B-series, also known as Butcher series, are an algebraic tool for analysing solutions to ordinary differential equations, including approximate solutions. Through the formulation and manipulation of these series, properties of numerical methods can be assessed. Runge–Kutta methods, in particular, depend on B-series for a clean and elegant approach to the derivation of high order and efficient methods. However, the utility of B-series goes much further and opens a path to the design and construction of highly accurate and efficient multivalue methods.
This book offers a self-contained...
B-series, also known as Butcher series, are an algebraic tool for analysing solutions to ordinary differential equations, including approximate soluti...
Lattice rules are a powerful and popular form of quasi-Monte Carlo rules based on multidimensional integration lattices. This book provides a comprehensive treatment of the subject with detailed explanations of the basic concepts and the current methods used in research. This comprises, for example, error analysis in reproducing kernel Hilbert spaces, fast component-by-component constructions, the curse of dimensionality and tractability, weighted integration and approximation problems, and applications of lattice rules.
Lattice rules are a powerful and popular form of quasi-Monte Carlo rules based on multidimensional integration lattices. This book provides a comprehe...
Lattice rules are a powerful and popular form of quasi-Monte Carlo rules based on multidimensional integration lattices. This book provides a comprehensive treatment of the subject with detailed explanations of the basic concepts and the current methods used in research. This comprises, for example, error analysis in reproducing kernel Hilbert spaces, fast component-by-component constructions, the curse of dimensionality and tractability, weighted integration and approximation problems, and applications of lattice rules.
Lattice rules are a powerful and popular form of quasi-Monte Carlo rules based on multidimensional integration lattices. This book provides a comprehe...
This book provides a comprehensive analysis of time domain boundary integral equations and their discretisation by convolution quadrature and the boundary element method.
Properties of convolution quadrature, based on both linear multistep and Runge–Kutta methods, are explained in detail, always with wave propagation problems in mind. Main algorithms for implementing the discrete schemes are described and illustrated by short Matlab codes; translation to other languages can be found on the accompanying GitHub page. The codes are used to present numerous...
This book provides a comprehensive analysis of time domain boundary integral equations and their discretisation by convolution quadrature and the...
This book is about the theory of so-called Schwarz methods for solving variational problems in a Hilbert space V arising from linear equations and their associated quadratic minimization problems. Schwarz methods are based on the construction of a sequence of approximate solutions by solving auxiliary variational problems on a set of (smaller, finite-dimensional) Hilbert spaces $V_i$ in a certain order, combining them, and using the combined approximations in an iterative procedure. The spaces $V_i$ form a so-called space splitting for V, they need not necessarily be...
This book is about the theory of so-called Schwarz methods for solving variational problems in a Hilbert space V arising from linear equa...
This book offers the first comprehensive account of how the logarithmic norm is used for matrices, nonlinear maps and linear differential operators, with a focus on initial and boundary value problems.
Complementing the usual operator norm, the logarithmic norm is a versatile tool which provides unique additional information on the magnitude of an operator. It is instrumental in the stability theory of dynamical systems and in the theory of elliptic operator equations.
The text adopts a unified approach to address a wide range of themes in applied mathematics. It explores the...
This book offers the first comprehensive account of how the logarithmic norm is used for matrices, nonlinear maps and linear differential operators...
