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Spatial Decomposition for Differential Equation Constrained Stochastic Programs
Šabartová, Zuzana ; Mrázková, Eva (oponent) ; Popela, Pavel (vedoucí práce)
Wide variety of optimum design problems in engineering leads to optimization models constrained by ordinary or partial differential equations (ODE or PDE). Numerical methods based on discretising domain are required to obtain a non-differential numerical description of the differential parts of constraints because the analytical solutions can be found only for simple problems. We chose the finite element method. The real problems are often large-scale and exceed computational capacity. Hence, we employ the progressive hedging algorithm (PHA) - an efficient scenario decomposition method for solving scenario-based stochastic programs, which can be implemented in parallel to reduce the computing time. A modified PHA was used for an original concept of spatial decomposition based on the mesh created for approximation of differential equation constraints. The algorithm consists of a few main steps: solve our problem with a raw discretization, decompose it into overlapping parts of the domain, and solve it again iteratively by the PHA with a finer discretization - using values from the raw discretization as boundary conditions until a given accuracy is reached. The spatial decomposition is applied to a basic test problem from the civil engineering area: design of beam cross section dimensions. The algorithms are implemented in GAMS software and finally results are evaluated with respect to a computational complexity and a length of overlap.
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Dynamika letu a řízení
Šabartová, Zuzana ; Tomášek, Petr (oponent) ; Nechvátal, Luděk (vedoucí práce)
Tato bakalářská práce se zabývá odvozením pohybových rovnic pro dynamickou soustavu letadla. Rovnice za předpokladu ustáleného přímočarého letu s malými odchylkami můžeme linearizovat a uplatnit teorii řízení. Výsledkem práce je vedle teoretické stránky také simulace pohybové odezvy na změnu polohy řídících prvků v prostředí MATLAB & Simulink u dvou vybraných letadel.
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Spatial Decomposition for Differential Equation Constrained Stochastic Programs
Šabartová, Zuzana ; Mrázková, Eva (oponent) ; Popela, Pavel (vedoucí práce)
Wide variety of optimum design problems in engineering leads to optimization models constrained by ordinary or partial differential equations (ODE or PDE). Numerical methods based on discretising domain are required to obtain a non-differential numerical description of the differential parts of constraints because the analytical solutions can be found only for simple problems. We chose the finite element method. The real problems are often large-scale and exceed computational capacity. Hence, we employ the progressive hedging algorithm (PHA) - an efficient scenario decomposition method for solving scenario-based stochastic programs, which can be implemented in parallel to reduce the computing time. A modified PHA was used for an original concept of spatial decomposition based on the mesh created for approximation of differential equation constraints. The algorithm consists of a few main steps: solve our problem with a raw discretization, decompose it into overlapping parts of the domain, and solve it again iteratively by the PHA with a finer discretization - using values from the raw discretization as boundary conditions until a given accuracy is reached. The spatial decomposition is applied to a basic test problem from the civil engineering area: design of beam cross section dimensions. The algorithms are implemented in GAMS software and finally results are evaluated with respect to a computational complexity and a length of overlap.
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Dynamika letu a řízení
Šabartová, Zuzana ; Tomášek, Petr (oponent) ; Nechvátal, Luděk (vedoucí práce)
Tato bakalářská práce se zabývá odvozením pohybových rovnic pro dynamickou soustavu letadla. Rovnice za předpokladu ustáleného přímočarého letu s malými odchylkami můžeme linearizovat a uplatnit teorii řízení. Výsledkem práce je vedle teoretické stránky také simulace pohybové odezvy na změnu polohy řídících prvků v prostředí MATLAB & Simulink u dvou vybraných letadel.
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