There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

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Iterative Methods for Sparse Linear Systems

Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.

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Tensor Decompositions and Applications

We present a new integral equation formulation of the polarizable continuum model (PCM) which allows one to treat in a single approach dielectrics of different nature: standard isotropic liquids, intrinsically anisotropic medialike liquid crystals and solid matrices, or ionic solutions. The present work shows that integral equation methods may be used with success also for the latter cases, which are usually studied with three-dimensional methods, by far less competitive in terms of computational effort. We present the theoretical bases which underlie the method and some numerical tests which show both a complete equivalence with standard PCM versions for isotropic solvents, and a good efficiency for calculations with anisotropic dielectrics.

A new integral equation formalism for the polarizable continuum model: Theoretical background and applications to isotropic and anisotropic dielectrics

1. Preliminaries.- 2. Introductory Model Problem.- 3. General Two-Grid Method.- 4. General Multi-Grid Iteration.- 5. Nested Iteration Technique.- 6. Convergence of the Two-Grid Iteration.- 7. Convergence of the Multi-Grid Iteration.- 8. Fourier Analysis.- 9. Nonlinear Multi-Grid Methods.- 10. Singular Perturbation Problems.- 11. Elliptic Systems.- 12. Eigenvalue Problems and Singular Equations.- 13. Continuation Techniques.- 14. Extrapolation and Defect Correction Techniques.- 15. Local Techniques.- 16. The Multi-Grid Method of the Second Kind.

Multi-Grid Methods and Applications

A class of matrices (
 $\Cal H$
 -matrices) is introduced which have the following properties (i) They are sparse in the sense that only few data are needed for their representation (ii) The matrix-vector multiplication is of almost linear complexity (iii) In general, sums and products of these matrices are no longer in the same set, but their truncations to the $\Cal H$
 -matrix format are again of almost linear complexity (iv) The same statement holds for the inverse of an $\Cal H$
 -matrix This paper is the first of a series and is devoted to the first introduction of the $\Cal H$
 -matrix concept Two concret formats are described The first one is the simplest possible Nevertheless, it allows the exact inversion of tridiagonal matrices The second one is able to approximate discrete integral operators

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A sparse matrix arithmetic based on H -matrices. Part I: introduction to H -matrices

Let the function f be defined on I=[a,b] and, possibly, be singular at an interior point c∈(a,b). Recall that the improper integral was defined by 

$$\int\limits_{a}^{b} {f\left( x \right)} dx: = \mathop{{\lim }}\limits_{{\mathop{{{{\varepsilon }_{1}} \to 0}}\limits_{{{{\varepsilon }_{1}} > 0}} }} \int\limits_{a}^{{c - {{\varepsilon }_{1}}}} {f\left( x \right)} dx + \mathop{{\lim }}\limits_{{\mathop{{{{\varepsilon }_{2}} \to 0}}\limits_{{{{\varepsilon }_{2}} > 0}} }} \int\limits_{{c + {{\varepsilon }_{2}}}}^{b} {f\left( x \right)dx,} $$

if both limits exist (cf. §6.1.3). By Remark 6.1.2a, the improper integral exists for f (x): = |x-c|s with s>-1. For $ f(x): = \frac{1}{{x - c}}({\text{i}}{\text{.e}}{\text{.,s = - 1}}) $ (i.e, s=-1) one obtains 

$$\int\limits_{a}^{{c - {{\varepsilon }_{1}}}} {\frac{1}{{x - c}}dx} + \int\limits_{{c + {{\varepsilon }_{2}}}}^{b} {\frac{1}{{x - c}}dx = \log \frac{{b - c}}{{c - a}} + \log \frac{{{{\varepsilon }_{1}}}}{{{{\varepsilon }_{2}}}}.} $$

(7.1.1)

Singular Integral Equations

In the second edition of this classic monograph, complete with four new chapters and updated references, readers will now have access to content describing and analysing classical and modern methods with emphasis on the algebraic structure of linear iteration, which is usually ignored in other literature. The necessary amount of work increases dramatically with the size of systems, so one has to search for algorithms that most efficiently and accurately solve systems of, e.g., several million equations. The choice of algorithms depends on the special properties the matrices in practice have. An important class of large systems arises from the discretization of partial differential equations. In this case, the matrices are sparse (i.e., they contain mostly zeroes) and well-suited to iterative algorithms. The first edition of this book grew out of a series of lectures given by the author at the Christian-Albrecht University of Kiel to students of mathematics. The second edition includes quite novel approaches

Iterative Solution of Large Sparse Systems of Equations

Part I: Algebraic Tensors.- Introduction.- Matrix Tools.- Algebraic Foundations of Tensor Spaces.- Part II: Functional Analysis of Tensor Spaces.- Banach Tensor Spaces.- General Techniques.- Minimal Subspaces.-Part III: Numerical Treatment.- r-Term Representation.- Tensor Subspace Represenation.- r-Term Approximation.- Tensor Subspace Approximation.-Hierarchical Tensor Representation.- Matrix Product Systems.- Tensor Operations.- Tensorisation.- Generalised Cross Approximation.- Applications to Elliptic Partial Differential Equations.- Miscellaneous Topics.- References.- Index.

Wolfgang Hackbusch

Papers

Multi-Grid Methods and Applications

A sparse matrix arithmetic based on H -matrices. Part I: introduction to H -matrices

Singular Integral Equations

Iterative Solution of Large Sparse Systems of Equations

Tensor Spaces and Numerical Tensor Calculus