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Chan Kai Yam. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 80-82). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Parameter Identification in Heat Conduction Systems --- p.1 / Chapter 1.2 --- Inverse Problems --- p.3 / Chapter 1.3 --- Challenges in Inverse Problems --- p.6 / Chapter 2 --- Tools in Parameter Identification --- p.9 / Chapter 2.1 --- Output Least Squares Method --- p.10 / Chapter 2.2 --- Tikhonov Regularization --- p.11 / Chapter 2.3 --- Our Approach --- p.14 / Chapter 3 --- Numerical Implementations --- p.20 / Chapter 3.1 --- Finite Element Discretization and Its Convergence --- p.20 / Chapter 3.2 --- Steepest Descent Method --- p.22 / Chapter 3.3 --- Multigrid Techniques --- p.26 / Chapter 4 --- Numerical Experiments --- p.29 / Chapter 4.1 --- One Dimensional Examples --- p.30 / Chapter 4.1.1 --- Selection of mk --- p.31 / Chapter 4.1.2 --- Selection of nk --- p.34 / Chapter 4.1.3 --- Selection of Number of Levels in the Coarse Grid Correction Step --- p.37 / Chapter 4.1.4 --- Convergence with Different Regularization Pa- rameters γ --- p.39 / Chapter 4.1.5 --- Convergence with Different Initial Guesses --- p.42 / Chapter 4.1.6 --- Comparisons between MG and SG Methods --- p.44 / Chapter 4.1.7 --- Comparisons between MG and RMG Methods --- p.46 / Chapter 4.1.8 --- More Examples --- p.49 / Chapter 4.1.9 --- Coarse Grid Correction in Another Approach --- p.60 / Chapter 4.2 --- Two Dimensional Examples --- p.71 / Chapter 4.3 --- Conclusions --- p.78 / Bibliography --- p.80

The effectiveness of the Additive Correction Multigrid (ACM) algorithm, a line-byline Tri-diagonal Matrix Algorithm (TDMA), and simple Gauss-Seidel (GS) iteration in numerical heat transfer analysis is investigated on a conventional single processor computer and on a distributed memory parallel computer. The performance of these methods is studied by solving a two-dimensional, steady heat conduction problem. The execution time of ACM on a single processor is proportional to the number of unknowns to the 1.5 power. This is in contrast to the execution time of the TDMA for which the execution time is proportional to the number of unknowns to the 2.0 power. The GS , TDMA and ACM algorithms are adapted to a model IPSC2 Intel hypercube which has a 32 processing nodes each with 8 MBytes oflocal memory. Because GS is a local method, it has almost perfect speed up, but it also converges more slowly than TDMA, The TDMA, on the other hand, is affected by domain decomposition to a greater extent than GS. As the number of processors used to solve the problem is increased, the execution times for GS and TDMA are essentially equal. Solving the model problem with 32 processors on a 192x192 grid resulted in parallel efficiencies of 95%, 80% and 78% for the GS, TDMA, and ACM algorithms, respectively. Though the parallel efficiency of ACM was the lowest of the three, the parallel ACM algorithm required an order of magnitude less time to solve the model than either parallel GS or parallel TDMA without multigrid.

The objective of this thesis is to develop a more efficient solver for a large system of linear equations arising from finite element discretization on unstructured tetrahedral meshes for a scalar elliptic partial differential equation of second order for pressure in a commercial computational fluid dynamics (CFD) simulation. Segregated solution methods (or pressure correction type methods) are a widely used approach to obtain solutions of Navier-Stokes equations during numerical simulation by many commercial CFD codes. At each time step, these simulations usually require the approximate solution of a series of scalar equations for velocity, pressure and temperature. Even if the simulation does not require high-accuracy approximations, the large systems of linear equations for pressure may not be efficiently solved. The matrices of these systems of linear equations of real-life industry problems often strongly violate weak diagonal dominance and the numerical simulation often requires solutions of very large systems with over a few hundred thousands degrees of freedom. These conditions produce very ill-conditioned systems of linear equations. Therefore, it is very difficult to solve such systems of linear equations efficiently using most currently available common iterative solvers. A survey of solvers for systems of linear equations was undertaken to determine the preferred solution methodology. An algebraic multigrid preconditioned conjugate gradient (AMGPCG) method solver was chosen for these problems. This solver uses the algebraic multigrid (AMG) cycle as a preconditioner for the conjugate gradient (CG) method. The disadvantages of the conventional AMG method are an expensive setup time and large memory requirements, particularly for three dimensional problems. The disadvantage of an expensive setup time needs to be overcome because the simulation usually requires only low-accuracy approximations for pressure. Also it is important to overcome the disadvantage of the large memory requirements for use in commercial software. In this work, an efficient AMGPCG solver is developed by overcoming the disadvantages of the conventional AMG method. The robustness of AMGPCG is shown theoretically so that the solver is always convergent. Optimum or close to optimum rates of convergence behavior for the solver are shown numerically so that the number of necessary iterations to obtain the estimated solution is approximately independent of mesh resolution. Furthermore, numerical experiments of solving pressure for some industry problems were carried out and compared with other efficient solvers including a fast commercial AMGPCG solver (SAMG, release 20b1). It was found that the developed AMGPCG solver was the fastest among these solvers for solving these problems and its algorithm has been numerically proven to be efficient. In addition, the memory requirement is at an acceptable level for commercial CFD codes.

Integral equations arise in many scienti c and engineering problems. A large class of initial and boundary value problems can be converted to Volterra or Fredholm integral equations. The potential theory contributed more than any eld to give rise to integral equations. Integral equations also has signi cant application in mathematical physics models, such as di rac- tion problems, scattering in quantum mechanics, conformal mapping and water waves. The Volterra's population growth model, biological species living together, propagation of stocked sh in a new lake, the heat transfer and the heat radiation are among many areas that are described by integral equations. For limited applicability of analytical techniques, the numer- ical solvers often are the only viable alternative. General computational techniques of solving integral equation involve discretization and generates equivalent system of linear equations. In most of the cases the discretization produces dense matrix. Multigrid methods are widely used to solve partial di erential equation. We discuss the multigrid algorithms to solve integral equations and propose usages of distributive relaxation and the Kaczmarz method. / Department of Mathematical Sciences

Thesis (Ph. D.)--Technische Universiteit Delft, 1989. / Summary also in Dutch. "Stellingen" (3 p.) inserted. Vita. Includes bibliographical references.

Thesis (M.S.)--Nuclear and Radiological Engineering, Georgia Institute of Technology, 2008. / Committee Chair: Stacey, Weston M.; Committee Co-Chair: de Oliveira, Cassiano R.E.; Committee Member: Hertel, Nolan; Committee Member: van Rooijen, Wilfred F.G.

Thesis (Ph.D.)--Memorial University of Newfoundland, 2004. / Bibliography: leaves 170-183.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

In this paper, we consider domain decomposition preconditioners for a system of linear algebraic equations arising from the <i>p</i>-version of the fem. We propose several multi-level preconditioners for the Dirichlet problems in the sub-domains in two and three dimensions. It is proved that the condition number of the preconditioned system is bounded by a constant independent of the polynomial degree. The proof uses interpretations of the <i>p</i>-version element stiffness matrix and mass matrix on [-1,1] as <i>h</i>-version stiffness matrix and weighted mass matrix. The analysis requires wavelet methods.