The inverses of block tridiagonal matrices

Ran, Ruisheng; Huang, Ting‐Zhu

doi:10.1016/j.amc.2005.11.098

Cited by 19 publications

(18 citation statements)

References 5 publications

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“…Computational complexity analysis [34] shows that inversion of a roughly sparse matrix with n p evaluation points and n e equations or dependent variables is On 2.5 p · n 3 e . For the ANN, with n q quadrature points, N basis functions per time step, the complexity is On p · n e · n q · N. This indicates that, for three-dimensional systems, as long as n q · N < n 1.5 p n 2 e per time step, the ANN will require roughly the same or less computational effort.…”

Section: E Time-integration Techniquesmentioning

confidence: 99%

Radial Basis Function Artificial Neural-Network-Inspired Numerical Solver

Wilkinson

Meade

2016

Journal of Aerospace Information Systems

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A framework for a mesh-free numerical solver of differential equations is presented in this paper. Development of the solver is derived from machine learning techniques using artificial neural networks with Gaussian radial basis functions for their neurons. The proposed method incrementally develops an approximation through the optimization of a scalar condensed form of the differential equations. Unlike traditional solvers that require grids, volumes, or meshes, along with corresponding connectivity data, the proposed framework requires only a list of independent variable values to approximate the solution. Because of this, there is no need for the derivation or inversion of system matrices. Results are presented demonstrating the stability and accuracy of the proposed method and it is demonstrated that the spatial error estimate can exceed that of traditional methods.H ISTORICALLY, the finite difference method (FDM), finite volume method (FVM), and finite element method (FEM) have all been used to numerically approximate the solution to differential equations that model problems of engineering interest, particularly in fluid dynamics [1]. Though conventional numerical methods have been used successfully on real world problems involving complex solution spaces (e.g., aircraft geometries, adaptive meshing, fluid-structure interaction), the coding, mesh creation, and adaptation requires increasing effort from the user. Because the cost of man hours now outweighs the cost of CPU hours, researchers have been investigating methods that sidestep the issues with meshing. Mesh-free methods, also known as meshless methods, are particularly attractive at solving these increasingly complex types of problems because they are relatively simple to program, accurate, and by definition require no explicit mesh connectivity.Liu [2] and Pepper [3] define a mesh-free method as one that establishes a system of algebraic equations for the entire problem domain without requiring mesh connectivity data, geometry Jacobians, or conformal maps to transform between the input and computational domains. Further, the element and volume edge crossing restrictions on the distribution of discrete evaluation points associated with meshed methods are eliminated. Specifically, a predefined mesh is not required for the dependent variable's approximation or interpolation. The problem domain and its boundary are represented by scattered points (nodes) and not used for discretization; these nodes carry the values of the dependent variable(s). There is no connectivity between these nodes and they can be arbitrarily distributed within the problem domain. This difference between the mesh and the nodes of the mesh-free approach is illustrated in Fig. 1. There currently exist several types and forms of mesh-free methods. One can separate them into two extremes based on formulation of the differential equations of interest: those that use the weak form and those that use the strong form.In the weak form, the governing differential equations and differential...

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Section: E Time-integration Techniquesmentioning

confidence: 99%

Radial Basis Function Artificial Neural-Network-Inspired Numerical Solver

Wilkinson

Meade

2016

Journal of Aerospace Information Systems

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“…(1), calculation of the GF is tantamount to inverting a block tridiagonal and nearly block Toeplitz matrix, and the primary computational benefits of the PL approximation lie in exploiting this structure. [18][19][20][21] Such a PL approach was recently combined with a recursive algorithm for inverting block tridiagonal matrices 19,22 to explore the decay of surface effects in carbon nanotubes 12 and graphene nanoribbons. 13 Analysis of this method reveals that it is only applicable to systems with a finite number of PLs and, more importantly, that it scales linearly with the number of PLs.…”

Section: Introductionmentioning

confidence: 99%

Probing the surface-to-bulk transition: A closed-form constant-scaling algorithm for computing subsurface Green functions

2011

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A closed-form algorithm for computing subsurface Green functions-the blocks of a material's Green function between the surface and the bulk-is presented, where we assume the system satisfies a common principal-layer approximation. By exploiting the block tridiagonal and nearly block Toeplitz structure of the Hamiltonian and overlap matrices, this method scales independently of the system size (constant scaling), allowing studies of large systems. As a proof-of-concept example, we investigate the decay of surface effects in an armchair graphene nanoribbon, demonstrating the persistence of surface effects hundreds of atomic layers (∼0.5 μm) away from a surface. We finally compare the surface-to-bulk transitions of finite and semi-infinite systems, finding that finite systems exhibit amplified surface effects.

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“…, tridiagonal/Toeplitz [5], block tridiagonal [4,[6][7][8][9][10][11][12], block Toeplitz [13] and block tridiagonal/block Toeplitz [14,15] matrices. Moreover, some applications, such as the electronic structure of materials [15][16][17][18], require only specific elements of the inverse matrix (as opposed to the entire matrix), resulting in additional efficiency [10,15,19].…”

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confidence: 99%

Crystal Growth and Characterization of (Bi_0.5Na_0.5)TiO₃–BaTiO₃ Single Crystals Obtained by a Top-Seeded Solution Growth Method under High-Pressure Oxygen Atmosphere

Onozuka

Kitanaka

Noguchi

et al. 2011

Jpn. J. Appl. Phys.

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We present an algorithm for computing any block of the inverse of a block tridiagonal, nearly block Toeplitz matrix (defined as a block tridiagonal matrix with a small number of deviations from the purely block Toeplitz structure). By exploiting both the block tridiagonal and the nearly block Toeplitz structures, this method scales independently of the total number of blocks in the matrix and linearly with the number of deviations. Numerical studies demonstrate this scaling and the advantages of our method over alternatives.

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The inverses of block tridiagonal matrices

Cited by 19 publications

References 5 publications

Radial Basis Function Artificial Neural-Network-Inspired Numerical Solver

Radial Basis Function Artificial Neural-Network-Inspired Numerical Solver

Probing the surface-to-bulk transition: A closed-form constant-scaling algorithm for computing subsurface Green functions

Crystal Growth and Characterization of (Bi_0.5Na_0.5)TiO₃–BaTiO₃ Single Crystals Obtained by a Top-Seeded Solution Growth Method under High-Pressure Oxygen Atmosphere

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The inverses of block tridiagonal matrices

Cited by 19 publications

References 5 publications

Radial Basis Function Artificial Neural-Network-Inspired Numerical Solver

Radial Basis Function Artificial Neural-Network-Inspired Numerical Solver

Probing the surface-to-bulk transition: A closed-form constant-scaling algorithm for computing subsurface Green functions

Crystal Growth and Characterization of (Bi0.5Na0.5)TiO3–BaTiO3 Single Crystals Obtained by a Top-Seeded Solution Growth Method under High-Pressure Oxygen Atmosphere

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Crystal Growth and Characterization of (Bi_0.5Na_0.5)TiO₃–BaTiO₃ Single Crystals Obtained by a Top-Seeded Solution Growth Method under High-Pressure Oxygen Atmosphere