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‣ Efficient methods for solving multi-rate partial differential equations in radio frequency applications

Oliveira, Jorge dos Santos Freitas de
Fonte: Instituto Politécnico de Leiria Publicador: Instituto Politécnico de Leiria
Tipo: Artigo de Revista Científica
Publicado em //2006 Português
Relevância na Pesquisa
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In telecommunication electronics, radio frequency applications are usually characterized by widely separated time scales. This multi-rate behavior arises in many kinds of circuits and increases considerably the computation costs of numerical simulations. In this paper we are mainly interested in electronic circuits driven by envelope modulated signals and we will show that the application of numerical methods based on a multi-rate partial differential equation analysis will lead to an efficient strategy for simulating this type of problems.

‣ Forward backward stochastic differential equations: existence, uniqueness, a large deviations principle and connections with partial differential equations

Gomes, André de Oliveira
Fonte: Universidade de Lisboa Publicador: Universidade de Lisboa
Tipo: Dissertação de Mestrado
Publicado em //2011 Português
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Tese de mestrado em Matemática, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2011; We consider Forward Backward Stochastic Differential Equations (FBSDEs for short) with different assumptions on its coefficients. In a first part we present results of existence, uniqueness and dependence upon initial conditions and on the coefficients. There are two main methodologies employed in this study. The first one presented is the Four Step Scheme, which makes very clear the connection of FBSDEs with quasilinear parabolic systems of Partial Differential Equations (PDEs for short). The weakness of this methodology is the smoothness and regularity assumptions recquired on the coefficients of the system, which motivate the employment of Banach`s Fixed Point Theorem in the study of existence and uniqueness results. This classic analytical tool requires less regularity on the coefficients, but gives only local existence of solution in a small time duration. In a second stage, with the help of the previous work with a running-down induction on time, we can assure the existence and uniqueness of solution for the FBSDE problem in global time. The second goal of this work is the study of the assymptotic behaviour of the FBSDEs solutions when the diffusion coefficient of the forward equation is multiplicatively perturbed with a small parameter that goes to zero. This question adresses the problem of the convergence of the classical/viscosity solutions of the quasilinear parabolic system of PDEs associated to the system. When this quasilinear parabolic system of PDEs takes the form of the backward Burgers Equation...

‣ Reliable Real-Time Optimization of Nonconvex Systems Described by Parametrized Partial Differential Equations

Oliveira, I.B.; Patera, Anthony T.
Fonte: MIT - Massachusetts Institute of Technology Publicador: MIT - Massachusetts Institute of Technology
Tipo: Artigo de Revista Científica Formato: 334725 bytes; application/pdf
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The solution of a single optimization problem often requires computationally-demanding evaluations; this is especially true in optimal design of engineering components and systems described by partial differential equations. We present a technique for the rapid and reliable optimization of systems characterized by linear-functional outputs of partial differential equations with affine parameter dependence. The critical ingredients of the method are: (i) reduced-basis techniques for dimension reduction in computational requirements; (ii) an "off-line/on-line" computational decomposition for the rapid calculation of outputs of interest and respective sensitivities in the limit of many queries; (iii) a posteriori error bounds for rigorous uncertainty and feasibility control; (iv) Interior Point Methods (IPMs) for efficient solution of the optimization problem; and (v) a trust-region Sequential Quadratic Programming (SQP) interpretation of IPMs for treatment of possibly non-convex costs and constraints.; Singapore-MIT Alliance (SMA)

‣ Reduced-Basis Methods for Inverse Problems in Partial Differential Equations

Nguyen, C.N.; Liu, Guirong; Patera, Anthony T.
Fonte: MIT - Massachusetts Institute of Technology Publicador: MIT - Massachusetts Institute of Technology
Tipo: Artigo de Revista Científica Formato: 15952 bytes; application/pdf
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We present a technique for the rapid, reliable, and accurate evaluation of functional outputs of parametrized elliptic partial differential equations. The essential ingredients are (i) rapidly globally convergent reduced-basis approximations – Galerkin projection onto a space WN spanned by the solutions of the governing partial differential equations at N selected points in parameter space; (ii) a posteriori error estimation - relaxations of the error-residual equation that provide sharp and inexpensive bounds for the error in the output of interest; and (iii) off-line/online computational procedures – methods that decouple the generation and projection stages of the approximation process. The operation count for the online stage – in which, given a new parameter, we calculate the output of interest and associated error bounds – depends only on N (typically very small) and the parametric dependencies of the problem. In this study, we first develop rigorous a posteriori error estimators for (affine in the parameter) noncoercive problems such as the Helmholtz (reduced-wave) equation. The critical ingredients are the residual, an appropriate bound conditioner, and a piecewise-constant lower bound for the inf-sup stability factor. In addition...

‣ Sistema p-Fuzzy aplicado às equações diferenciais parciais; Model P-Fuzzy applied to partial differential equations

Ferreira, Daniela Portes Leal
Fonte: Universidade Federal de Uberlândia Publicador: Universidade Federal de Uberlândia
Tipo: Dissertação
Português
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Descrever matematicamente os fenômenos naturais para fazer previsões e tomar decisões é um dos grandes desafios da matemática. Vários fenômenos podem ser descritos através de equações diferenciais parciais, entretanto muitos desses fenômenos apresentam variáveis linguísticas, isto é, informações vagas e imprecisas. Essa característica dificulta a modelagem do fenômeno através das equações diferenciais, já que estas dependem da precisão dos parâmetros utilizados. A proposta deste trabalho é demonstrar a viabilidade e aplicabilidade dos sistemas parcialmente fuzzy (p-fuzzy) na modelagem de fenômenos descritos por equações diferenciais parciais. Os sistemas p-fuzzy foram obtidos utilizando a função ANFIS do Matlab, que a partir de um conjunto de dados identifica as funções de pertinência e os parâmetros do sistema baseado em regras fuzzy. Analisando os resultados alcançados concluímos que a utilização dos sistemas pfuzzy é uma ferramenta útil para a modelagem de fenômenos particulares que envolvem taxas de variações parciais, inclusive com evolução no tempo. __________________________________________________________________________________________ ABSTRACT; Describing mathematically natural phenomena in order to make predictions and decisions is one of the biggest challenges in mathematics. Several phenomena can be described through partial differential equations...

‣ Average and deviation for slow-fast stochastic partial differential equations

Wang, W.; Roberts, A.
Fonte: Academic Press Inc Publicador: Academic Press Inc
Tipo: Artigo de Revista Científica
Publicado em //2012 Português
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Averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. This article derives an averaged equation for a class of stochastic partial differential equations without any Lipschitz assumption on the slow modes. The rate of convergence in probability is obtained as a byproduct. Importantly, the stochastic deviation between the original equation and the averaged equation is also studied. A martingale approach proves that the deviation is described by a Gaussian process. This gives an approximation to errors of order O(ε) instead of order O(√ε) attained in previous averaging.; W. Wang, A.J. Roberts

‣ Long-time analysis of Hamiltonian partial differential equations and their discretizations; Langzeitverhalten Hamiltonscher partieller Differentialgleichungen und ihrer Diskretisierungen

Gauckler, Ludwig
Fonte: Universidade de Tubinga Publicador: Universidade de Tubinga
Tipo: Dissertação
Português
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Hamiltonian partial differential equations are partial differential equations, that can be written in the form of a Hamiltonian system as for instance the equations of motion in classical mechanics but on an infinite dimensional phase space. Important examples are Schrödinger equations and wave equations which attract much interest because of both, their beautiful mathematical structure but also their applications in physics, for instance in quantum mechanics. In the theory of (finite or infinite dimensional) Hamiltonian systems invariants or conserved quantities play a dominant role. These quantities are conserved along a solution of such equations and represent important physical properties such as energy conservation, but they are also fundamental in a mathematical analysis of the equations, for instance to show well-posedness. From the point of view of numerical analysis the following question is then inevitable: What is the behaviour of invariants of Hamiltonian partial differential equations along a numerical solution of such equations? This is a fundamental problem in the field of geometric numerical integration which is concerned with the construction and the analysis of structure-preserving algorithms for differential equations. This question turns out to be closely related to a question in perturbation theory concerning the exact solution of Hamiltonian partial differential equations: How does a small (nonlinear) perturbation change the dynamics of a linear Hamiltonian partial differential equation? This thesis contributes to the answers of both questions. We show that exact invariants of a linear Hamiltonian partial differential equation are approximately conserved along solutions of a nonlinearly perturbed version of the equation on remarkably long time intervals. This is done with the help of a modulated Fourier expansion of the solution. It turns out that this technique also allows to study rigorously the first question...

‣ On Symbolic Solutions of Algebraic Partial Differential Equations

Grasegger, G; Sendra Pons, Juan Rafael; Lastra Sedano, Alberto; Winkler, Fraz
Fonte: Springer Publicador: Springer
Tipo: info:eu-repo/semantics/article; info:eu-repo/semantics/submittedVersion Formato: application/pdf
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The final version of this paper appears in Grasegger G., Lastra A., Sendra J.R. and Winkler F. (2014). On symbolic solutions of algebraic partial differential equations, Proc. CASC 2014 SpringerVerlag LNCS 8660 pp. 111-120. DOI 10.1007/978-3-319-10515-4_9 and it is available at at Springer via http://DOI 10.1007/978-3-319-10515-4_9; In this paper we present a general procedure for solving rst-order autonomous algebraic partial di erential equations in two independent variables. The method uses proper rational parametrizations of algebraic surfaces and generalizes a similar procedure for rst-order autonomous ordinary di erential equations. We will demonstrate in examples that, depending on certain steps in the procedure, rational, radical or even non-algebraic solutions can be found. Solutions computed by the procedure will depend on two arbitrary independent constants.

‣ Aspects of overdetermined systems of partial differential equations in projective and conformal geometry

Randall, Matthew
Fonte: Universidade Nacional da Austrália Publicador: Universidade Nacional da Austrália
Tipo: Thesis (PhD); Doctor of Philosophy (PhD)
Português
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This thesis discusses aspects of overdetermined systems of partial differential equations (PDEs) in projective and conformal geometry. The first part deals with projective differential geometry. A projective surface is a 2-dimensional smooth manifold equipped with a projective structure i.e. a class of torsion-free affine connections that have the same geodesics as unparameterised curves. Given any projective surface we can ask whether it admits a torsion-free affine connection (in its projective class) that has skew-symmetric Ricci tensor. This is equivalent to solving a particular overdetermined system of semi-linear partial differential equations. It turns out that there are local obstructions to solving the system of PDEs in two dimensions. These obstructions are constructed out of local invariants of the projective structure. We give examples of projective surfaces that admit skew-symmetric Ricci tensor and examples that do not because of nonvanishing obstructions. We relate projective surfaces admitting skew-symmetric Ricci tensor to 3-webs in 2 dimensions. We also give examples of projective structures in higher dimensions that admit skew-symmetric Ricci tensor. The second part of the thesis deals with conformal differential geometry. On Mobius surfaces introduced in [5]...

‣ Um estudo sobre o espalhamento da dengues usando equações diferenciais parciais e logica fuzzy; A study of the spread of dengue using partial differential equations and fuzzy logic

Luciana Takata Gomes
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Dissertação de Mestrado Formato: application/pdf
Publicado em 29/06/2009 Português
Relevância na Pesquisa
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A doença a ser analisada é a dengue e, com este intuito, são criados alguns modelos matemáticos para simular sua evolução no distrito sul da cidade de Campinas. Divide-se a população humana local em três compartimentos, de acordo com o estado dos indivíduos - suscetível, infectante ou recuperado. A interação destas diferentes populações de humanos com a de mosquitos Aedes aegypti determina o comportamento da doença no domínio especificado. As variáveis de estado do modelo são as populações de humanos e a população de mosquitos, cuja divisão em compartimentos depende do modelo adotado. Seus valores são determinísticos e representam a densidade das populações em cada ponto do domínio. O trabalho contempla informações de especialistas a respeito do comportamento da doença e das condições para a proliferação e espalhamento do mosquito vetor. Tais condições, consideradas de natureza incerta, acabam por determinar o risco de contração da doença e, consequentemente, parâmetros dos modelos. A modelagem resulta em sistemas de Equações Diferencias Parciais, com alguns de seus parâmetros incertos. Para a obtenção de soluções (valores das variáveis em questão ao longo do tempo e sobre o domínio espacial citado)...

‣ 18.303 Linear Partial Differential Equations, Fall 2005; Linear Partial Differential Equations

Hancock, Matthew James, 1975-
Fonte: MIT - Massachusetts Institute of Technology Publicador: MIT - Massachusetts Institute of Technology
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The classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. Methods of solution, such as separation of variables, Fourier series and transforms, eigenvalue problems. Green's function methods are emphasized. 18.04 or 18.112 are useful, as well as previous acquaintance with the equations as they arise in scientific applications.

‣ 18.303 Linear Partial Differential Equations, Fall 2004; Linear Partial Differential Equations

Hancock, Matthew James, 1975-
Fonte: MIT - Massachusetts Institute of Technology Publicador: MIT - Massachusetts Institute of Technology
Português
Relevância na Pesquisa
78.18772%
The classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. Methods of solution, such as separation of variables, Fourier series and transforms, eigenvalue problems. Green's function methods are emphasized. 18.04 or 18.112 are useful, as well as previous acquaintance with the equations as they arise in scientific applications.

‣ Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations

Prud'homme, C.; Rovas, D.V.; Veroy, K.; Machiels, L.; Maday, Y.; Patera, Anthony T.; Turinici, G.
Fonte: MIT - Massachusetts Institute of Technology Publicador: MIT - Massachusetts Institute of Technology
Tipo: Artigo de Revista Científica Formato: 554202 bytes; application/pdf
Português
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98.14538%
We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations -- Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation -- relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures -- methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage -- in which, given a new parameter value, we calculate the output of interest and associated error bound -- depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.; Singapore-MIT Alliance (SMA)

‣ Reliable Real-Time Solution of Parametrized Elliptic Partial Differential Equations: Application to Elasticity

Veroy, K.; Leurent, T.; Prud'homme, C.; Rovas, D.V.; Patera, Anthony T.
Fonte: MIT - Massachusetts Institute of Technology Publicador: MIT - Massachusetts Institute of Technology
Tipo: Artigo de Revista Científica Formato: 319291 bytes; application/pdf
Português
Relevância na Pesquisa
98.2491%
The optimization, control, and characterization of engineering components or systems require fast, repeated, and accurate evaluation of a partial-differential-equation-induced input-output relationship. We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic partial differential equations with affine parameter dependence. The method has three components: (i) rapidly convergent reduced{basis approximations; (ii) a posteriori error estimation; and (iii) off-line/on-line computational procedures. These components -- integrated within a special network architecture -- render partial differential equation solutions truly "useful": essentially real{time as regards operation count; "blackbox" as regards reliability; and directly relevant as regards the (limited) input-output data required.; Singapore-MIT Alliance (SMA)

‣ Stochastic Partial Differential Equations on Evolving Surfaces and Evolving Riemannian Manifolds

Elliott, C. M.; Hairer, M.; Scott, M. R.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 29/08/2012 Português
Relevância na Pesquisa
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We formulate stochastic partial differential equations on Riemannian manifolds, moving surfaces, general evolving Riemannian manifolds (with appropriate assumptions) and Riemannian manifolds with random metrics, in the variational setting of the analysis to stochastic partial differential equations. Considering mainly linear stochastic partial differential equations, we establish various existence and uniqueness theorems.; Comment: 42 pages, 1 figure

‣ On Degenerate Partial Differential Equations

Chen, Gui-Qiang G.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 15/05/2010 Português
Relevância na Pesquisa
78.440654%
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial differential equations, are presented, which arise naturally in some longstanding, fundamental problems in fluid mechanics and differential geometry. The solution to these fundamental problems greatly requires a deep understanding of nonlinear degenerate partial differential equations. Our emphasis is on exploring and/or developing unified mathematical approaches, as well as new ideas and techniques. The potential approaches we have identified and/or developed through these examples include kinetic approaches, free boundary approaches, weak convergence approaches, and related nonlinear ideas and techniques. We remark that most of the important problems for nonlinear degenerate partial differential equations are truly challenging and still widely open, which require further new ideas, techniques, and approaches, and deserve our special attention and further efforts.

‣ Space-like Weingarten surfaces in the three-dimensional Minkowski space and their natural partial differential equations

Ganchev, Georgi; Mihova, Vesselka
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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On any space-like W-surface in the three-dimensional Minkowski space we introduce locally natural principal parameters and prove that such a surface is determined uniquely up to motion by a special invariant function, which satisfies a natural non-linear partial differential equation. This result can be interpreted as a solution to the Lund-Regge reduction problem for space-like W-surfaces in Minkowski space. We apply this theory to linear fractional space-like W-surfaces and obtain the natural non-linear partial differential equations describing them. We obtain a characterization of space-like surfaces, whose curvatures satisfy a linear relation, by means of their natural partial differential equations. We obtain the ten natural PDE's describing all linear fractional space-like W-surfaces.; Comment: 16 pages; 5 references added. arXiv admin note: substantial text overlap with arXiv:1105.3652

‣ The inverse problem of the calculus of variations for systems of second-order partial differential equations in the plane

Biesecker, Matt
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 15/10/2009 Português
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A complete solution to the multiplier version of the inverse problem of the calculus of variations is given for a class of hyperbolic systems of second-order partial differential equations in two independent variables. The necessary and sufficient algebraic and differential conditions for the existence of a variational multiplier are derived. It is shown that the number of independent variational multipliers is determined by the nullity of a completely algebraic system of equations associated to the given system of partial differential equations. An algorithm for solving the inverse problem is demonstrated on several examples. Systems of second-order partial differential equations in two independent and dependent variables are studied and systems which have more than one variational formulation are classified up to contact equivalence.

‣ On some partial differential equation models in socio-economic contexts - analysis and numerical simulations

Pietschmann, Jan-Frederik
Fonte: University of Cambridge; Departent of Applied Mathhematics and Theoretical Physics; Trinity College Publicador: University of Cambridge; Departent of Applied Mathhematics and Theoretical Physics; Trinity College
Tipo: Thesis; doctoral; PhD
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This thesis deals with the analysis and numerical simulation of different partial differential equation models arising in socioeconomic sciences. It is divided into two parts: The first part deals with a mean-field price formation model introduced by Lasry and Lions in 2007. This model describes the dynamic behaviour of the price of a good being traded between a group of buyers and a group of vendors. Existence (locally in time) of smooth solutions is established, and obstructions to proving a global existence result are examined. Also, properties of a regularised version of the model are explored and numerical examples are shown. Furthermore, the possibility of reconstructing the initial datum given a number of observations, regarding the price and the transaction rate, is considered. Using a variational approach, the problem can be expressed as a non-linear constrained minimization problem. We show that the initial datum is uniquely determined by the price (identifiability). Furthermore, a numerical scheme is implemented and a variety of examples are presented. The second part of this thesis treats two different models describing the motion of (large) human crowds. For the first model, introduced by R.L. Hughes in 2002, several regularised versions are considered. Existence and uniqueness of entropy solutions are proven using the technique of vanishing viscosity. In one space dimension...

‣ Solving Partial Differential Equations Using Artificial Neural Networks

Rudd, Keith
Fonte: Universidade Duke Publicador: Universidade Duke
Tipo: Dissertação
Publicado em //2013 Português
Relevância na Pesquisa
88.22243%

This thesis presents a method for solving partial differential equations (PDEs) using articial neural networks. The method uses a constrained backpropagation (CPROP) approach for preserving prior knowledge during incremental training for solving nonlinear elliptic and parabolic PDEs adaptively, in non-stationary environments. Compared to previous methods that use penalty functions or Lagrange multipliers,

CPROP reduces the dimensionality of the optimization problem by using direct elimination, while satisfying the equality constraints associated with the boundary and initial conditions exactly, at every iteration of the algorithm. The effectiveness of this method is demonstrated through several examples, including nonlinear elliptic

and parabolic PDEs with changing parameters and non-homogeneous terms. The computational complexity analysis shows that CPROP compares favorably to existing methods of solution, and that it leads to considerable computational savings when subject to non-stationary environments.

The CPROP based approach is extended to a constrained integration (CINT) method for solving initial boundary value partial differential equations (PDEs). The CINT method combines classical Galerkin methods with CPROP in order to constrain the ANN to approximately satisfy the boundary condition at each stage of integration. The advantage of the CINT method is that it is readily applicable to PDEs in irregular domains and requires no special modification for domains with complex geometries. Furthermore...