Spectral features of matrix-sequences, GLT, symbol, and application in preconditioning Krylov methods, image deblurring, and multigrid algorithms.

The final purpose of any scientific discipline can be regarded as the solution of real-world problems. With this aim, a mathematical modeling of the considered phenomenon is often compulsory. Closed-form solutions of the arising functional equations are usually not available and numerical discretization techniques are required. In this setting, the discretization of an infinite-dimensional linear equation via some linear approximation method, leads to a sequence of linear systems of increasing dimension whose coefficient matrices could inherit a structure from the continuous problem. For instance, the numerical approximation by local methods of constant or nonconstant coefficients systems of Partial Differential Equations (PDEs) over multidimensional domains, gives rise to multilevel block Toeplitz or to Generalized Locally Toeplitz (GLT) sequences, respectively. In the context of structured matrices, the convergence properties of iterative methods, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol, a function whose role relies in describing the asymptotical distribution of the spectrum. This thesis can be seen as a byproduct of the combined use of powerful tools like symbol, spectral distribution, and GLT, when dealing with the numerical solution of structured linear systems. We approach such an issue both from a theoretical and practical viewpoint. On the one hand, we enlarge some known spectral distribution tools by proving the eigenvalue distribution of matrix-sequences obtained as combination of some algebraic operations on multilevel block Toeplitz matrices. On the other hand, we take advantage of the obtained results for designing efficient preconditioning techniques. Moreover, we focus on the numerical solution of structured linear systems coming from the following applications: image deblurring, fractional diffusion equations, and coupled PDEs. A spectral analysis of the arising structured sequences allows us either to study the convergence and predict the behavior of preconditioned Krylov and multigrid methods applied to the coefficient matrices, or to design effective preconditioners and multigrid solvers for the associated linear systems

A Cartesian Cut‐Stencil Method for the Finite Difference Solution of PDEs in Complex Domains

A new finite difference formulation, referred to as the Cartesian cut-stencil finite difference method (FDM), for discretization of partial differential equations (PDEs) in any complex physical domain is proposed in this dissertation. The method employs unique localized 1-D quadratic transformation functions to map non-uniform (uncut or cut) physical stencils to a uniform computational stencil. The transformation functions are uniquely determined by the coordinates of the points on the physical stencil. In its basic formulation, 2nd-order central differencing is used to approximate derivatives in the transformed PDEs. The resulting finite difference equations can be solved by classical iterative methods. In the case of a boundary node with a Dirichlet boundary condition, the prescribed value can be used directly in the calculations on the corresponding stencil adjacent to the boundary. However, for Neumann boundary nodes, discretization of the normal derivative in the Neumann condition is accomplished using one-sided approximations, producing an approximate value for the solution variable at the boundary. Then, the cut-stencil method allows stencils adjacent to boundaries to be treated in the same way as interior stencils, thus enabling finite difference calculations on arbitrarily complex domains. This new formulation can be combined with the higher-order compact Padé-Hermitian method to produce higher-order cut-stencil schemes. Three different Cartesian cut-stencil formulations based on local 4th-order approximations are proposed and analyzed. It has been shown that global 4th-order accuracy can be achieved when the same order of accuracy is implemented at Neumann boundaries. Comparison of numerical results for some manufactured problems with the exact solution verifies the capability of the cut-stencil method to deal with PDEs in regular and irregular shaped domains. Cartesian cut-stencil FDM solutions are also obtained for some classical engineering benchmark problems, including Prandtl’s stress function, steady or unsteady heat conduction and flow in a lid-driven cavity. This dissertation demonstrates that the Cartesian cut-stencil finite difference method has many desirable features of a high-end numerical simulation code including simplicity in formulation, meshing and coding, higher-order accuracy, high-fidelity solutions, reliable error estimator, applicable in different science and engineering fields, and can solve complicated nonlinear PDEs in complex geometries

Spectral features of matrix-sequences, GLT, symbol, and application in preconditioning Krylov methods, image deblurring, and multigrid algorithms.

The final purpose of any scientific discipline can be regarded as the solution of real-world problems. With this aim, a mathematical modeling of the considered phenomenon is often compulsory. Closed-form solutions of the arising functional equations are usually not available and numerical discretization techniques are required. In this setting, the discretization of an infinite-dimensional linear equation via some linear approximation method, leads to a sequence of linear systems of increasing dimension whose coefficient matrices could inherit a structure from the continuous problem. For instance, the numerical approximation by local methods of constant or nonconstant coefficients systems of Partial Differential Equations (PDEs) over multidimensional domains, gives rise to multilevel block Toeplitz or to Generalized Locally Toeplitz (GLT) sequences, respectively. In the context of structured matrices, the convergence properties of iterative methods, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol, a function whose role relies in describing the asymptotical distribution of the spectrum. This thesis can be seen as a byproduct of the combined use of powerful tools like symbol, spectral distribution, and GLT, when dealing with the numerical solution of structured linear systems. We approach such an issue both from a theoretical and practical viewpoint. On the one hand, we enlarge some known spectral distribution tools by proving the eigenvalue distribution of matrix-sequences obtained as combination of some algebraic operations on multilevel block Toeplitz matrices. On the other hand, we take advantage of the obtained results for designing efficient preconditioning techniques. Moreover, we focus on the numerical solution of structured linear systems coming from the following applications: image deblurring, fractional diffusion equations, and coupled PDEs. A spectral analysis of the arising structured sequences allows us either to study the convergence and predict the behavior of preconditioned Krylov and multigrid methods applied to the coefficient matrices, or to design effective preconditioners and multigrid solvers for the associated linear systems

Error Reduction Program

The details of a study to select, incorporate and evaluate the best available finite difference scheme to reduce numerical error in combustor performance evaluation codes are described. The combustor performance computer programs chosen were the two dimensional and three dimensional versions of Pratt & Whitney's TEACH code. The criteria used to select schemes required that the difference equations mirror the properties of the governing differential equation, be more accurate than the current hybrid difference scheme, be stable and economical, be compatible with TEACH codes, use only modest amounts of additional storage, and be relatively simple. The methods of assessment used in the selection process consisted of examination of the difference equation, evaluation of the properties of the coefficient matrix, Taylor series analysis, and performance on model problems. Five schemes from the literature and three schemes developed during the course of the study were evaluated. This effort resulted in the incorporation of a scheme in 3D-TEACH which is usuallly more accurate than the hybrid differencing method and never less accurate

A 3-D RBF-FD solver for modeling the atmospheric global electric circuit with topography (GEC-RBFFD v1.0)

A numerical model based on radial basis functiongenerated finite differences (RBF-FD) is developed for simulating the global electric circuit (GEC) within the Earth's atmosphere, represented by a 3-D variable coefficient linearelliptic partial differential equation (PDE) in a sphericallyshaped volume with the lower boundary being the Earth's topography and the upper boundary a sphere at 60 km. To ourknowledge, this is (1) the first numerical model of the GECto combine the Earth's topography with directly approximating the differential operators in 3-D space and, related to this,(2) the first RBF-FD method to use irregular 3-D stencils fordiscretization to handle the topography. It benefits from themesh-free nature of RBF-FD, which is especially suitable formodeling high-dimensional problems with irregular boundaries. The RBF-FD elliptic solver proposed here makes nolimiting assumptions on the spatial variability of the coefficients in the PDE (i.e., the conductivity profile), the righthand side forcing term of the PDE (i.e., distribution of current sources) or the geometry of the lower boundary.This work was supported by NSF awards AGS-1135446 and DMS-094581. The National Center for Atmospheric Research is sponsored by the NSF.Publicad

Explicit alternating direction methods for problems in fluid dynamics

Recently an iterative method was formulated employing a new splitting strategy for the solution of tridiagonal systems of difference equations. The method was successful in solving the systems of equations arising from one dimensional initial boundary value problems, and a theoretical analysis for proving the convergence of the method for systems whose constituent matrices are positive definite was presented by Evans and Sahimi [22]. The method was known as the Alternating Group Explicit (AGE) method and is referred to as AGE-1D. The explicit nature of the method meant that its implementation on parallel machines can be very promising. The method was also extended to solve systems arising from two and three dimensional initial-boundary value problems, but the AGE-2D and AGE-3D algorithms proved to be too demanding in computational cost which largely reduces the advantages of its parallel nature. In this thesis, further theoretical analyses and experimental studies are pursued to establish the convergence and suitability of the AGE-1D method to a wider class of systems arising from univariate and multivariate differential equations with symmetric and non symmetric difference operators. Also the possibility of a Chebyshev acceleration of the AGE-1D algorithm is considered. For two and three dimensional problems it is proposed to couple the use of the AGE-1D algorithm with an ADI scheme or an ADI iterative method in what is called the Explicit Alternating Direction (EAD) method. It is then shown through experimental results that the EAD method retains the parallel features of the AGE method and moreover leads to savings of up to 83 % in the computational cost for solving some of the model problems. The thesis also includes applications of the AGE-1D algorithm and the EAD method to solve some problems of fluid dynamics such as the linearized Shallow Water equations, and the Navier Stokes' equations for the flow in an idealized one dimensional Planetary Boundary Layer. The thesis terminates with conclusions and suggestions for further work together with a comprehensive bibliography and an appendix containing some selected programs