres of the quasi-homogeneous planar polynomial differential systems of degree three. We show that at most one limit cycle can bifurcate from the periodic orbits of a centre of a cubic homogeneous polynomial system using the averaging theory of first order.
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Stanistreet, Timothy Francis. "Numerical methods for first order partial differential equations describing steady-state forming processes." Thesis, Imperial College London, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.398232.
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Jonasson, Jens. "Systems of Linear First Order Partial Differential Equations Admitting a Bilinear Multiplication of Solutions." Doctoral thesis, Linköping : Department of Mathematics, Linköpings universitet, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-9949.
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Studener, Stephan [Verfasser]. "Embedded Control and Parameter Estimation Algorithms for Transport Process Systems : modeled by first-order Partial Differential Equations / Stephan Studener." Aachen : Shaker, 2011. http://d-nb.info/1069049832/34.
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Gorgone, Matteo. "Symmetries, Equivalence and Decoupling of First Order PDE's." Doctoral thesis, Università di Catania, 2017. http://hdl.handle.net/10761/3901.
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The present Ph.D. Thesis is concerned with first order PDE's and to the structural conditions allowing for their transformation into an equivalent, and somehow simpler, form.
Most of the results are framed in the context of the classical theory of the Lie symmetries of differential equations, and on the analysis of some invariant quantities.
The thesis is organized in 5 main sections. The first two Chapters present the basic elements of the Lie theory and some introductory facts about first order PDE's, with special emphasis on quasilinear ones.
Chapter 3 is devoted to investigate equivalence transformations, i.e., point transformations suitable to deal with classes of differential equations involving arbitrary elements.
The general framework of equivalence transformations is then applied to a class of systems of first order PDE's, consisting of a linear conservation law and four general balance laws involving some arbitrary continuously differentiable functions, in order to identify the elements of the class that can be mapped to a system of autonomous conservation laws.
Chapter 4 is concerned with the transformation of nonlinear first order systems of differential equations to a simpler form. At first, the reduction to an equivalent first order autonomous and homogeneous quasilinear form is considered. A theorem providing necessary conditions is given, and the reduction to quasilinear form is performed by constructing the canonical variables associated to the Lie point symmetries admitted by the nonlinear system.
Then, a general nonlinear system of first order PDE's involving the derivatives of the unknown variables in polynomial form is considered, and a theorem giving necessary and sufficient conditions in order to map it to an autonomous system polynomially homogeneous in the derivatives is established. Several classes of first order Monge-Ampere systems, either with constant coefficients or with coefficients depending on the field variables, provided that the coefficients entering their equations satisfy some constraints, are reduced to quasilinear (or linear) form.
Chapter 5 faces the decoupling problem of general quasilinear first order systems. Starting from the direct decoupling problem of hyperbolic quasilinear first order systems in two independent variables and two or three dependent variables, we observe that the decoupling conditions can be written in terms of the eigenvalues and eigenvectors of the coefficient matrix.
This allows to obtain a completely general result. At first, general autonomous and homogeneous quasilinear first order systems (either hyperbolic or not) are discussed, and the necessary and sufficient conditions for the decoupling in two or more subsystems proved. Then, the analysis is extended to the case of nonhomogeneous and/or nonautonomous systems. The conditions, as one expects, involve just the properties of the eigenvalues and the eigenvectors (together with the generalized eigenvectors, if needed) of the coefficient matrix; in particular, the conditions for the full decoupling of a hyperbolic system in non-interacting subsystems have a physical interpretation since require the vanishing both of the change of characteristic speeds of a subsystem across a wave of the other subsystems, and of the interaction coefficients between waves of different subsystems.
Moreover, when the required decoupling conditions are satisfied, we have also the differential constraints whose integration provides the variable transformation leading to the (partially or fully) decoupled system. All the results are extended to the decoupling of nonhomogeneous and/or nonautonomous quasilinear first order systems.
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Strogies, Nikolai. "Optimization of nonsmooth first order hyperbolic systems." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2016. http://dx.doi.org/10.18452/17633.
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Wir betrachten Optimalsteuerungsprobleme, die von partiellen Differentialgleichungen beziehungsweise Variationsungleichungen mit Differentialoperatoren erster Ordnung abhängen. Wir führen die Reformulierung eines Tagebauplanungsproblems, das auf stetigen Funktionen beruht, ein. Das Resultat ist ein Optimalsteuerungsproblem für Viskositätslösungen einer Eikonalgleichung. Die Existenz von Lösungen dieses und bestimmter Hilfsprobleme, die von semilinearen PDG‘s mit künstlicher Viskosität abhängen, wird bewiesen, Stationaritätsbedingungen hergeleitet und ein schwaches Konsistenzresultat für stationäre Punkte präsentiert. Des Weiteren betrachten wir Optimalsteuerungsprobleme, die von stationären Variationsungleichungen erster Art mit linearen Differentialoperatoren erster Ordnung abhängen. Wir diskutieren Lösbarkeit und Stationaritätskonzepte für diese Probleme. Für letzteres vergleichen wir Ergebnisse, die entweder durch die Anwendung von Penalisierungs- und Regularisierungsansätzen direkt auf Ebene von Differentialoperatoren erster Ordnung oder als Grenzwertprozess von Stationaritätssystemen für viskositätsregularisierte Optimalsteuerungsprobleme unter passenden Annahmen erhalten werden. Um die Konsistenz von ursprünglichem und regularisierten Problemen zu sichern, wird ein bekanntes Ergebnis für Lösungen von VU’s mit degeneriertem Differentialoperator erweitert. In beiden Fällen ist die erhaltene Stationarität schwächer als W-stationarität. Die theoretischen Ergebnisse werden anhand numerischer Beispiele verifiziert. Wir erweitern diese Ergebnisse auf Optimalsteuerungsprobleme bezüglich zeitabhängiger VU’s mit Differentialoperatoren erster Ordnung. Hierfür wird die Existenz von Lösungen bewiesen und erneut ein Stationaritätssystem mit Hilfe verschwindender Viskositäten unter bestimmten Beschränktheitsannahmen hergeleitet. Die erhaltenen Ergebnisse werden anhand von numerischen Beispielen verifiziert.We consider problems of optimal control subject to partial differential equations and variational inequality problems with first order differential operators. We introduce a reformulation of an open pit mine planning problem that is based on continuous functions. The resulting formulation is a problem of optimal control subject to viscosity solutions of a partial differential equation of Eikonal Type. The existence of solutions to this problem and auxiliary problems of optimal control subject to regularized, semilinear PDE’s with artificial viscosity is proven. For the latter a first order optimality condition is established and a mild consistency result for the stationary points is proven. Further we study certain problems of optimal control subject to time-independent variational inequalities of the first kind with linear first order differential operators. We discuss solvability and stationarity concepts for such problems. In the latter case, we compare the results obtained by either utilizing penalization-regularization strategies directly on the first order level or considering the limit of systems for viscosity-regularized problems under suitable assumptions. To guarantee the consistency of the original and viscosity-regularized problems of optimal control, we extend known results for solutions to variational inequalities with degenerated differential operators. In both cases, the resulting stationarity concepts are weaker than W-stationarity. We validate the theoretical findings by numerical experiments for several examples. Finally, we extend the results from the time-independent to the case of problems of optimal control subject to VI’s with linear first order differential operators that are time-dependent. After establishing the existence of solutions to the problem of optimal control, a stationarity system is derived by a vanishing viscosity approach under certain boundedness assumptions and the theoretical findings are validated by numerical experiments.
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Schnücke, Gero [Verfasser], Christian [Gutachter] Klingenberg, and Manfred [Gutachter] Dobrowolski. "Arbitrary Lagrangian-Eulerian Discontinous Galerkin methods for nonlinear time-dependent first order partial differential equations / Gero Schnücke ; Gutachter: Christian Klingenberg, Manfred Dobrowolski." Würzburg : Universität Würzburg, 2016. http://d-nb.info/1117477290/34.
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Ježková, Jitka. "Modelování dopravního toku." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2015. http://www.nusl.cz/ntk/nusl-232180.
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Tato diplomová práce prezentuje problematiku dopravního toku a jeho modelování. Zabývá se především několika LWR modely, které následně rozebírá a hledá řešení pro počáteční úlohy. Ukazuje se, že ne pro všechny počáteční úlohy lze řešení definovat na celém prostoru, ale jen v určitém okolí počáteční křivky. Proto je dále odvozena metoda výpočtu velikosti tohoto okolí a to nejen zcela obecně, ale i pro dané modely. Teoretický rozbor LWR modelů a řešení počátečních úloh jsou demonstrovány několika příklady, které zřetelně ukazují, jak se dopravní tok simulovaný danými modely chová.
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Shu, Yupeng. "Numerical Solutions of Generalized Burgers' Equations for Some Incompressible Non-Newtonian Fluids." ScholarWorks@UNO, 2015. http://scholarworks.uno.edu/td/2051.
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The author presents some generalized Burgers' equations for incompressible and isothermal flow of viscous non-Newtonian fluids based on the Cross model, the Carreau model, and the Power-Law model and some simple assumptions on the flows. The author numerically solves the traveling wave equations for the Cross model, the Carreau model, the Power-Law model by using industrial data. The author proves existence and uniqueness of solutions to the traveling wave equations of each of the three models. The author also provides numerical estimates of the shock thickness as well as maximum strain $\varepsilon_{11}$ for each of the fluids.
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Rangelova, Marina. "Error estimation for fourth order partial differential equations." Ann Arbor, Mich. : ProQuest, 2007. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3258675.
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Thesis (Ph.D. in Computational and Applied Mathematics)--S.M.U., 2007.Title from PDF title page (viewed Mar. 18, 2008). Source: Dissertation Abstracts International, Volume: 68-03, Section: B, page: 1675. Adviser: Peter Moore. Includes bibliographical references.
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Cheema, Tasleem Akhter. "Higher-order finite-difference methods for partial differential equations." Thesis, Brunel University, 1997. http://bura.brunel.ac.uk/handle/2438/7131.
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This thesis develops two families of numerical methods, based upon rational approximations having distinct real poles, for solving first- and second-order parabolic/ hyperbolic partial differential equations. These methods are thirdand fourth-order accurate in space and time, and do not require the use of complex arithmetic. In these methods first- and second-order spatial derivatives are approximated by finite-difference approximations which produce systems of ordinary differential equations expressible in vector-matrix forms. Solutions of these systems satisfy recurrence relations which lead to the development of parallel algorithms suitable for computer architectures consisting of three or four processors. Finally, the methods are tested on advection, advection-diffusion and wave equations with constant coefficients.
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Kassam, Aly-Khan. "High order timestepping for stiff semilinear partial differential equations." Thesis, University of Oxford, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.403758.
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Bowen, Matthew K. "High-order finite difference methods for partial differential equations." Thesis, Loughborough University, 2005. https://dspace.lboro.ac.uk/2134/13492.
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General n-point formulae for difference operators and their errors are derived in terms of elementary symmetric functions. These are used to derive high-order, compact and parallelisable finite difference schemes for the decay-advection-diffusion and linear damped Korteweg-de Vnes equations. Stability calculations are presented and the speed and accuracy of the schemes is compared to that of other finite difference methods in common use. Appendices contain useful tables of difference operators and errors and present a stability proof for quadratic inequalities. For completeness, the appendices conclude with the standard Thomas method for solving tri-diagonal systems.
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Duke, Elizabeth R. "Solving higher order dynamic equations on time scales as first order systems." Huntington, WV : [Marshall University Libraries], 2006. http://www.marshall.edu/etd/descript.asp?ref=653.
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Ugail, Hassan. "Generalized partial differential equations for interactive design." World Scientific Publishing Company, 2007. http://hdl.handle.net/10454/2642.
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This paper presents a method for interactive design by means of extending the PDE
based approach for surface generation. The governing partial differential equation is
generalized to arbitrary order allowing complex shapes to be designed as single patch
PDE surfaces. Using this technique a designer has the flexibility of creating and manipulating
the geometry of shape that satisfying an arbitrary set of boundary conditions.
Both the boundary conditions which are defined as curves in 3-space and the spine of the
corresponding PDE are utilized as interactive design tools for creating and manipulating
geometry intuitively. In order to facilitate interactive design in real time, a compact
analytic solution for the chosen arbitrary order PDE is formulated. This solution scheme
even in the case of general boundary conditions satisfies exactly the boundary conditions
where the resulting surface has an closed form representation allowing real time
shape manipulation. In order to enable users to appreciate the powerful shape design
and manipulation capability of the method, we present a set of practical examples.
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Taj, Malik Shahadat Ali. "Higher order parallel splitting methods for parabolic partial differential equations." Thesis, Brunel University, 1995. http://bura.brunel.ac.uk/handle/2438/5780.
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The thesis develops two families of numerical methods, based upon new rational approximations to the matrix exponential function, for solving second-order parabolic partial differential equations. These methods are L-stable, third- and fourth-order accurate in space and time, and do not require the use of complex arithmetic. In these methods second-order spatial derivatives are approximated by new difference approximations. Then parallel algorithms are developed and tested on one-, two- and three-dimensional heat equations, with constant coefficients, subject to homogeneous boundary conditions with discontinuities between initial and boundary conditions. The schemes are seen to have high accuracy. A family of cubic polynomials, with a natural number dependent coefficients, is also introduced. Each member of this family has real zeros. Third- and fourth-order methods are also developed for one-dimensional heat equation subject to time-dependent boundary conditions, approximating the integral term in a new way, and tested on a variety of problems from the literature.
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MONTEIRO, GABRIEL DE LIMA. "WEAK SOLUTIONS FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS OF SECOND ORDER." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2018. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=36023@1.
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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIROCOORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
PROGRAMA DE SUPORTE À PÓS-GRADUAÇÃO DE INSTS. DE ENSINO
PROGRAMA DE EXCELENCIA ACADEMICA
Esse trabalho tem como objetivo ser uma introdução ao estudo da existência e unicidade de soluções fracas para equações diferenciais parciais elípticas. Começamos definindo o espaço de Sobolev para, a partir da definição, provarmos algumas propriedades básicas que nos ajudarão no estudo das equações diferenciais parciais elípticas. Finalizamos com o desenvolvimento do Teorema de Lax-Milgram e de Stampacchia que permitirão o uso de técnicas de Análise Funcional para estudarmos alguns exemplos de equações elípticas.
This dissertation aims to be an introduction to the study of the existence and uniqueness of weak solutions for elliptic partial differential equations. We begin by defining the Sobolev spaces and proving some basics properties that will assist in the study of the elliptical equations. Lastly, we develop the Theorems of Lax-Milgram and Stampacchia that allow the use of Functional Analysis for the studying of some examples of elliptic equations.
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Eftang, Jens Lohne. "Reduced basis methods for parametrized partial differential equations." Doctoral thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2011. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-12550.
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Alnafisah, Yousef Ali. "First-order numerical schemes for stochastic differential equations using coupling." Thesis, University of Edinburgh, 2016. http://hdl.handle.net/1842/20420.
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We study a new method for the strong approximate solution of stochastic differential equations using coupling and we prove order one error bounds for the new scheme in Lp space assuming the invertibility of the diffusion matrix. We introduce and implement two couplings called the exact and approximate coupling for this scheme obtaining good agreement with the theoretical bound. Also we describe a method for non-invertibility case (Combined method) and we investigate its convergence order which will give O(h3/4 √log(h)j) under some conditions. Moreover we compare the computational results for the combined method with its theoretical error bound and we have obtained a good agreement between them. In the last part of this thesis we work out the performance of the multilevel Monte Carlo method using the new scheme with the exact coupling and we compare the results with the trivial coupling for the same scheme.
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Postell, Floyd Vince. "High order finite difference methods." Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/28876.
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Pitts, George Gustav. "Domain decomposition and high order discretization of elliptic partial differential equations." Diss., Virginia Tech, 1994. http://hdl.handle.net/10919/39143.
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Atwell, Jeanne A. "Proper Orthogonal Decomposition for Reduced Order Control of Partial Differential Equations." Diss., Virginia Tech, 2000. http://hdl.handle.net/10919/26985.
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Numerical models of PDE systems can involve
very large matrix equations, but
feedback controllers for these systems must be computable
in real time to be implemented on physical systems.
Classical control design methods produce
controllers of the same order as the numerical models.
Therefore, emph{reduced} order control design is vital for practical
controllers. The main contribution of this research is a method
of control order reduction
that uses a newly developed low order basis. The low order basis
is obtained
by applying Proper Orthogonal Decomposition (POD) to a set of
functional gains, and is referred to as the functional gain
POD basis.
Low order controllers resulting from the
functional gain POD basis
are compared with low order controllers resulting from
more commonly used time snapshot POD bases, with the two dimensional
heat equation as a test problem. The functional gain
POD basis avoids subjective criteria associated
with the time snapshot POD basis and provides
an equally effective low order controller with larger stability
radii. An efficient and effective methodology is introduced for
using a low order basis in reduced order compensator design. This
method combines
"design-then-reduce" and "reduce-then-design" philosophies.
The desirable qualities of the resulting
reduced order compensator are verified by application
to Burgers' equation in numerical experiments.Ph. D.
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Pitts, George G. "Domain decomposition and high order discretization of elliptic partial differential equations." Diss., Virginia Tech, 1994. http://hdl.handle.net/10919/39143.
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Numerical solutions of partial differential equations (PDEs) resulting from problems in both the engineering and natural sciences result in solving large sparse linear systems Au = b. The construction of such linear systems and their solutions using either direct or iterative methods are topics of continuing research. The recent advent of parallel computer architectures has resulted in a search for good parallel algorithms to solve such systems, which in turn has led to a recent burgeoning of research into domain decomposition algorithms. Domain decomposition is a procedure which employs subdivision of the solution domain into smaller regions of convenient size or shape and, although such partitionings have proven to be quite effective on serial computers, they have proven to be even more effective on parallel computers.
Recent work in domain decomposition algorithms has largely been based on second order accurate discretization techniques. This dissertation describes an algorithm for the numerical solution of general two-dimensional linear elliptic partial differential equations with variable coefficients which employs both a high order accurate discretization and a Krylov subspace iterative solver in which a preconditioner is developed using domain decomposition. Most current research into such algorithms has been based on symmetric systems; however, variable PDE coefficients generally result in a nonsymmetric A, and less is known about the use of preconditioned Krylov subspace iterative methods for the solution of nonsymmetric systems.
The use of the high order accurate discretization together with a domain decomposition based preconditioner results in an iterative technique with both high accuracy and rapid convergence. Supporting theory for both the discretization and the preconditioned iterative solver is presented. Numerical results are given on a set of test problems of varying complexity demonstrating the robustness of the algorithm. It is shown that, if only second order accuracy is required, the algorithm becomes an extremely fast direct solver. Parallel performance of the algorithm is illustrated with results from a shared memory multiproces-SOr.Ph. D.
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Van, der Walt Jan Harm. "Generalized solutions of systems of nonlinear partial differential equations." Thesis, Pretoria : [s.n.], 2009. http://upetd.up.ac.za/thesis/available/etd-05242009-122628.
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Khavanin, Mohammad. "The Method of Mixed Monotony and First Order Delay Differential Equations." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/96643.
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In this paper I extend the method of mixed monotony, to construct monotone sequences that converge to the unique solution of an initial value delay differential equation.En este artículo se prueba una generalización del método de monotonía mixta, para construir sucesiones monótonas que convergen a la solución única de una ecuación diferencial de retraso con valor inicial.
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Rejoub, Riad A. "Projective and non-projective systems of first order nonlinear differential equations." Scholarly Commons, 1992. https://scholarlycommons.pacific.edu/uop_etds/2228.
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It is well established that many physical and chemical phenomena such as those in chemical reaction kinetics, laser cavities, rotating fluids, and in plasmas and in solid state physics are governed by nonlinear differential equations whose solutions are of variable character and even may lack regularities. Such systems are usually first studied qualitatively by examining their temporal behavior near singular points of their phase portrait.
In this work we will be concerned with systems governed by the time evolution equations [see PDF for mathematical formulas]
The xi may generally be considered to be concentrations of species in a chemical reaction, in which case the k's are rate constants. In some cases the xi may be considered to be position and momentum variables in a mechanical system. We will divide the equations into two classes: those in which the evolution can be carried out by the action of one of Lie's transformation groups of the plane, and those for which this is not possible. Members of the first class can be integrated by quadrature either directly or by use of an integrating factor; those in the second class cannot. Of those in the first class the most interesting evolve by transformations of the projective group, and these, as well as the equations that cannot be integrated by quadrature, we study in some detail. We seek a qualitative analysis of systems which have no linear terms in their evolution equations when the origin from which the xi are measured is a critical point. The standard, linear, phase plane analysis is of course not adequate for our purposes.
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Du, Zhihua [Verfasser]. "Boundary Value Problems for Higher Order Complex Partial Differential Equations / Zhihua Du." Berlin : Freie Universität Berlin, 2008. http://d-nb.info/1022870912/34.
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Tråsdahl, Øystein. "High order methods for partial differential equations: geometry representation and coordinate transformations." Doctoral thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2012. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-17077.
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Zigic, Jovan. "Optimization Methods for Dynamic Mode Decomposition of Nonlinear Partial Differential Equations." Thesis, Virginia Tech, 2021. http://hdl.handle.net/10919/103862.
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Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD.Master of Science
The Navier-Stokes (NS) equations are the primary mathematical model for understanding the behavior of fluids. The existence and smoothness of the NS equations is considered to be one of the most important open problems in mathematics, and challenges in their numerical simulation is a barrier to understanding the physical phenomenon of turbulence. Due to the difficulty of studying this problem directly, simpler problems in the form of nonlinear partial differential equations (PDEs) that exhibit similar properties to the NS equations are studied as preliminary steps towards building a wider understanding of the field. Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD.
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Smith, James. "Global time estimates for solutions to higher order strictly hyperbolic partial differential equations." Thesis, Imperial College London, 2006. http://hdl.handle.net/10044/1/1267.
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In this thesis we consider the Cauchy problem for general higher order constant coefficient strictly hyperbolic PDEs with lower order terms and show how the behaviour of the characteristic roots determine the rate of decay in the associated Lp-Lq estimates. In particular, we show under what conditions the solution behaves like that of the standard wave equation, the wave equation with dissipation or the Klein-Gordon equation. We explain the various factors involved, such as the presence of multiple roots, the size of the sets of multiplicity and the order with which characteristics meet the real axis, yield different rates of decay. As an example, we show how the results obtained can be applied to the Fokker-Planck equation. In the second part, we derive Lp-Lq estimates for wave equations with a bounded time dependent coefficient. A classification of the oscillating behaviour of the coefficient is given and related to the estimate which can be obtained.
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Li, Chih-i. Bruno Oscar P. Bruno Oscar P. "High-order solution of elliptic partial differential equations in domains containing conical singularities /." Diss., Pasadena, Calif. : Caltech, 2009. http://resolver.caltech.edu/CaltechETD:etd-08042008-005339.
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Jurás, Martin. "Geometric Aspects of Second-Order Scalar Hyperbolic Partial Differential Equations in the Plane." DigitalCommons@USU, 1997. https://digitalcommons.usu.edu/etd/7139.
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The purpose of this dissertation is to address various geometric aspects of second-order scalar hyperbolic partial differential equations in two independent variables and one dependent variable
F(x, y, u, u_x, u_y, u_xx, u_xy, u_yy )= 0 (1)
We find a characterization of hyperbolic Darboux integrable equations at level k (1) in terms of the vanishing of the generalized Laplace invariants and provide an invariant characterization of various cases in the Goursat general classification of hyperbolic Darboux integrable equations (1). In particular we give a contact invariant characterization of equations integrable by the methods of general and intermediate integrals. New relative invariants that control the existence of the first integrals of the characteristic Pfaffian systems are found and used to obtain an invariant characterization for the class of -Gordon equations. A notion of a hyperbolic Darboux system is introduced and we show by examples that the classical Laplace transformation is just a special case of a diffeomorphism of hyperbolic Darboux systems. We also construct new examples of homomorphisms between certain hyperbolic systems. We characterize Monge-Ampere equations and explicitly exhibit two invariants whose vanishing is a necessary and sufficient condition for the equation to be of the Monge-Ampere type. The solution to the inverse problem of the calculus of variations for hyperbolic equations (1) in terms of the generalized Laplace invariants is presented. We also obtain some partial results on symplectic conservation laws. We characterize, up to contact equivalence, some classical equations using the generalized Laplace invariants. These results contain characterizations of the wave, Liouville, Klein-Gordon, and certain types of Euler-Poisson equations.
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Yang, Lixiang. "Modeling Waves in Linear and Nonlinear Solids by First-Order Hyperbolic Differential Equations." The Ohio State University, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=osu1303846979.
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Pipilis, Konstantinos Georgiou. "Higher order moving finite element methods for systems described by partial differential-algebraic equations." Thesis, Imperial College London, 1990. http://hdl.handle.net/10044/1/7510.
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Kutahyalioglu, Aysen. "Oscillation Of Second Order Dynamic Equations On Time Scales." Master's thesis, METU, 2004. http://etd.lib.metu.edu.tr/upload/12605380/index.pdf.
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During the last decade, the use of time scales as a means of unifying and
extending results about various types of dynamic equations has proven to be
both prolific and fruitful. Many classical results from the theories of differential
and difference equations have time scale analogues.
In this thesis we derive new oscillation criteria for second order dynamic equations
on time scales.
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Pefferly, Robert J. "Finite difference approximations of second order quasi-linear elliptic and hyperbolic stochastic partial differential equations." Thesis, University of Edinburgh, 2001. http://hdl.handle.net/1842/11244.
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This thesis covers topics such as finite difference schemes, mean-square convergence, modelling, and numerical approximations of second order quasi-linear stochastic partial differential equations (SPDE) driven by white noise in less than three space dimensions. The motivation for discussing and expanding these topics lies in their implications in such physical phenomena as signal and information flow, gravitational and electromagnetic fields, large scale weather systems, and macro-computer networks. Chapter 2 delves into the hyperbolic SPDE in one space and one time dimension. This is an important equation to such fields as signal processing, communications, and information theory where singularities propagate throughout space as a function of time. Chapter 3 discusses some concepts and implications of elliptic SPDE's driven by additive noise. These systems are key for understanding steady state phenomena. Chapter 4 presents some numerical work regarding elliptic SPDE's driven by multiplicative and general noise. These SPDE's are open topics in the theoretical literature, hence numerical work provides significant insight into the nature of the process. Chapter 5 presents some numerical work regarding quasi-geostrophic geophysical fluid dynamics involving stochastic noise and demonstrates how these systems can be represented as a combination of elliptic and hyperbolic components.
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Brito, Loeza Carlos Francisco. "Fast numerical algorithms for high order partial differential equations with applications to image restoration techniques." Thesis, University of Liverpool, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.526786.
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Dai, Ruxin. "Richardson Extrapolation-Based High Accuracy High Efficiency Computation for Partial Differential Equations." UKnowledge, 2014. http://uknowledge.uky.edu/cs_etds/20.
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In this dissertation, Richardson extrapolation and other computational techniques are used to develop a series of high accuracy high efficiency solution techniques for solving partial differential equations (PDEs).
A Richardson extrapolation-based sixth-order method with multiple coarse grid (MCG) updating strategy is developed for 2D and 3D steady-state equations on uniform grids. Richardson extrapolation is applied to explicitly obtain a sixth-order solution on the coarse grid from two fourth-order solutions with different related scale grids. The MCG updating strategy directly computes a sixth-order solution on the fine grid by using various combinations of multiple coarse grids. A multiscale multigrid (MSMG) method is used to solve the linear systems resulting from fourth-order compact (FOC) discretizations. Numerical investigations show that the proposed methods compute high accuracy solutions and have better computational efficiency and scalability than the existing Richardson extrapolation-based sixth order method with iterative operator based interpolation.
Completed Richardson extrapolation is explored to compute sixth-order solutions on the entire fine grid. The correction between the fourth-order solution and the extrapolated sixth-order solution rather than the extrapolated sixth-order solution is involved in the interpolation process to compute sixth-order solutions for all fine grid points. The completed Richardson extrapolation does not involve significant computational cost, thus it can reach high accuracy and high efficiency goals at the same time.
There are three different techniques worked with Richardson extrapolation for computing fine grid sixth-order solutions, which are the iterative operator based interpolation, the MCG updating strategy and the completed Richardson extrapolation. In order to compare the accuracy of these Richardson extrapolation-based sixth-order methods, truncation error analysis is conducted on solving a 2D Poisson equation. Numerical comparisons are also carried out to verify the theoretical analysis.
Richardson extrapolation-based high accuracy high efficiency computation is extended to solve unsteady-state equations. A higher-order alternating direction implicit (ADI) method with completed Richardson extrapolation is developed for solving unsteady 2D convection-diffusion equations. The completed Richardson extrapolation is used to improve the accuracy of the solution obtained from a high-order ADI method in spatial and temporal domains simultaneously. Stability analysis is given to show the effects of Richardson extrapolation on stable numerical solutions from the underlying ADI method.
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Theljani, Anis. "Partial differential equations methods and regularization techniques for image inpainting." Thesis, Mulhouse, 2015. http://www.theses.fr/2015MULH0278/document.
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Cette thèse concerne le problème de désocclusion d'images, au moyen des équations aux dérivées partielles. Dans la première partie de la thèse, la désocclusion est modélisée par un problème de Cauchy qui consiste à déterminer une solution d'une équation aux dérivées partielles avec des données aux bords accessibles seulement sur une partie du bord de la partie à recouvrir. Ensuite, on a utilisé des algorithmes de minimisation issus de la théorie des jeux, pour résoudre ce problème de Cauchy. La deuxième partie de la thèse est consacrée au choix des paramètres de régularisation pour des EDP d'ordre deux et d'ordre quatre. L'approche développée consiste à construire une famille de problèmes d'optimisation bien posés où les paramètres sont choisis comme étant une fonction variable en espace. Ceci permet de prendre en compte les différents détails, à différents échelles dans l'image. L'apport de la méthode est de résoudre de façon satisfaisante et objective, le choix du paramètre de régularisation en se basant sur des indicateurs d'erreur et donc le caractère à posteriori de la méthode (i.e. indépendant de la solution exacte, en générale inconnue). En outre, elle fait appel à des techniques classiques d'adaptation de maillage, qui rendent peu coûteuses les calculs numériques. En plus, un des aspects attractif de cette méthode, en traitement d'images est la récupération et la détection de contours et de structures finesImage inpainting refers to the process of restoring a damaged image with missing information. Different mathematical approaches were suggested to deal with this problem. In particular, partial differential diffusion equations are extensively used. The underlying idea of PDE-based approaches is to fill-in damaged regions with available information from their surroundings. The first purpose of this Thesis is to treat the case where this information is not available in a part of the boundary of the damaged region. We formulate the inpainting problem as a nonlinear boundary inverse problem for incomplete images. Then, we give a Nash-game formulation of this Cauchy problem and we present different numerical which show the efficiency of the proposed approach as an inpainting method.Typically, inpainting is an ill-posed inverse problem for it most of PDEs approaches are obtained from minimization of regularized energies, in the context of Tikhonov regularization. The second part of the thesis is devoted to the choice of regularization parameters in second-and fourth-order energy-based models with the aim of obtaining as far as possible fine features of the initial image, e.g., (corners, edges, … ) in the inpainted region. We introduce a family of regularized functionals with regularization parameters to be selected locally, adaptively and in a posteriori way allowing to change locally the initial model. We also draw connections between the proposed method and the Mumford-Shah functional. An important feature of the proposed method is that the investigated PDEs are easy to discretize and the overall adaptive approach is easy to implement numerically
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Luo, BiYong. "Shooting method-based algorithms for solving control problems associated with second-order hyperbolic partial differential equations." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp05/NQ66358.pdf.
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Yano, Masayuki Ph D. Massachusetts Institute of Technology. "An optimization framework for adaptive higher-order discretizations of partial differential equations on anisotropic simplex meshes." Thesis, Massachusetts Institute of Technology, 2012. http://hdl.handle.net/1721.1/76090.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2012.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 271-281).
Improving the autonomy, efficiency, and reliability of partial differential equation (PDE) solvers has become increasingly important as powerful computers enable engineers to address modern computational challenges that require rapid characterization of the input-output relationship of complex PDE governed processes. This thesis presents work toward development of a versatile PDE solver that accurately predicts engineering quantities of interest to user-prescribed accuracy in a fully automated manner. We develop an anisotropic adaptation framework that works with any localizable error estimate, handles any discretization order, permits arbitrarily oriented anisotropic elements, robustly treats irregular features, and inherits the versatility of the underlying discretization and error estimate. Given a discretization and any localizable error estimate, the framework iterates toward a mesh that minimizes the error for a given number of degrees of freedom by considering a continuous optimization problem of the Riemannian metric field. The adaptation procedure consists of three key steps: sampling of the anisotropic error behavior using element-wise local solves; synthesis of the local errors to construct a surrogate error model based on an affine-invariant metric interpolation framework; and optimization of the surrogate model to drive the mesh toward optimality. The combination of the framework with a discontinuous Galerkin discretization and an a posteriori output error estimate results in a versatile PDE solver for reliable output prediction. The versatility and effectiveness of the adaptive framework are demonstrated in a number of applications. First, the optimality of the method is verified against anisotropic polynomial approximation theory in the context of L2 projection. Second, the behavior of the method is studied in the context of output-based adaptation using advection-diffusion problems with manufactured primal and dual solutions. Third, the framework is applied to the steady-state Euler and Reynolds-averaged Navier-Stokes equations. The results highlight the importance of adaptation for high-order discretizations and demonstrate the robustness and effectiveness of the proposed method in solving complex aerodynamic flows exhibiting a wide range of scales. Fourth, fully-unstructured space-time adaptivity is realized, and its competitiveness is assessed for wave propagation problems. Finally, the framework is applied to enable spatial error control of parametrized PDEs, producing universal optimal meshes applicable for a wide range of parameters.
by Masayuki Yano.
Ph.D.
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Kelome, Djivèdé Armel. "Viscosity solutions of second order equations in a separable Hilbert space and applications to stochastic optimal control." Diss., Georgia Institute of Technology, 2002. http://hdl.handle.net/1853/29159.
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Kubeisa, S., Hassan Ugail, and M. J. Wilson. "Interactive design using higher order PDE's." Springer Berlin, 2004. http://hdl.handle.net/10454/2659.
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YesThis paper extends the PDE method of surface generation. The governing partial differential equation is generalised to sixth order to increase its flexibility. The PDE is solved analytically, even in the case of general boundary conditions, making the method fast. The boundary conditions, which control the surface shape, are specified interactively, allowing intuitive manipulation of generic shapes. A compact user interface is presented which makes use of direct manipulation and other techniques for 3D interaction.
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Svärd, Magnus. "Stable high-order finite difference methods for aerodynamics /." Uppsala : Acta Universitatis Upsaliensis : Univ.-bibl. [distributör], 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-4621.
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Teka, Kubrom Hisho. "The obstacle problem for second order elliptic operators in nondivergence form." Diss., Kansas State University, 2012. http://hdl.handle.net/2097/14035.
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Doctor of PhilosophyDepartment of Mathematics
Ivan Blank
We study the obstacle problem with an elliptic operator in nondivergence form with principal coefficients in VMO. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary, and we show existence of blowup limits, a basic measure stability result, and a measure-theoretic version of the Caffarelli alternative proven in Caffarelli's 1977 paper ``The regularity of free boundaries in higher dimensions." Finally, we show that blowup limits are in general not unique at free boundary points.
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Malroy, Eric Thomas. "Solution of the ideal adiabatic stirling model with coupled first order differential equations by the Pasic method." Ohio : Ohio University, 1998. http://www.ohiolink.edu/etd/view.cgi?ohiou1176410606.
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Huré, Come. "Numerical methods and deep learning for stochastic control problems and partial differential equations." Thesis, Sorbonne Paris Cité, 2019. http://www.theses.fr/2019USPCC052.
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La thèse porte sur les schémas numériques pour les problèmes de décisions Markoviennes (MDPs), les équations aux dérivées partielles (EDPs), les équations différentielles stochastiques rétrogrades (ED- SRs), ainsi que les équations différentielles stochastiques rétrogrades réfléchies (EDSRs réfléchies). La thèse se divise en trois parties.La première partie porte sur des méthodes numériques pour résoudre les MDPs, à base de quan- tification et de régression locale ou globale. Un problème de market-making est proposé: il est résolu théoriquement en le réécrivant comme un MDP; et numériquement en utilisant le nouvel algorithme. Dans un second temps, une méthode de Markovian embedding est proposée pour réduire des prob- lèmes de type McKean-Vlasov avec information partielle à des MDPs. Cette méthode est mise en œuvre sur trois différents problèmes de type McKean-Vlasov avec information partielle, qui sont par la suite numériquement résolus en utilisant des méthodes numériques à base de régression et de quantification.Dans la seconde partie, on propose de nouveaux algorithmes pour résoudre les MDPs en grande dimension. Ces derniers reposent sur les réseaux de neurones, qui ont prouvé en pratique être les meilleurs pour apprendre des fonctions en grande dimension. La consistance des algorithmes proposés est prouvée, et ces derniers sont testés sur de nombreux problèmes de contrôle stochastique, ce qui permet d’illustrer leurs performances.Dans la troisième partie, on s’intéresse à des méthodes basées sur les réseaux de neurones pour résoudre les EDPs, EDSRs et EDSRs réfléchies. La convergence des algorithmes proposés est prouvée; et ces derniers sont comparés à d’autres algorithmes récents de la littérature sur quelques exemples, ce qui permet d’illustrer leurs très bonnes performancesThe present thesis deals with numerical schemes to solve Markov Decision Problems (MDPs), partial differential equations (PDEs), quasi-variational inequalities (QVIs), backward stochastic differential equations (BSDEs) and reflected backward stochastic differential equations (RBSDEs). The thesis is divided into three parts.The first part focuses on methods based on quantization, local regression and global regression to solve MDPs. Firstly, we present a new algorithm, named Qknn, and study its consistency. A time-continuous control problem of market-making is then presented, which is theoretically solved by reducing the problem to a MDP, and whose optimal control is accurately approximated by Qknn. Then, a method based on Markovian embedding is presented to reduce McKean-Vlasov control prob- lem with partial information to standard MDP. This method is applied to three different McKean- Vlasov control problems with partial information. The method and high accuracy of Qknn is validated by comparing the performance of the latter with some finite difference-based algorithms and some global regression-based algorithm such as regress-now and regress-later.In the second part of the thesis, we propose new algorithms to solve MDPs in high-dimension. Neural networks, combined with gradient-descent methods, have been empirically proved to be the best at learning complex functions in high-dimension, thus, leading us to base our new algorithms on them. We derived the theoretical rates of convergence of the proposed new algorithms, and tested them on several relevant applications.In the third part of the thesis, we propose a numerical scheme for PDEs, QVIs, BSDEs, and RBSDEs. We analyze the performance of our new algorithms, and compare them to other ones available in the literature (including the recent one proposed in [EHJ17]) on several tests, which illustrates the efficiency of our methods to estimate complex solutions in high-dimension.Keywords: Deep learning, neural networks, Stochastic control, Markov Decision Process, non- linear PDEs, QVIs, optimal stopping problem BSDEs, RBSDEs, McKean-Vlasov control, perfor- mance iteration, value iteration, hybrid iteration, global regression, local regression, regress-later, quantization, limit order book, pure-jump controlled process, algorithmic-trading, market-making, high-dimension
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Haque, Md Z. "An adaptive finite element method for systems of second-order hyperbolic partial differential equations in one space dimension." Ann Arbor, Mich. : ProQuest, 2008. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3316356.
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Thesis (Ph.D. in Computational and Applied Mathematics)--S.M.U.Title from PDF title page (viewed Mar. 16, 2009). Source: Dissertation Abstracts International, Volume: 69-08, Section: B Adviser: Peter K. Moore. Includes bibliographical references.
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Fan, Guodong. "Model Order Reduction of Multi-Dimensional Partial Differential Equations for Electrochemical-Thermal Modeling of Large-Format Lithium-ion Batteries." The Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1468917668.
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