Stochastic Momentum Methods For Optimal Control Problems Governed By Convection-diffusion Equations With Uncertain Coefficients

2022-1-6
Toraman, Sıtkı Can
Many 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.

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Citation Formats
S. C. Toraman, “Stochastic Momentum Methods For Optimal Control Problems Governed By Convection-diffusion Equations With Uncertain Coefficients,” M.S. - Master of Science, Middle East Technical University, 2022.