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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 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.
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.
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.
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 includes graphical explanations and computed
examples using publicly available data sets in nearly every chapter
to highlight similarities and differences among the algorithms.
Interesting, real-world mathematical modeling 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 modeling paradigms: the
macroscopic, in which the authors describe phenomena in terms of
time evolution via ordinary differential equations, and the
microscopic, which requires knowledge of random events and
probability. The text emphasizes the development of computational
skills to construct predictive models and analyze the results. To
elucidate the concepts, a wealth of examples and portions of
MATLAB? code used by the authors are included. Computational
Mathematical Modeling: An Integrated Approach Across Scales:* Is
designed for classroom use, has been extensively tested by the
authors, and has homework problems carefully designed to develop
students' computational skills.* Is based on an unorthodox
combination of deterministic and probablistic methodologies that
are naturally bridged through examples.* Painlessly introduces
students to advanced themes in a natural progression.* Includes
suggestions for further reading.
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