Published by De Gruyter

Radon Series on Computational and Applied Mathematics

Edited by: Otmar Scherzer, Hansjörg Albrecher, Ronald H. W. Hoppe, Karl Kunisch, Harald Niederreiter and Christian Schmeiser

ISSN: 1865-3707

Overview
Volumes

The series is devoted to the publication of high-level monographs, surveys and proceedings which cover the whole spectrum of computational and applied mathematics.

The books of this series are addressed to both specialists and advanced students.

Interested authors may submit book proposals to the Managing Editor or to any member of the Editorial Board.

Managing Editor
Otmar Scherzer, RICAM, Linz, Austria; University of Vienna, Austria

Editorial Board
Hansjörg Albrecher, University of Lausanne, Switzerland
Ronald H. W. Hoppe, University of Houston, USA
Karl Kunisch, RICAM, Linz, Austria; University of Graz, Austria
Harald Niederreiter, RICAM, Linz, Austria
Christian Schmeiser, University of Vienna, Austria

Book 2022

Optimization and Control for Partial Differential Equations

Uncertainty quantification, open and closed-loop control, and shape optimization

Roland Herzog, Matthias Heinkenschloss, Dante Kalise, Georg Stadler, Emmanuel Trélat

Volume 29 in this series

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This book highlights new developments in the wide and growing field of partial differential equations (PDE)-constrained optimization. Optimization problems where the dynamics evolve according to a system of PDEs arise in science, engineering, and economic applications and they can take the form of inverse problems, optimal control problems or optimal design problems. This book covers new theoretical, computational as well as implementation aspects for PDE-constrained optimization problems under uncertainty, in shape optimization, and in feedback control, and it illustrates the new developments on representative problems from a variety of applications.

DOI:: https://doi.org/10.1515/9783110695984
ISBN:: 9783110695984
ISBN:: 9783110695960
ISBN:: 9783110696004
Subject:: Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2020

Multiphysics Phase-Field Fracture

Modeling, Adaptive Discretizations, and Solvers

Thomas Wick

Volume 28 in this series

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This monograph is centered on mathematical modeling, innovative numerical algorithms and adaptive concepts to deal with fracture phenomena in multiphysics. State-of-the-art phase-field fracture models are complemented with prototype explanations and rigorous numerical analysis. These developments are embedded into a carefully designed balance between scientific computing aspects and numerical modeling of nonstationary coupled variational inequality systems. Therein, a focus is on nonlinear solvers, goal-oriented error estimation, predictor-corrector adaptivity, and interface conditions. Engineering applications show the potential for tackling practical problems within the fields of solid mechanics, porous media, and fluidstructure interaction.

DOI:: https://doi.org/10.1515/9783110497397
ISBN:: 9783110497397
ISBN:: 9783110496567
ISBN:: 9783110494198
Subject:: Engineering
Subject:: Mechanical Engineering
Subject:: Mechanics
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Subject:: Physics
Subject:: Mechanics and Fluid Dynamics
Publisher:: De Gruyter

Book 2020

Multivariate Algorithms and Information-Based Complexity

Fred J. Hickernell, Peter Kritzer

Volume 27 in this series

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The contributions by leading experts in this book focus on a variety of topics of current interest related to information-based complexity, ranging from function approximation, numerical integration, numerical methods for the sphere, and algorithms with random information, to Bayesian probabilistic numerical methods and numerical methods for stochastic differential equations.

DOI:: https://doi.org/10.1515/9783110635461
ISBN:: 9783110635461
ISBN:: 9783110633115
ISBN:: 9783110633153
Subject:: Computer Sciences
Subject:: Algorithms
Subject:: Mathematics
Subject:: Probability and Statistics
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2020

Discrepancy Theory

Dmitriy Bilyk, Josef Dick, Friedrich Pillichshammer

Volume 26 in this series

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The contributions in this book focus on a variety of topics related to discrepancy theory, comprising Fourier techniques to analyze discrepancy, low discrepancy point sets for quasi-Monte Carlo integration, probabilistic discrepancy bounds, dispersion of point sets, pair correlation of sequences, integer points in convex bodies, discrepancy with respect to geometric shapes other than rectangular boxes, and also open problems in discrepany theory.

DOI:: https://doi.org/10.1515/9783110652581
ISBN:: 9783110652581
ISBN:: 9783110651157
ISBN:: 9783110651201
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Discrete Mathematics
Subject:: Probability and Statistics
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

Book 2019

Space-Time Methods

Applications to Partial Differential Equations

Ulrich Langer, Olaf Steinbach

Volume 25 in this series

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This volume provides an introduction to modern space-time discretization methods such as finite and boundary elements and isogeometric analysis for time-dependent initial-boundary value problems of parabolic and hyperbolic type. Particular focus is given on stable formulations, error estimates, adaptivity in space and time, efficient solution algorithms, parallelization of the solution pipeline, and applications in science and engineering.

DOI:: https://doi.org/10.1515/9783110548488
ISBN:: 9783110548488
ISBN:: 9783110547993
ISBN:: 9783110547870
Subject:: Mathematics
Subject:: Differential Equations and Dynamical Systems
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2019

Maxwell’s Equations

Analysis and Numerics

Ulrich Langer, Dirk Pauly, Sergey Repin

Volume 24 in this series

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This volume collects longer articles on the analysis and numerics of Maxwell’s equations. The topics include functional analytic and Hilbert space methods, compact embeddings, solution theories and asymptotics, electromagnetostatics, time-harmonic Maxwell’s equations, time-dependent Maxwell’s equations, eddy current approximations, scattering and radiation problems, inverse problems, finite element methods, boundary element methods, and isogeometric analysis.

DOI:: https://doi.org/10.1515/9783110543612
ISBN:: 9783110543612
ISBN:: 9783110542691
ISBN:: 9783110542646
Subject:: Mathematics
Subject:: Differential Equations and Dynamical Systems
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2019

Combinatorics and Finite Fields

Difference Sets, Polynomials, Pseudorandomness and Applications

Kai-Uwe Schmidt, Arne Winterhof

Volume 23 in this series

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Combinatorics and finite fields are of great importance in modern applications such as in the analysis of algorithms, in information and communication theory, and in signal processing and coding theory. This book contains survey articles on topics such as difference sets, polynomials, and pseudorandomness.

DOI:: https://doi.org/10.1515/9783110642094
ISBN:: 9783110642094
ISBN:: 9783110641790
ISBN:: 9783110641967
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Combinatorics and Graph Theory
Subject:: Discrete Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2019

The Radon Transform

The First 100 Years and Beyond

Ronny Ramlau, Otmar Scherzer

Volume 22 in this series

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In 1917, Johann Radon published his fundamental work, where he introduced what is now called the Radon transform. Including important contributions by several experts, this book reports on ground-breaking developments related to the Radon transform throughout these years, and also discusses novel mathematical research topics and applications for the next century.

DOI:: https://doi.org/10.1515/9783110560855
ISBN:: 9783110560855
ISBN:: 9783110559514
ISBN:: 9783110559415
Subject:: Mathematics
Subject:: Analysis
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2018

Hamilton-Jacobi-Bellman Equations

Numerical Methods and Applications in Optimal Control

Dante Kalise, Karl Kunisch, Zhiping Rao, Marianne Akian, Jan Blechschmidt, Nikolai D. Botkin, Max Jensen, Axel Kröner, Athena Picarelli, Iain Smears, Karsten Urban, Mickaël D. Chekroun, Roland Herzog, Ilja Kalmykov, Johannes Diepolder, Eric Fodjo, Honghu Liu, Christoph Reisinger, Julen Rotaetxe Arto, Sebastian Steck, Varvara L. Turova

Volume 21 in this series

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Optimal feedback control arises in different areas such as aerospace engineering, chemical processing, resource economics, etc. In this context, the application of dynamic programming techniques leads to the solution of fully nonlinear Hamilton-Jacobi-Bellman equations. This book presents the state of the art in the numerical approximation of Hamilton-Jacobi-Bellman equations, including post-processing of Galerkin methods, high-order methods, boundary treatment in semi-Lagrangian schemes, reduced basis methods, comparison principles for viscosity solutions, max-plus methods, and the numerical approximation of Monge-Ampère equations. This book also features applications in the simulation of adaptive controllers and the control of nonlinear delay differential equations.

Contents
From a monotone probabilistic scheme to a probabilistic max-plus algorithm for solving Hamilton–Jacobi–Bellman equations
Improving policies for Hamilton–Jacobi–Bellman equations by postprocessing
Viability approach to simulation of an adaptive controller
Galerkin approximations for the optimal control of nonlinear delay differential equations
Efficient higher order time discretization schemes for Hamilton–Jacobi–Bellman equations based on diagonally implicit symplectic Runge–Kutta methods
Numerical solution of the simple Monge–Ampere equation with nonconvex Dirichlet data on nonconvex domains
On the notion of boundary conditions in comparison principles for viscosity solutions
Boundary mesh refinement for semi-Lagrangian schemes
A reduced basis method for the Hamilton–Jacobi–Bellman equation within the European Union Emission Trading Scheme

DOI:: https://doi.org/10.1515/9783110543599
ISBN:: 9783110543599
ISBN:: 9783110542714
ISBN:: 9783110542639
Subject:: Mathematics
Subject:: Differential Equations and Dynamical Systems
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2017

Fluid-Structure Interaction

Modeling, Adaptive Discretisations and Solvers

Stefan Frei, Bärbel Holm, Thomas Richter, Thomas Wick, Huidong Yang, Steffen Basting, Philipp Birken, Sunica Canic, Claudia Colciago, Simone Deparis, Davide Forti, Roland Glowinski, Johan Hoffman, M.Sc. Daniel Jodlbauer, Ulrich Langer, Azahar Monge, Annalisa Quaini, M.Sc. Mahnaz Shokrpour Roudbari, M.Sc. Görkem Simsek, E. Harald van Brummelen, Kristoffer G. van der Zee

Volume 20 in this series

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This monograph discusses modeling, adaptive discretisation techniques and the numerical solution of fluid structure interaction. An emphasis in part I lies on innovative discretisation and advanced interface resolution techniques. The second part covers the efficient and robust numerical solution of fluid-structure interaction.

In part III, recent advances in the application fields vascular flows, binary-fluid-solid interaction, and coupling to fractures in the solid part are presented. Moreover each chapter provides a comprehensive overview in the respective topics including many references to concurring state-of-the art work.

Contents
Part I: Modeling and discretization
On the implementation and benchmarking of an extended ALE method for FSI problems
The locally adapted parametric finite element method for interface problems on triangular meshes
An accurate Eulerian approach for fluid-structure interactions

Part II: Solvers
Numerical methods for unsteady thermal fluid structure interaction
Recent development of robust monolithic fluid-structure interaction solvers
A monolithic FSI solver applied to the FSI 1,2,3 benchmarks

Part III: Applications
Fluid-structure interaction for vascular flows: From supercomputers to laptops
Binary-fluid–solid interaction based on the Navier–Stokes–Cahn–Hilliard Equations
Coupling fluid-structure interaction with phase-field fracture: Algorithmic details

DOI:: https://doi.org/10.1515/9783110494259
ISBN:: 9783110494259
ISBN:: 9783110495270
ISBN:: 9783110492583
Subject:: Mathematics
Subject:: Differential Equations and Dynamical Systems
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Subject:: Physics
Subject:: Mechanics and Fluid Dynamics
Publisher:: De Gruyter

Book 2018

Tensor Numerical Methods in Scientific Computing

Boris N. Khoromskij

Volume 19 in this series

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The most difficult computational problems nowadays are those of higher dimensions. This research monograph offers an introduction to tensor numerical methods designed for the solution of the multidimensional problems in scientific computing. These methods are based on the rank-structured approximation of multivariate functions and operators by using the appropriate tensor formats. The old and new rank-structured tensor formats are investigated. We discuss in detail the novel quantized tensor approximation method (QTT) which provides function-operator calculus in higher dimensions in logarithmic complexity rendering super-fast convolution, FFT and wavelet transforms.

This book suggests the constructive recipes and computational schemes for a number of real life problems described by the multidimensional partial differential equations. We present the theory and algorithms for the sinc-based separable approximation of the analytic radial basis functions including Green’s and Helmholtz kernels. The efficient tensor-based techniques for computational problems in electronic structure calculations and for the grid-based evaluation of long-range interaction potentials in multi-particle systems are considered. We also discuss the QTT numerical approach in many-particle dynamics, tensor techniques for stochastic/parametric PDEs as well as for the solution and homogenization of the elliptic equations with highly-oscillating coefficients.

Contents
Theory on separable approximation of multivariate functions
Multilinear algebra and nonlinear tensor approximation
Superfast computations via quantized tensor approximation
Tensor approach to multidimensional integrodifferential equations

DOI:: https://doi.org/10.1515/9783110365917
ISBN:: 9783110365917
ISBN:: 9783110391398
ISBN:: 9783110370133
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

Book 2017

Variational Methods

In Imaging and Geometric Control

Maïtine Bergounioux, Gabriel Peyré, Christoph Schnörr, Jean-Baptiste Caillau, Thomas Haberkorn

Volume 18 in this series

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With a focus on the interplay between mathematics and applications of imaging, the first part covers topics from optimization, inverse problems and shape spaces to computer vision and computational anatomy. The second part is geared towards geometric control and related topics, including Riemannian geometry, celestial mechanics and quantum control.

Contents:
Part I
Second-order decomposition model for image processing: numerical experimentation
Optimizing spatial and tonal data for PDE-based inpainting
Image registration using phase・amplitude separation
Rotation invariance in exemplar-based image inpainting
Convective regularization for optical flow
A variational method for quantitative photoacoustic tomography with piecewise constant coefficients
On optical flow models for variational motion estimation
Bilevel approaches for learning of variational imaging models
Part II
Non-degenerate forms of the generalized Euler・Lagrange condition for state-constrained optimal control problems
The Purcell three-link swimmer: some geometric and numerical aspects related to periodic optimal controls
Controllability of Keplerian motion with low-thrust control systems
Higher variational equation techniques for the integrability of homogeneous potentials
Introduction to KAM theory with a view to celestial mechanics
Invariants of contact sub-pseudo-Riemannian structures and Einstein・Weyl geometry
Time-optimal control for a perturbed Brockett integrator
Twist maps and Arnold diffusion for diffeomorphisms
A Hamiltonian approach to sufficiency in optimal control with minimal regularity conditions: Part I
Index

DOI:: https://doi.org/10.1515/9783110430394
ISBN:: 9783110430394
ISBN:: 9783110439236
ISBN:: 9783110430493
Subject:: Computer Sciences
Subject:: Algorithms
Subject:: Engineering
Subject:: Electrical Engineering
Subject:: Control Engineering, Metrology and Testing
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2017

Topological Optimization and Optimal Transport

In the Applied Sciences

Maïtine Bergounioux, Édouard Oudet, Martin Rumpf, Guillaume Carlier, Thierry Champion, Filippo Santambrogio

Volume 17 in this series

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By discussing topics such as shape representations, relaxation theory and optimal transport, trends and synergies of mathematical tools required for optimization of geometry and topology of shapes are explored. Furthermore, applications in science and engineering, including economics, social sciences, biology, physics and image processing are covered.

Contents
Part I

Geometric issues in PDE problems related to the infinity Laplace operator
Solution of free boundary problems in the presence of geometric uncertainties
Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies
High-order topological expansions for Helmholtz problems in 2D
On a new phase field model for the approximation of interfacial energies of multiphase systems
Optimization of eigenvalues and eigenmodes by using the adjoint method
Discrete varifolds and surface approximation

Part II

Weak Monge–Ampere solutions of the semi-discrete optimal transportation problem
Optimal transportation theory with repulsive costs
Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations
On the Lagrangian branched transport model and the equivalence with its Eulerian formulation
On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows
Pressureless Euler equations with maximal density constraint: a time-splitting scheme
Convergence of a fully discrete variational scheme for a thin-film equatio
Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance

DOI:: https://doi.org/10.1515/9783110430417
ISBN:: 9783110430417
ISBN:: 9783110439267
ISBN:: 9783110430509
Subject:: Computer Sciences
Subject:: Algorithms
Subject:: Engineering
Subject:: Electrical Engineering
Subject:: Control Engineering, Metrology and Testing
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2014

Algebraic Curves and Finite Fields

Cryptography and Other Applications

Harald Niederreiter, Alina Ostafe, Daniel Panario, Arne Winterhof, Jeffrey D. Achter, Alp Bassa, Peter Beelen, Jung Hee Cheon, Carlet Claude, Massimo Giulietti, Sylvain Guilley, Tor Helleseth, Taechan Kim, Gabor Korchmaros, Nhut Nguyen, Florian Pausinger, Alexander Pott, Rachel Pries, Kai-Uwe Schmidt, Yongsoo Song, Ron Steinfeld, Alev Topuzoglu, Gabriel D. Villa-Salvador, Yue Zhou

Volume 16 in this series

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Algebra and number theory have always been counted among the most beautiful and fundamental mathematical areas with deep proofs and elegant results. However, for a long time they were not considered of any substantial importance for real-life applications. This has dramatically changed with the appearance of new topics such as modern cryptography, coding theory, and wireless communication. Nowadays we find applications of algebra and number theory frequently in our daily life. We mention security and error detection for internet banking, check digit systems and the bar code, GPS and radar systems, pricing options at a stock market, and noise suppression on mobile phones as most common examples.

This book collects the results of the workshops "Applications of algebraic curves" and "Applications of finite fields" of the RICAM Special Semester 2013. These workshops brought together the most prominent researchers in the area of finite fields and their applications around the world. They address old and new problems on curves and other aspects of finite fields, with emphasis on their diverse applications to many areas of pure and applied mathematics.

DOI:: https://doi.org/10.1515/9783110317916
ISBN:: 978-3-11-031788-6
ISBN:: 978-3-11-031791-6
ISBN:: 9783110379556
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2014

Uniform Distribution and Quasi-Monte Carlo Methods

Discrepancy, Integration and Applications

Peter Kritzer, Harald Niederreiter, Friedrich Pillichshammer, Arne Winterhof, Christoph Aistleitner, Jozsef Beck, Dmitriy Bilyk, Josef Dick, Michael Drmota, Henri Faure, Peter Hellekalek, Gerhard Larcher, Gunther Leobacher, Dirk Nuyens, Henryk Wozniakowski

Volume 15 in this series

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This book is summarizing the results of the workshop "Uniform Distribution and Quasi-Monte Carlo Methods" of the RICAM Special Semester on "Applications of Algebra and Number Theory" in October 2013.

The survey articles in this book focus on number theoretic point constructions, uniform distribution theory, and quasi-Monte Carlo methods. As deterministic versions of the Monte Carlo method, quasi-Monte Carlo rules enjoy increasing popularity, with many fruitful applications in mathematical practice, as for example in finance, computer graphics, and biology.

The goal of this book is to give an overview of recent developments in uniform distribution theory, quasi-Monte Carlo methods, and their applications, presented by leading experts in these vivid fields of research.

DOI:: https://doi.org/10.1515/9783110317930
ISBN:: 978-3-11-031789-3
ISBN:: 978-3-11-031793-0
ISBN:: 9783110375039
Subject:: Mathematics
Subject:: Probability and Statistics
Publisher:: De Gruyter

Book 2013

Direct and Inverse Problems in Wave Propagation and Applications

Ivan G. Graham, Ulrich Langer, Jens M. Melenk, Mourad Sini

Volume 14 in this series

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This book is the third volume of three volume series recording the "Radon Special Semester 2011 on Multiscale Simulation & Analysis in Energy and the Environment" taking place in Linz, Austria, October 3-7, 2011. This book surveys recent developments in the analysis of wave propagation problems. The topics covered include aspects of the forward problem and problems in inverse problems, as well as applications in the earth sciences.

Wave propagation problems are ubiquitous in environmental applications such as seismic analysis, acoustic and electromagnetic scattering. The design of efficient numerical methods for the forward problem, in which the scattered field is computed from known geometric configurations is very challenging due to the multiscale nature of the problems. Even more challenging are inverse problems where material parameters and configurations have to be determined from measurements in conjunction with the forward problem. This book contains review articles covering several state-of-the-art numerical methods for both forward and inverse problems.

This collection of survey articles focusses on the efficient computation of wave propagation and scattering is a core problem in numerical mathematics, which is currently of great research interest and is central to many applications in energy and the environment. Two generic applications which resonate strongly with the central aims of the Radon Special Semester 2011 are forward wave propagation in heterogeneous media and seismic inversion for subsurface imaging. As an example of the first application, modelling of absorption and scattering of radiation by clouds, aerosol and precipitation is used as a tool for interpretation of (e.g.) solar, infrared and radar measurements, and as a component in larger weather/climate prediction models in numerical weather forecasting. As an example of the second application, inverse problems in wave propagation in heterogeneous media arise in the problem of imaging the subsurface below land or marine deposits.

The book records the achievements of Workshop 3 "Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment". It brings together key numerical mathematicians whose interest is in the analysis and computation of wave propagation and scattering problems, and in inverse problems, together with practitioners from engineering and industry whose interest is in the applications of these core problems.

DOI:: https://doi.org/10.1515/9783110282283
ISBN:: 978-3-11-028223-8
ISBN:: 978-3-11-028228-3
Subject:: Geosciences
Subject:: Geography
Subject:: Atmospheric Science and Climatology
Subject:: Geology and Mineralogy
Subject:: Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book Open Access 2013

Large Scale Inverse Problems

Computational Methods and Applications in the Earth Sciences

Mike Cullen, Melina A Freitag, Stefan Kindermann, Robert Scheichl

Volume 13 in this series

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This book is the second volume of a three volume series recording the "Radon Special Semester 2011 on Multiscale Simulation & Analysis in Energy and the Environment" that took placein Linz, Austria, October 3-7, 2011. This volume addresses the common ground in the mathematical and computational procedures required for large-scale inverse problems and data assimilation in forefront applications.

The solution of inverse problems is fundamental to a wide variety of applications such as weather forecasting, medical tomography, and oil exploration. Regularisation techniques are needed to ensure solutions of sufficient quality to be useful, and soundly theoretically based. This book addresses the common techniques required for all the applications, and is thus truly interdisciplinary.

This collection of survey articles focusses on the large inverse problems commonly arising in simulation and forecasting in the earth sciences. For example, operational weather forecasting models have between 10⁷ and 10⁸ degrees of freedom. Even so, these degrees of freedom represent grossly space-time averaged properties of the atmosphere. Accurate forecasts require accurate initial conditions. With recent developments in satellite data, there are between 10⁶ and 10⁷ observations each day. However, while these also represent space-time averaged properties, the averaging implicit in the measurements is quite different from that used in the models. In atmosphere and ocean applications, there is a physically-based model available which can be used to regularise the problem. We assume that there is a set of observations with known error characteristics available over a period of time. The basic deterministic technique is to fit a model trajectory to the observations over a period of time to within the observation error. Since the model is not perfect the model trajectory has to be corrected, which defines the data assimilation problem. The stochastic view can be expressed by using an ensemble of model trajectories, and calculating corrections to both the mean value and the spread which allow the observations to be fitted by each ensemble member. In other areas of earth science, only the structure of the model formulation itself is known and the aim is to use the past observation history to determine the unknown model parameters.

The book records the achievements of Workshop 2 "Large-Scale Inverse Problems and Applications in the Earth Sciences". It involves experts in the theory of inverse problems together with experts working on both theoretical and practical aspects of the techniques by which large inverse problems arise in the earth sciences.

DOI:: https://doi.org/10.1515/9783110282269
ISBN:: 978-3-11-028222-1
ISBN:: 978-3-11-028226-9
Subject:: Geosciences
Subject:: Atmospheric Science and Climatology
Subject:: Geophysics
Subject:: Geology and Mineralogy
Subject:: Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2013

Simulation of Flow in Porous Media

Applications in Energy and Environment

Peter Bastian, Johannes Kraus, Robert Scheichl, Mary Wheeler

Volume 12 in this series

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Subsurface flow problems are inherently multiscale in space due to the large variability of material properties and in time due to the coupling of many different physical processes, such as advection, diffusion, reaction and phase exchange. Subsurface flow models still need considerable development. For example, nonequilibrium effects, entrapped air, anomalous dispersion and hysteresis effects can still not be adequately described. Moreover, parameters of the models are diffcult to access and often uncertain. Computational issues in subsurface flows include the treatment of strong heterogeneities and anisotropies in the models, the effcient solution of transport-reaction problems with many species, treatment of multiphase-multicomponent flows and the coupling of subsurface flow models to surface flow models given by shallow water or Stokes equations. With respect to energy and the environment, in particular the modelling and simulation of radioactive waste management and sequestration of CO₂ underground have gained high interest in the community in recent years. Both applications provide unique challenges ranging from modelling of clay materials to treating very large scale models with high-performance computing.

This book brings together key numerical mathematicians whose interest is in the analysis and computation of multiscale subsurface flow and practitioners from engineering and industry whose interest is in the applications of these core problems.

DOI:: https://doi.org/10.1515/9783110282245
ISBN:: 978-3-11-028221-4
ISBN:: 978-3-11-028224-5
Subject:: Geosciences
Subject:: Geophysics
Subject:: Materials Sciences
Subject:: Porous Materials
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Subject:: Physics
Subject:: Mechanics and Fluid Dynamics
Publisher:: De Gruyter

Book 2013

Finite Fields and Their Applications

Character Sums and Polynomials

Pascale Charpin, Alexander Pott, Arne Winterhof, Guang Gong, Katalin Gyarmati, Fernando Hernando, Sophie Huczynska, Dieter Jungnickel, Gohar M. Kyureghyan, Gary McGuire, Harald Niederreiter, Alina Ostafe, Igor E. Shparlinski

Volume 11 in this series

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This book is based on the invited talks of the "RICAM-Workshop on Finite Fields and Their Applications: Character Sums and Polynomials" held at the Federal Institute for Adult Education (BIfEB) in Strobl, Austria, from September 2-7, 2012.

Finite fields play important roles in many application areas such as coding theory, cryptography, Monte Carlo and quasi-Monte Carlo methods, pseudorandom number generation, quantum computing, and wireless communication. In this book we will focus on sequences, character sums, and polynomials over finite fields in view of the above mentioned application areas:

Chapters 1 and 2 deal with sequences mainly constructed via characters and analyzed using bounds on character sums.
Chapters 3, 5, and 6 deal with polynomials over finite fields.
Chapters 4 and 9 consider problems related to coding theory studied via finite geometry and additive combinatorics, respectively.
Chapter 7 deals with quasirandom points in view of applications to numerical integration using quasi-Monte Carlo methods and simulation.
Chapter 8 studies aspects of iterations of rational functions from which pseudorandom numbers for Monte Carlo methods can be derived.

The goal of this book is giving an overview of several recent research directions as well as stimulating research in sequences and polynomials under the unified framework of character theory.

DOI:: https://doi.org/10.1515/9783110283600
ISBN:: 978-3-11-028240-5
ISBN:: 978-3-11-028360-0
Subject:: Computer Sciences
Subject:: Theoretical Computer Sciences
Publisher:: De Gruyter

Book 2012

Regularization Methods in Banach Spaces

Thomas Schuster, Barbara Kaltenbacher, Bernd Hofmann, Kamil S. Kazimierski

Volume 10 in this series

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Regularization methods aimed at finding stable approximate solutions are a necessary tool to tackle inverse and ill-posed problems. Inverse problems arise in a large variety of applications ranging from medical imaging and non-destructive testing via finance to systems biology. Many of these problems belong to the class of parameter identification problems in partial differential equations (PDEs) and thus are computationally demanding and mathematically challenging. Hence there is a substantial need for stable and efficient solvers for this kind of problems as well as for a rigorous convergence analysis of these methods.

This monograph consists of five parts. Part I motivates the importance of developing and analyzing regularization methods in Banach spaces by presenting four applications which intrinsically demand for a Banach space setting and giving a brief glimpse of sparsity constraints. Part II summarizes all mathematical tools that are necessary to carry out an analysis in Banach spaces. Part III represents the current state-of-the-art concerning Tikhonov regularization in Banach spaces. Part IV about iterative regularization methods is concerned with linear operator equations and the iterative solution of nonlinear operator equations by gradient type methods and the iteratively regularized Gauß-Newton method. Part V finally outlines the method of approximate inverse which is based on the efficient evaluation of the measured data with reconstruction kernels.

DOI:: https://doi.org/10.1515/9783110255720
ISBN:: 978-3-11-025524-9
ISBN:: 978-3-11-025572-0
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

Book 2010

Theoretical Foundations and Numerical Methods for Sparse Recovery

Massimo Fornasier

Volume 9 in this series

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The present collection is the very first contribution of this type in the field of sparse recovery. Compressed sensing is one of the important facets of the broader concept presented in the book, which by now has made connections with other branches such as mathematical imaging, inverse problems, numerical analysis and simulation.

The book consists of four lecture notes of courses given at the Summer School on "Theoretical Foundations and Numerical Methods for Sparse Recovery" held at the Johann Radon Institute for Computational and Applied Mathematics in Linz, Austria, in September 2009. This unique collection will be of value for a broad community and may serve as a textbook for graduate courses.

From the contents:

"Compressive Sensing and Structured Random Matrices" by Holger Rauhut

"Numerical Methods for Sparse Recovery" by Massimo Fornasier

"Sparse Recovery in Inverse Problems" by Ronny Ramlau and Gerd Teschke

"An Introduction to Total Variation for Image Analysis" by Antonin Chambolle, Vicent Caselles, Daniel Cremers, Matteo Novaga and Thomas Pock

DOI:: https://doi.org/10.1515/9783110226157
ISBN:: 978-3-11-022614-0
ISBN:: 978-3-11-022615-7
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2009

Advanced Financial Modelling

Hansjörg Albrecher, Wolfgang J. Runggaldier, Walter Schachermayer

Volume 8 in this series

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This book is a collection of state–of–the–art surveys on various topics in mathematical finance, with an emphasis on recent modelling and computational approaches. The volume is related to a 'Special Semester on Stochastics with Emphasis on Finance' that took place from September to December 2008 at the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences in Linz, Austria.

DOI:: https://doi.org/10.1515/9783110213140
ISBN:: 978-3-11-021313-3
ISBN:: 978-3-11-021314-0
Subject:: Business and Economics
Subject:: Mathematics and Statistics for Economists
Subject:: Mathematics
Subject:: Mathematics
Subject:: Probability and Statistics
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

Book 2009

Robust Static Super-Replication of Barrier Options

Jan H. Maruhn

Volume 7 in this series

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Static hedge portfolios for barrier options are very sensitive with respect to changes of the volatility surface. To prevent potentially significant hedging losses this book develops a static super-replication strategy with market-typical robustness against volatility, skew and liquidity risk as well as model errors. Empirical results and various numerical examples confirm that the static superhedge successfully eliminates the risk of a changing volatility surface. Combined with associated sub-replication strategies this leads to robust price bounds for barrier options which are also relevant in the context of dynamic hedging. The mathematical techniques used to prove appropriate existence, duality and convergence results range from financial mathematics, stochastic and semi-infinite optimization, convex analysis and partial differential equations to semidefinite programming.

DOI:: https://doi.org/10.1515/9783110208511
ISBN:: 978-3-11-020468-1
ISBN:: 978-3-11-020851-1
Subject:: Mathematics
Subject:: Probability and Statistics
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

Book 2008

Iterative Regularization Methods for Nonlinear Ill-Posed Problems

Barbara Kaltenbacher, Andreas Neubauer, Otmar Scherzer

Volume 6 in this series

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Nonlinear inverse problems appear in many applications, and typically they lead to mathematical models that are ill-posed, i.e., they are unstable under data perturbations. Those problems require a regularization, i.e., a special numerical treatment. This book presents regularization schemes which are based on iteration methods, e.g., nonlinear Landweber iteration, level set methods, multilevel methods and Newton type methods.

DOI:: https://doi.org/10.1515/9783110208276
ISBN:: 978-3-11-020420-9
ISBN:: 978-3-11-020827-6
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2009

Robust Algebraic Multilevel Methods and Algorithms

Johannes Kraus, Svetozar Margenov

Volume 5 in this series

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This book deals with algorithms for the solution of linear systems of algebraic equations with large-scale sparse matrices, with a focus on problems that are obtained after discretization of partial differential equations using finite element methods. The authors provide a systematic presentation of the recent advances in robust algebraic multilevel methods and algorithms, e.g., the preconditioned conjugate gradient method, algebraic multilevel iteration (AMLI) preconditioners, the classical algebraic multigrid (AMG) method and its recent modifications, namely AMG using element interpolation (AMGe) and AMG based on smoothed aggregation.

The first six chapters can serve as a short introductory course on the theory of AMLI methods and algorithms. The next part of the monograph is devoted to more advanced topics, including the description of new generation AMG methods, AMLI methods for discontinuous Galerkin systems, looking-free algorithms for coupled problems etc., ending with important practical issues of implementation and challenging applications. This second part is addressed to some more experienced students and practitioners and can be used to complete a more advanced course on robust AMLI and AMG methods and their efficient application.

This book is intended for mathematicians, engineers, natural scientists etc.

DOI:: https://doi.org/10.1515/9783110214833
ISBN:: 978-3-11-019365-7
ISBN:: 978-3-11-021483-3
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2008

A Posteriori Estimates for Partial Differential Equations

Sergey I. Repin

Volume 4 in this series

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This book deals with the reliable verification of the accuracy of approximate solutions which is one of the central problems in modern applied analysis. After giving an overview of the methods developed for models based on partial differential equations, the author derives computable a posteriori error estimates by using methods of the theory of partial differential equations and functional analysis. These estimates are applicable to approximate solutions computed by various methods.

DOI:: https://doi.org/10.1515/9783110203042
ISBN:: 978-3-11-019153-0
ISBN:: 978-3-11-020304-2
Subject:: Mathematics
Subject:: Analysis
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

Book 2007

Gröbner Bases in Control Theory and Signal Processing

Hyungju Park, Georg Regensburger

Volume 3 in this series

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This volume contains survey and original articles presenting the state of the art on the application of Gröbner bases in control theory and signal processing.

The contributions are based on talks delivered at the Special Semester on Gröbner Bases and Related Methods at the Johann Radon Institute of Computational and Applied Mathematics (RICAM), Linz, Austria, in May 2006.

DOI:: https://doi.org/10.1515/9783110909746
ISBN:: 978-3-11-019333-6
ISBN:: 978-3-11-090974-6
Subject:: Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2007

Gröbner Bases in Symbolic Analysis

Markus Rosenkranz, Dongming Wang

Volume 2 in this series

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This volume contains survey articles and original research papers, presenting the state of the art on applying the symbolic approach of Gröbner bases and related methods to differential and difference equations. The contributions are based on talks delivered at the Special Semester on Gröbner Bases and Related Methods hosted by the Johann Radon Institute of Computational and Applied Mathematics, Linz, Austria, in May 2006.

DOI:: https://doi.org/10.1515/9783110922752
ISBN:: 978-3-11-019323-7
ISBN:: 978-3-11-092275-2
Subject:: Mathematics
Subject:: Algebra and Number Theory
Publisher:: De Gruyter

Book 2007

Lectures on Advanced Computational Methods in Mechanics

Johannes Kraus, Ulrich Langer

Volume 1 in this series

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This book contains four survey papers related to different topics in computational mechanics, in particular (1) novel discretization and solver techniques in mechanics and (2) inverse, control, and optimization problems in mechanics. These topics were considered in lectures, seminars, tutorials, and workshops at the Special Semester on Computational Mechanics held at the Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, in December 2005.

DOI:: https://doi.org/10.1515/9783110927092
ISBN:: 978-3-11-019556-9
ISBN:: 978-3-11-092709-2
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter