Erkki Somersalo's Books

Bayesian Scientific Computing

By: Daniela Calvetti,Erkki Somersalo

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The once esoteric idea of embedding scientific computing into a probabilistic framework, mostly along the lines of the Bayesian paradigm, has recently enjoyed wide popularity and found its way into numerous applications. This book provides an insider’s view of how to combine two mature fields, scientific computing and Bayesian inference, into a powerful language leveraging the capabilities of both components for computational efficiency, high resolution power and uncertainty quantification ability. The impact of Bayesian scientific computing has been particularly significant in the area of computational inverse problems where the data are often scarce or of low quality, but some characteristics of the unknown solution may be available a priori. The ability to combine the flexibility of the Bayesian probabilistic framework with efficient numerical methods has contributed to the popularity of Bayesian inversion, with the prior distribution being the counterpart of classical regularization. However, the interplay between Bayesian inference and numerical analysis is much richer than providing an alternative way to regularize inverse problems, as demonstrated by the discussion of time dependent problems, iterative methods, and sparsity promoting priors in this book. The quantification of uncertainty in computed solutions and model predictions is another area where Bayesian scientific computing plays a critical role. This book demonstrates that Bayesian inference and scientific computing have much more in common than what one may expect, and gradually builds a natural interface between these two areas.

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Page Count

295

Publisher

Springer Nature

Published Date

ISBN 10

3031238249

ISBN 13

9783031238246

The Less Is More Linear Algebra of Vector Spaces and Matrices

By: Daniela Calvetti,Erkki Somersalo

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The Less Is More Linear Algebra of Vector Spaces and Matrices

By: Daniela Calvetti,Erkki Somersalo

Designed for a proof-based course on linear algebra, this rigorous and concise textbook intentionally introduces vector spaces, inner products, and vector and matrix norms before Gaussian elimination and eigenvalues so students can quickly discover the singular value decomposition (SVD)—arguably the most enlightening and useful of all matrix factorizations. Gaussian elimination is then introduced after the SVD and the four fundamental subspaces and is presented in the context of vector spaces rather than as a computational recipe. This allows the authors to use linear independence, spanning sets and bases, and the four fundamental subspaces to explain and exploit Gaussian elimination and the LU factorization, as well as the solution of overdetermined linear systems in the least squares sense and eigenvalues and eigenvectors. This unique textbook also includes examples and problems focused on concepts rather than the mechanics of linear algebra. The problems at the end of each chapter that and in an associated website encourage readers to explore how to use the notions introduced in the chapter in a variety of ways. Additional problems, quizzes, and exams will be posted on an accompanying website and updated regularly. The Less Is More Linear Algebra of Vector Spaces and Matrices is for students and researchers interested in learning linear algebra who have the mathematical maturity to appreciate abstract concepts that generalize intuitive ideas. The early introduction of the SVD makes the book particularly useful for those interested in using linear algebra in applications such as scientific computing and data science. It is appropriate for a first proof-based course in linear algebra.

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Page Count

181

Publisher

SIAM

Published Date

ISBN 10

1611977401

ISBN 13

9781611977400

Computational Mathematical Modeling

An Integrated Approach Across Scales

By: Daniela Calvetti,Erkki Somersalo

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Computational Mathematical Modeling

An Integrated Approach Across Scales

By: Daniela Calvetti,Erkki Somersalo

Interesting real-world mathematical modelling problems are complex and can usually be studied at different scales. The scale at which the investigation is carried out is one of the factors that determines the type of mathematics most appropriate to describe the problem. The book concentrates on two modelling paradigms: the macroscopic, in which phenomena are described in terms of time evolution via ordinary differential equations; and the microscopic, which requires knowledge of random events and probability. The exposition is based on this unorthodox combination of deterministic and probabilistic methodologies, and emphasizes the development of computational skills to construct predictive models. To elucidate the concepts, a wealth of examples, self-study problems, and portions of MATLAB code used by the authors are included. This book, which has been extensively tested by the authors for classroom use, is intended for students in mathematics and the physical sciences at the advanced undergraduate level and above.

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229

Publisher

SIAM

Published Date

ISBN 10

1611972477

ISBN 13

9781611972474

Mathematics of Data Science: A Computational Approach to Clustering and Classification

By: Daniela Calvetti,Erkki Somersalo

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Mathematics of Data Science: A Computational Approach to Clustering and Classification

By: Daniela Calvetti,Erkki Somersalo

This textbook provides a solid mathematical basis for understanding popular data science algorithms for clustering and classification and shows that an in-depth understanding of the mathematics powering these algorithms gives insight into the underlying data. It presents a step-by-step derivation of these algorithms, outlining their implementation from scratch in a computationally sound way. Mathematics of Data Science: A Computational Approach to Clustering and Classification proposes different ways of visualizing high-dimensional data to unveil hidden internal structures, and nearly every chapter includes graphical explanations and computed examples using publicly available data sets to highlight similarities and differences among the algorithms. This self-contained book is geared toward advanced undergraduate and beginning graduate students in the mathematical sciences, engineering, and computer science and can be used as the main text in a semester course. Researchers in any application area where data science methods are used will also find the book of interest. No advanced mathematical or statistical background is assumed.

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Page Count

199

Publisher

SIAM

Published Date

ISBN 10

1611976375

ISBN 13

9781611976373

Mathematical Modelling

By: Seppo Pohjolainen,Matti Heiliö,Timo Lähivaara,Erkki Laitinen,Timo Mantere,Jorma Merikoski,Kimmo Raivio,Risto Silvennoinen,Antti Suutala,Tanja Tarvainen,Timo Tiihonen,Jukka Tuomela,Esko Turunen,Marko Vauhkonen

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Mathematical Modelling

This book provides a thorough introduction to the challenge of applying mathematics in real-world scenarios. Modelling tasks rarely involve well-defined categories, and they often require multidisciplinary input from mathematics, physics, computer sciences, or engineering. In keeping with this spirit of modelling, the book includes a wealth of cross-references between the chapters and frequently points to the real-world context. The book combines classical approaches to modelling with novel areas such as soft computing methods, inverse problems, and model uncertainty. Attention is also paid to the interaction between models, data and the use of mathematical software. The reader will find a broad selection of theoretical tools for practicing industrial mathematics, including the analysis of continuum models, probabilistic and discrete phenomena, and asymptotic and sensitivity analysis.

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Page Count

247

Publisher

Springer

Published Date

ISBN 10

3319278363

ISBN 13

9783319278360

Computational Mathematical Modeling

An Integrated Approach Across Scales

By: Daniela Calvetti,Erkki Somersalo

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Computational Mathematical Modeling

An Integrated Approach Across Scales

By: Daniela Calvetti,Erkki Somersalo

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229

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SIAM

Published Date

ISBN 10

1611972477

ISBN 13

9781611972474

Inverse Problems in Mathematical Physics

Proceedings of the Lapland Conference on Inverse Problems Held at Saariselkä, Finland, 14-20 June 1992

By: Lassi Päivärinta,Erkki Somersalo

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Inverse Problems in Mathematical Physics

Proceedings of the Lapland Conference on Inverse Problems Held at Saariselkä, Finland, 14-20 June 1992

By: Lassi Päivärinta,Erkki Somersalo

The book contains presentations of recent and ongoing research on inverse problems and its application to engineering and physical sciences. The articles are structured around three closely related topics: Inverse scattering problems, inverse boundary value problems, and inverse spectral problems. The applications range from quantum and electromagnetic scattering to medical imaging, geophysical sounding of the Earth, and non-destructive material evaluation. The book gives an up-to-date presentation of the most recent developments in these rapidlychanging and evolving fields of applied research. The contributors of the volume give extra emphysis to the pedagogical aspects of their presentation to make this collection eysily accessible to graduate students as well as to people working on nearby fields of research.

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Page Count

256

Publisher

Springer Verlag

Published Date

ISBN 10

0387571957

ISBN 13

9780387571959

Book's by Erkki Somersalo

Bayesian Scientific Computing

Bayesian Scientific Computing

The Less Is More Linear Algebra of Vector Spaces and Matrices

The Less Is More Linear Algebra of Vector Spaces and Matrices

Computational Mathematical Modeling

Computational Mathematical Modeling

Mathematics of Data Science: A Computational Approach to Clustering and Classification

Mathematics of Data Science: A Computational Approach to Clustering and Classification

Mathematical Modelling

Mathematical Modelling

Computational Mathematical Modeling

Computational Mathematical Modeling

Inverse Problems in Mathematical Physics

Inverse Problems in Mathematical Physics

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