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As the title of the book indicates, this is primarily a book on
partial differential equations (PDEs) with two definite slants:
toward inverse problems and to the inclusion of fractional
derivatives. The standard paradigm, or direct problem, is to take a
PDE, including all coefficients and initial/boundary conditions,
and to determine the solution. The inverse problem reverses this
approach asking what information about coefficients of the model
can be obtained from partial information on the solution. Answering
this question requires knowledge of the underlying physical model,
including the exact dependence on material parameters. The last
feature of the approach taken by the authors is the inclusion of
fractional derivatives. This is driven by direct physical
applications: a fractional derivative model often allows greater
adherence to physical observations than the traditional integer
order case. The book also has an extensive historical section and
the material that can be called ""fractional calculus"" and
ordinary differential equations with fractional derivatives. This
part is accessible to advanced undergraduates with basic knowledge
on real and complex analysis. At the other end of the spectrum, lie
nonlinear fractional PDEs that require a standard graduate level
course on PDEs.
As the title of the book indicates, this is primarily a book on
partial differential equations (PDEs) with two definite slants:
toward inverse problems and to the inclusion of fractional
derivatives. The standard paradigm, or direct problem, is to take a
PDE, including all coefficients and initial/boundary conditions,
and to determine the solution. The inverse problem reverses this
approach asking what information about coefficients of the model
can be obtained from partial information on the solution. Answering
this question requires knowledge of the underlying physical model,
including the exact dependence on material parameters. The last
feature of the approach taken by the authors is the inclusion of
fractional derivatives. This is driven by direct physical
applications: a fractional derivative model often allows greater
adherence to physical observations than the traditional integer
order case. The book also has an extensive historical section and
the material that can be called ""fractional calculus"" and
ordinary differential equations with fractional derivatives. This
part is accessible to advanced undergraduates with basic knowledge
on real and complex analysis. At the other end of the spectrum, lie
nonlinear fractional PDEs that require a standard graduate level
course on PDEs.
Inverse problems such as imaging or parameter identification deal
with the recovery of unknown quantities from indirect observations,
connected via a model describing the underlying context. While
traditionally inverse problems are formulated and investigated in a
static setting, we observe a significant increase of interest in
time-dependence in a growing number of important applications over
the last few years. Here, time-dependence affects a) the unknown
function to be recovered and / or b) the observed data and / or c)
the underlying process. Challenging applications in the field of
imaging and parameter identification are techniques such as
photoacoustic tomography, elastography, dynamic computerized or
emission tomography, dynamic magnetic resonance imaging,
super-resolution in image sequences and videos, health monitoring
of elastic structures, optical flow problems or magnetic particle
imaging to name only a few. Such problems demand for innovation
concerning their mathematical description and analysis as well as
computational approaches for their solution.
Inverse problems such as imaging or parameter identification deal
with the recovery of unknown quantities from indirect observations,
connected via a model describing the underlying context. While
traditionally inverse problems are formulated and investigated in a
static setting, we observe a significant increase of interest in
time-dependence in a growing number of important applications over
the last few years. Here, time-dependence affects a) the unknown
function to be recovered and / or b) the observed data and / or c)
the underlying process. Challenging applications in the field of
imaging and parameter identification are techniques such as
photoacoustic tomography, elastography, dynamic computerized or
emission tomography, dynamic magnetic resonance imaging,
super-resolution in image sequences and videos, health monitoring
of elastic structures, optical flow problems or magnetic particle
imaging to name only a few. Such problems demand for innovation
concerning their mathematical description and analysis as well as
computational approaches for their solution.
This book is devoted to the study of coupled partial differential
equation models, which describe complex dynamical systems occurring
in modern scientific applications such as fluid/flow-structure
interactions. The first chapter provides a general description of a
fluid-structure interaction, which is formulated within a realistic
framework, where the structure subject to a frictional damping
moves within the fluid. The second chapter then offers a
multifaceted description, with often surprising results, of the
case of the static interface; a case that is argued in the
literature to be a good model for small, rapid oscillations of the
structure. The third chapter describes flow-structure interaction
where the compressible Navier-Stokes equations are replaced by the
linearized Euler equation, while the solid is taken as a nonlinear
plate, which oscillates in the surrounding gas flow. The final
chapter focuses on a the equations of nonlinear acoustics coupled
with linear acoustics or elasticity, as they arise in the context
of high intensity ultrasound applications.
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.
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