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Belirsiz Katsayılı Konveksi̇yon-Di̇füzyon Denklemlerinin Yönettiği Optimal Kontrol Problemleri Için Stokastik Momentum Yöntemleri
No full textMany physical phenomena such as the flow of an aircraft, heating process, or wave propagation are modeled mathematically by differential equations, in particular partial differential equations (PDEs). Analytical solutions to PDEs are often unknown or very hard to obtain. Because of that, we simulate such systems by numerical methods such as finite difference, finite volume, finite element, etc. When we want to control the behavior of certain system components, such as the shape of a wing of an aircraft or an applied heat distribution, it becomes equivalent to optimizing certain parameters of the underlying PDEs. Optimization of real-world systems in this way is called PDE-constrained optimization or optimal control problems. To have a more accurate mathematical model, we employ uncertain coefficients in PDEs since nature has different sources of intrinsic randomness. In this thesis, we study a numerical investigation of a strongly convex and smooth tracking-type functional subject to a convection-diffusion equation with random coefficients. In spatial dimension, we use the Finite Element Method (FEM), in probability dimension, we use the Monte Carlo (MC) method, and as an optimization method, we use the stochastic gradient (SG) method, where the true gradient is replaced by a stochastic one to minimize
the expected value over a random function. To accelerate the convergence of the stochastic approach, momentum terms, i.e., Polyak’s and Nesterov’s momentums, are added. A full error analysis including Monte Carlo, finite element, and stochastic momentum gradient iteration errors are done. Numerical examples are presented to illustrate the performance of the proposed stochastic approximations in the PDE-constrained optimization setting.Bir uçağın uçuşu, ısıtma işlemi veya dalga yayılımı gibi birçok fiziksel olay, matematiksel olarak diferansiyel denklemler, özellikle kısmi türevli diferansiyel denklemler ile modellenir. Kısmi türevli diferansiyel denklemlere yönelik analitik çözümler genellikle bilinmemektedir veya elde edilmesi çok zordur. Bu nedenle, bu tür sistemler sonlu fark, sonlu hacim veya sonlu eleman ve benzeri gibi sayısal yöntemlerle çözülebilir. Bir uçağın kanadının şekli veya uygulanan bir ısı dağıtımı gibi belirli sistem bile¸senlerinin davranışını kontrol etmek istediğimizde bu, kısmi türevli diferansiyel denklemin belirli parametrelerini optimize etmeye karşılık gelir. Gerçek dünya sistemlerinin bu optimizasyonuna kısmi türevli diferansiyel denklemler tarafından kısıtlı en iyileme problemleri veya optimal kontrol problemleri denir. Daha doğru bir matematiksel modele sahip olmak için, doğa farklı içsel rastgelelik kaynaklarına sahip olduğundan, kısmı türevli diferansiyel denklemler rastgelelik içeren parametrelerle ifade edilebilir. Bu tezde, rastgele katsayılı bir konveksiyon-difüzyon denklemine tabi olan güçlü dışbükey ve düzgün izleme tipi bir fonksiyonelin sayısal bir araştırması ele alınmıştır. Fiziksel boyutta sonlu elemanlar, olasılık boyutunda Monte Carlo ayrıklaştırma yöntemleri olarak kullanılırken, optimizasyon yöntemi olarak gerçek gradyanın stokastik varyantı ile değiştirildiği stokastik gradyan yöntemini kullanılmaktadır. Stokastik yaklaşımın yakınsamasını hızlandırmak için momentum terimleri, yani Polyak ve Nesterov’un momentumları eklenmiştir. Monte Carlo, sonlu eleman ve stokastik momentum gradyan yineleme hatalarını içeren tam bir hata analizi yapılmaktadır, Son olarak kısmi türevli diferansiyel denklemler tarafından kısıtlı optimizasyon dizeneğinde önerilen stokastik yaklaşımların performansını göstermek için sayısal örnekler sunulmaktadır.M.S. - Master of Scienc
Rastgele Katsayılı Kısmi Diferansiyel Denklem Tabanlı Modeller için Stokastik Süreksiz Galerkin Yöntemleri
No full textUncertainty, such as uncertain parameters, arises from many complex physical systems in engineering and science, e.g., fluid dynamics, heat transfer, chemically reacting systems, underwater pollution, radiation transport, and oil field reservoirs. It is well known that these systems can be modeled by partial differential equations (PDEs) with random input data. However, the information available on the input data is very limited, which causes a high level of uncertainty in approximating the solution to these problems. Therefore, the idea of uncertainty quantification (UQ) has become a powerful tool to model such physical problems in the last decade.
In this thesis, the aim is the development, analysis, and application of stochastic discontinuous Galerkin method for partial differential equation (PDE)--based models with random coefficients. As a model, we first focus on the single convection diffusion equation containing uncertainty. To identify the random coefficients, we use the well–known technique Karhunen Loève (KL) expansion. Stochastic Galerkin (SG) approach, turning the original problem containing uncertainties into a large system of deterministic problems, is applied to discretize the stochastic domain, while a discontinuous Galerkin method is preferred for the spatial discretization due to its better convergence behaviour for convection dominated PDEs. A priori and a posteriori error estimates are also derived. SG method generally results in a large coupled system of linear equations, the solution of which is computationally difficult to compute using standard solvers. Therefore, we provide low-rank iterative solvers for efficient computing of such solutions, which compute low-rank approximations to the solutions of those systems. Moreover, to overcome boundary and/or interior layers, localized regions where the derivative of the solution is large, an efficient adaptive algorithm is presented for the numerical solution of the parametric convection diffusion equations. On the other hand, certain parameters of a model are needed to be optimized in order to reach the desired target, for instance, the location where the oil is inserted into the medium, the temperature of a melting/heating process, or the shape of the aircraft wings. Therefore, we extend our findings to optimization problems and consider optimal control problems governed by convection diffusion equations involving random inputs.Belirsiz parametreler gibi belirsizlik, mühendislik ve bilimdeki birçok karmaşık fiziksel sistemden kaynaklanır; örneğin, akışkanlar dinamiği, ısı transferi, kimyasal olarak reaksiyona giren sistemler, su altı kirliliği, radyasyon taşınımı ve petrol sahası rezervuarları. Bu sistemlerin rasgele girdi verileri ile kısmi diferansiyel denklemler ile modellenebileceği iyi bilinmektedir. Bununla birlikte, girdi verilerinde mevcut olan bilgiler çok sınırlıdır ve bu da bu problemlerin çözümüne yaklaşmada yüksek düzeyde belirsizliğe neden olur. Bu nedenle, belirsizlik ölçümü fikri, son on yılda bu tür fiziksel problemleri modellemek için güçlü bir araç haline geldi.
Bu tezde amaç, rastgele katsayılara sahip kısmi diferansiyel denklem (PDE) tabanlı modeller için stokastik süreksiz Galerkin yönteminin geliştirilmesi, analizi ve uygulanmasıdır. Bir model olarak, önce belirsizlik içeren tek konveksiyon difüzyon denklemine odaklanıyoruz. Rastgele katsayıları belirlemek için iyi bilinen Karhunen Loève (KL) genişletme tekniğini kullanıyoruz. Belirsizlikler içeren orijinal problemi büyük bir deterministik problemler sistemine dönüştüren Stokastik Galerkin (SG) yaklaşımı, stokastik alanı ayrıklaştırmak için uygulanırken, konveksiyon ağırlıklı PDE'ler için daha iyi yakınsama davranışı nedeniyle uzaysal ayrıklaştırma için süreksiz bir Galerkin yöntemi tercih edilir. Priori ve posteriori hata tahminleri de türetilir. SG yöntemi genellikle, çözümünün standart çözücüler kullanılarak hesaplanması zor olan büyük bir birleşik lineer denklem sistemiyle sonuçlanır. Bu sebeple, bu sistemlerin çözümlerine düşük kerteli yaklaşımlar hesaplayan bu tür çözümlerin verimli bir şekilde hesaplanması için düşük kerteli yinelemeli çözücüler sağlıyoruz. Ayrıca, sınır ve/veya iç katmanların, çözümün türevinin büyük olduğu yerel bölgelerin üstesinden gelmek için, parametrik konveksiyon difüzyon denklemlerinin sayısal çözümü için etkili bir uyarlamalı algoritma sunulmuştur. Öte yandan, istenen hedefe ulaşmak için bir modelin belirli parametrelerinin, örneğin yağın ortama eklendiği konum, bir eritme/ısıtma işleminin sıcaklığı veya uçak kanatlarının şekli gibi bazı parametrelerin optimize edilmesi gerekir. Bu nedenle, bulgularımızı optimizasyon problemlerine genişlettik ve rastgele girdiler içeren konveksiyon difüzyon denklemleri tarafından yönetilen optimal kontrol problemlerini ele aldık.Ph.D. - Doctoral Progra
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Residual based a posteriori error estimation for Dirichlet boundary control problems
Get PDFWe study a residual–based a posteriori error estimate for the solution of Dirichlet boundary control problem governed by a convection diffusion equation on a two dimensional convex polygonal domain, using the local discontinuous Galerkin (LDG) method with upwinding for the convection term. With the usage of LDG method, the control variable naturally exists in the variational form due to its mixed finite element structure. We also demonstrate the application of our a posteriori error estimator for the adaptive solution of these optimal control problems
Konveksiyon ağırlıklı eniyilemeli kontrol problemlerinin uyarlamalı kesintili Galerkin yöntemleriyle çözümü.
No full textMany real-life applications such as the shape optimization of technological devices, the identification of parameters in environmental processes and flow control problems lead to optimization problems governed by systems of convection di usion partial di erential equations (PDEs). When convection dominates di usion, the solutions of these PDEs typically exhibit layers on small regions where the solution has large gradients. Hence, it requires special numerical techniques, which take into account the structure of the convection. The integration of discretization and optimization is important for the overall e ciency of the solution process. Discontinuous Galerkin (DG) methods became recently as an alternative to the finite di erence, finite volume and continuous finite element methods for solving wave dominated problems like convection di usion equations since they possess higher accuracy. This thesis will focus on analysis and application of DG methods for linear-quadratic convection dominated optimal control problems. Because of the inconsistencies of the standard stabilized methods such as streamline upwind Petrov Galerkin (SUPG) on convection di usion optimal control problems, the discretize-then-optimize and the optimize-then-discretize do not commute. However, the upwind symmetric interior penalty Galerkin (SIPG) method leads to the same discrete optimality systems. The other DG methods such as nonsymmetric interior penalty Galerkin (NIPG) and incomplete interior penalty Galerkin (IIPG) method also yield the same discrete optimality systems when penalization constant is taken large enough. We will study a posteriori error estimates of the upwind SIPG method for the distributed unconstrained and control constrained optimal control problems. In convection dominated optimal control problems with boundary and/or interior layers, the oscillations are propagated downwind and upwind direction in the interior domain, due the opposite sign of convection terms in state and adjoint equations. Hence, we will use residual based a posteriori error estimators to reduce these oscillations around the boundary and/or interior layers. Finally, theoretical analysis will be confirmed by several numerical examples with and without control constraintsM.S. - Master of Scienc
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Goal–oriented a posteriori error estimation for Dirichlet boundary control problems
No full textWe study goal-oriented a posteriori error estimates for the numerical approximation of Dirichlet boundary control problem governed by a convection diffusion equation with pointwise control constraints on a two dimensional convex polygonal domain. The local discontinuous Galerkin method is used as a discretization technique since the control variable is involved in a variational form in a natural sense. We derive primal–dual weighted error estimates for the objective functional with an error term representing the mismatch in the complementary system due to the discretization. Numerical examples are presented to illustrate the performance of the proposed estimator
Goal–oriented a posteriori error estimation for Dirichlet boundary control problems
No full textWe study goal-oriented a posteriori error estimates for the numerical approximation ofDirichlet boundary control problem governed by a convection diffusion equation withpointwise control constraints on a two dimensional convex polygonal domain. The localdiscontinuous Galerkin method is used as a discretization technique since the controlvariable is involved in a variational form in a natural sense. We derive primal–dualweightederrorestimatesfortheobjectivefunctionalwithanerrortermrepresentingthemismatch in the complementary system due to the discretization. Numerical examplesare presented to illustrate the performance of the proposed estimator
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