Progressive construction of a parametric reduced-order model for PDE-constrained optimization

An adaptive approach to using reduced-order models as surrogates in PDE-constrained optimization is introduced that breaks the traditional offline-online framework of model order reduction. A sequence of optimization problems constrained by a given Reduced-Order Model (ROM) is defined with the goal of converging to the solution of a given PDE-constrained optimization problem. For each reduced optimization problem, the constraining ROM is trained from sampling the High-Dimensional Model (HDM) at the solution of some of the previous problems in the sequence. The reduced optimization problems are equipped with a nonlinear trust-region based on a residual error indicator to keep the optimization trajectory in a region of the parameter space where the ROM is accurate. A technique for incorporating sensitivities into a Reduced-Order Basis (ROB) is also presented, along with a methodology for computing sensitivities of the reduced-order model that minimizes the distance to the corresponding HDM sensitivity, in a suitable norm. The proposed reduced optimization framework is applied to subsonic aerodynamic shape optimization and shown to reduce the number of queries to the HDM by a factor of 4-5, compared to the optimization problem solved using only the HDM, with errors in the optimal solution far less than 0.1%

Shape Determination for Deformed Cavities

A realistic superconducting RF cavity has its shape deformed comparing to its designed shape due to the loose tolerance in the fabrication process and the frequency tuning for its accelerating mode. A PDE-constrained optimization problem is proposed to determine the deformation of the cavity. A reduce space method is used to solve the PDE-constrained optimization problem where design sensitivities were computed using a continuous adjoint approach. A proof-of-concept example is given in which the deformation parameters of a single cavity-cell with two different types of deformation were computed

Directory Based Cache Coherency Protocols for Shared Memory Multiprocessors

Directory based cache coherency protocols can be used to build large scale, weakly ordered, shared memory multiprocessors. The salient feature of these protocols is that they are interconnection network independent, making them more scaleable than snoopy bus protocols. The major criticisms of previously defined directory protocols point to the size of memory heeded to store the directory and the amount of communication across the interconnection network required to maintain coherence. This thesis tries solving these problems by changing the entry format of the global table, altering the architecture of the global table, and developing new protocols. Some alternative directory entry formats are described, including a special entry format for implementing queueing semaphores. Evaluation of the various entry formats is done with probabilistic models of shared cache blocks and software simulation. A variable length global table organization is presented which can be used to reduce the size of the global table, regardless of the entry format. Its performance is analyzed using software simulation. A protocol which maintains a linked list of processors which have a particular block cached is presented. Several variations of this protocol induce less interconnection network traffic than traditional protocols

Hessian-based model reduction with applications to initial-condition inverse problems

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2007.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 85-89).(cont.) Reduced-order models that are able to approximate output quantities of interest of high-fidelity computational models over a wide range of input parameters play an important role in making tractable large-scale optimal design, optimal control, and inverse problem applications. We consider the problem of determining a reduced model of an initial value problem that spans all important initial conditions, and pose the task of determining appropriate training sets for reduced-basis construction as a sequence of optimization problems. We show that, under certain assumptions, these optimization problems have an explicit solution in the form of an eigenvalue problem, yielding an efficient Hessian based model reduction algorithm that scales well to systems with states of high dimension. Furthermore, tight upper bounds are given for the error in the outputs of the reduced models. The reduction methodology is demonstrated for several linear systems, including a large-scale contaminant transport problem. Models constructed with the Hessian-based approach are used to solve an initial condition inverse problem, and the resulting initial condition estimates compare favorably to those computed with high-fidelity models and low-rank approximations. Initial condition estimates are then formed with limited observational data to demonstrate that predictions of system state using reduced models are possible given relatively short measurement time windows. We show that reduced state can be used to approximate full state given an appropriate reduced basis, meaning that approximate forward simulations of large-scale systems can be computed in reduced space.by Omar Shahid Bashir.S.M

Smooth and Robust Solutions for Dirichlet Boundary Control of Fluid-Solid Conjugate Heat Transfer Problems

This work offers new computational methods for the optimal control of the conjugate heat transfer (CHT) problem in thermal science. Conjugate heat transfer has many important industrial applications, such as heat exchange processes in power plants and cooling in electronic packaging industry, and has been a staple of computational methods in thermal science for many years. This work considers the Dirichlet boundary control of fluid-solid CHT problems. The CHT system falls into the category of multi-physics problems. Its domain typically consists of two parts, namely, a solid region subject to thermal heating or cooling and a conjugate fluid region responsible for thermal convection transport. These two different physical systems are strongly coupled through the thermal boundary condition at the fluid-solid interface. The objective in the CHT boundary control problem is to select optimally the fluid inflow profile that minimizes an objective function that involves the sum of the mismatch between the temperature distribution in the system and a prescribed temperature profile and the cost of the control. This objective is realized by minimizing a nonlinear objective function of the boundary control and the fluid temperature variables, subject to partial differential equations (PDE) constraints governed by the coupled heat diffusion equation in the solid region and mass, momentum and energy conservation equations in the fluid region. Although CHT has received extensive attention as a forward problem, the optimal Dirichlet velocity boundary control for the coupled CHT process to our knowledge is only very sparsely studied analytically or computationally in the literature [131]. Therefore, in Part I, we describe the formulation of the optimal control problem and introduce the building blocks for the finite element modeling of the CHT problem, namely, the diffusion equation for the solid temperature, the convection-diffusion equation for the fluid temperature, the incompressible viscous Navier-Stokes equations for the fluid velocity and pressure, and the model verification of CHT simulations. In Part II, we provide theoretical analysis to explain the nonsmoothness issue which has been observed in this study and in Dirichlet boundary control of Navier-Stokes flows by other scientists. Based on these findings, we use either explicit or implicit numerical smoothing to resolve the nonsmoothness issue. Moreover, we use the numerical continuation on regularization parameters to alleviate the difficulty of locating the global minimum in one shot for highly nonlinear optimization problems even when the initial guess is far from optimal. Two suites of numerical experiments have been provided to demonstrate the feasibility, effectiveness and robustness of the optimization scheme. In Part III, we demonstrate the strategy of achieving parallel scalable algorithms for CHT models in Simulations of Reactor Thermal Hydraulics. Our motivation originates from the observation that parallel processing is necessary for optimal control problems of very large scale, when the simulation of the underlying physics (or PDE constraints) involves millions or billions of degrees of freedom. To achieve the overall scalability of optimal control problems governed by PDE constraints, scalable components that resolve the PDE constraints and their adjoints are the key. In this Part, first we provide the strategy of designing parallel scalable solvers for each building blocks of the CHT modeling, namely, for the discrete diffusive operator, the discrete convection-diffusion operator, and the discrete Navier-Stokes operator. Second, we demonstrate a pair of effective, robust, parallel, and scalable solvers built with collaborators for simulations of reactor thermal hydraulics. Finally, in the the section of future work, we outline the roadmap of parallel and scalable solutions for Dirichlet boundary control of fluid-solid conjugate heat transfer processes

Nonlinear model reduction for uncertainty quantification in large-scale inverse problems : application to nonlinear convection-diffusion-reaction equation

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2008.Includes bibliographical references (p. 147-152).There are multiple instances in science and engineering where quantities of interest are evaluated by solving one or several nonlinear partial differential equations (PDEs) that are parametrized in terms of a set of inputs. Even though well-established numerical techniques exist for solving these problems, their computational cost often precludes their use in cases where the outputs of interest must be evaluated repeatedly for different values of the input parameters such as probabilistic analysis applications. In this thesis we present a model reduction methodology that combines efficient representation of the nonlinearities in the governing PDE with an efficient model-constrained, greedy algorithm for sampling the input parameter space. The nonlinearities in the PDE are represented using a coefficient-function approximation that enables the development of an efficient offline-online computational procedure where the online computational cost is independent of the size of the original high-fidelity model. The input space sampling algorithm used for generating the reduced space basis adaptively improves the quality of the reduced order approximation by solving a PDE-constrained continuous optimization problem that targets the output error between the reduced and full order models in order to determine the optimal sampling point at every greedy cycle. The resulting model reduction methodology is applied to a highly nonlinear combustion problem governed by a convection-diffusion-reaction PDE with up to 3 input parameters. The reduced basis approximation developed for this problem is up to 50, 000 times faster to solve than the original high-fidelity finite element model with an average relative error in prediction of outputs of interest of 2.5 - 10-6 over the input parameter space. The reduced order model developed in this thesis is used in a novel probabilistic methodology for solving inverse problems.(cont) The extreme computational cost of the Bayesian framework approach for inferring the values of the inputs that generated a given set of empirically measured outputs often precludes its use in practical applications. In this thesis we show that using a reduced order model for running the Markovby David Galbally.S.M