|
Showing 1 - 9 of
9 matches in All Departments
Modeling and Inverse Problems in the Presence of Uncertainty
collects recent research-including the authors' own substantial
projects-on uncertainty propagation and quantification. It covers
two sources of uncertainty: where uncertainty is present primarily
due to measurement errors and where uncertainty is present due to
the modeling formulation itself. After a useful review of relevant
probability and statistical concepts, the book summarizes
mathematical and statistical aspects of inverse problem
methodology, including ordinary, weighted, and generalized
least-squares formulations. It then discusses asymptotic theories,
bootstrapping, and issues related to the evaluation of correctness
of assumed form of statistical models. The authors go on to present
methods for evaluating and comparing the validity of
appropriateness of a collection of models for describing a given
data set, including statistically based model selection and
comparison techniques. They also explore recent results on the
estimation of probability distributions when they are embedded in
complex mathematical models and only aggregate (not individual)
data are available. In addition, they briefly discuss the optimal
design of experiments in support of inverse problems for given
models. The book concludes with a focus on uncertainty in model
formulation itself, covering the general relationship of
differential equations driven by white noise and the ones driven by
colored noise in terms of their resulting probability density
functions. It also deals with questions related to the
appropriateness of discrete versus continuum models in transitions
from small to large numbers of individuals. With many examples
throughout addressing problems in physics, biology, and other
areas, this book is intended for applied mathematicians interested
in deterministic and/or stochastic models and their interactions.
It is also s
A Modern Framework Based on Time-Tested MaterialA Functional
Analysis Framework for Modeling, Estimation and Control in Science
and Engineering presents functional analysis as a tool for
understanding and treating distributed parameter systems. Drawing
on his extensive research and teaching from the past 20 years, the
author explains how functional analysis can be the basis of modern
partial differential equation (PDE) and delay differential equation
(DDE) techniques. Recent Examples of Functional Analysis in
Biology, Electromagnetics, Materials, and MechanicsThrough numerous
application examples, the book illustrates the role that functional
analysis-a classical subject-continues to play in the rigorous
formulation of modern applied areas. The text covers common
examples, such as thermal diffusion, transport in tissue, and beam
vibration, as well as less traditional ones, including HIV models,
uncertainty in noncooperative games, structured population models,
electromagnetics in materials, delay systems, and PDEs in control
and inverse problems. For some applications, computational aspects
are discussed since many problems necessitate a numerical approach.
Modeling and Inverse Problems in the Presence of Uncertainty
collects recent research-including the authors' own substantial
projects-on uncertainty propagation and quantification. It covers
two sources of uncertainty: where uncertainty is present primarily
due to measurement errors and where uncertainty is present due to
the modeling formulation itself. After a useful review of relevant
probability and statistical concepts, the book summarizes
mathematical and statistical aspects of inverse problem
methodology, including ordinary, weighted, and generalized
least-squares formulations. It then discusses asymptotic theories,
bootstrapping, and issues related to the evaluation of correctness
of assumed form of statistical models. The authors go on to present
methods for evaluating and comparing the validity of
appropriateness of a collection of models for describing a given
data set, including statistically based model selection and
comparison techniques. They also explore recent results on the
estimation of probability distributions when they are embedded in
complex mathematical models and only aggregate (not individual)
data are available. In addition, they briefly discuss the optimal
design of experiments in support of inverse problems for given
models. The book concludes with a focus on uncertainty in model
formulation itself, covering the general relationship of
differential equations driven by white noise and the ones driven by
colored noise in terms of their resulting probability density
functions. It also deals with questions related to the
appropriateness of discrete versus continuum models in transitions
from small to large numbers of individuals. With many examples
throughout addressing problems in physics, biology, and other
areas, this book is intended for applied mathematicians interested
in deterministic and/or stochastic models and their interactions.
It is also s
A Modern Framework Based on Time-Tested MaterialA Functional
Analysis Framework for Modeling, Estimation and Control in Science
and Engineering presents functional analysis as a tool for
understanding and treating distributed parameter systems. Drawing
on his extensive research and teaching from the past 20 years, the
author explains how functional analysis can be the basis of modern
partial differential equation (PDE) and delay differential equation
(DDE) techniques. Recent Examples of Functional Analysis in
Biology, Electromagnetics, Materials, and MechanicsThrough numerous
application examples, the book illustrates the role that functional
analysis-a classical subject-continues to play in the rigorous
formulation of modern applied areas. The text covers common
examples, such as thermal diffusion, transport in tissue, and beam
vibration, as well as less traditional ones, including HIV models,
uncertainty in noncooperative games, structured population models,
electromagnetics in materials, delay systems, and PDEs in control
and inverse problems. For some applications, computational aspects
are discussed since many problems necessitate a numerical approach.
The research detailed in this monograph was originally motivated by
our interest in control problems involving partial and delay
differential equations. Our attempts to apply control theory
techniques to such prob lems in several areas of science convinced
us that in the need for better and more detailed models of
distributed/ continuum processes in biology and mechanics lay a
rich, interesting, and challenging class of fundamen tal questions.
These questions, which involve science and mathematics, are typical
of those arising in inverse or parameter estimation problems. Our
efforts on inverse problems for distributed parameter systems,
which are infinite dimensional in the most common realizations,
began about seven years ago at a time when rapid advances in
computing capabilities and availability held promise for
significant progress in the development of a practically useful as
well as theoretically sound methodology for such problems. Much of
the research reported in our presentation was not begun when we
outlined the plans for this monograph some years ago. By publishing
this monograph now, when only a part of the originally intended
topics are covered (see Chapter VII in this respect), we hope to
stimulate the research and interest of others in an area of
scientific en deavor which has exceeded even our optimistic
expectations with respect to excitement, opportunity, and
stimulation. The computer revolution alluded to above and the
development of new codes allow one to solve rather routinely
certain estimation problems that would have been out of the
question ten years ago."
These notes are based on (i) a series of lectures that I gave at
the 14th Biennial Seminar of the Canadian Mathematical Congress
held at the University of Western Ontario August 12-24, 1973 and
(li) some of my lectures in a modeling course that I have cotaught
in the Division of Bio-Medical Sciences at Brown during the past
several years. An earlier version of these notes appeared in the
Center for Dynamical Systems Lectures Notes series (CDS LN 73-1,
November 1973). I have in this revised and extended version of
those earlier notes incorporated a number of changes based both on
classroom experience and on my research efforts with several
colleagues during the intervening period. The narrow viewpoint of
the present notes (use of optimization and control theory in
biomedical problems) reflects more the scope of the CMC lectures
given in August, 1973 than the scope of my own interests. Indeed,
my real interests have included the modeling process itself as well
as the contributions made by investiga tors who employ the
techniques and ideas of control theory, systems analysis, dif
ferential equations, and stochastic processes. Some of these
contributions have quite naturally involved application of optimal
control theory. But in my opinion many of the interesting efforts
being made in modeling in the biomedical sciences encompass much
more than the use of control theory."
Through several case study problems from industrial and scientific
research laboratory applications, Mathematical and Experimental
Modeling of Physical and Biological Processes provides students
with a fundamental understanding of how mathematics is applied to
problems in science and engineering. For each case study problem,
the authors discuss why a model is needed and what goals can be
achieved with the model. Exploring what mathematics can reveal
about applications, the book focuses on the design of appropriate
experiments to validate the development of mathematical models. It
guides students through the modeling process, from empirical
observations and formalization of properties to model analysis and
interpretation of results. The authors also describe the hardware
and software tools used to design the experiments so
faculty/students can duplicate them. Integrating real-world
applications into the traditional mathematics curriculum, this
textbook deals with the formulation and analysis of mathematical
models in science and engineering. It gives students an
appreciation of the use of mathematics and encourages them to
further study the applied topics. Real experimental data for
projects can be downloaded from CRC Press Online.
The research detailed in this monograph was originally motivated by
our interest in control problems involving partial and delay
differential equations. Our attempts to apply control theory
techniques to such prob lems in several areas of science
convinced us that in the need for better and more detailed models
of distributed/ continuum processes in biology and mechanics lay a
rich, interesting, and challenging class of fundamen tal
questions. These questions, which involve science and mathematics,
are typical of those arising in inverse or parameter estimation
problems. Our efforts on inverse problems for distributed parameter
systems, which are infinite dimensional in the most common
realizations, began about seven years ago at a time when rapid
advances in computing capabilities and availability held promise
for significant progress in the development of a practically useful
as well as theoretically sound methodology for such problems. Much
of the research reported in our presentation was not begun when we
outlined the plans for this monograph some years ago. By publishing
this monograph now, when only a part of the originally intended
topics are covered (see Chapter VII in this respect), we hope to
stimulate the research and interest of others in an area of
scientific en deavor which has exceeded even our optimistic
expectations with respect to excitement, opportunity, and
stimulation. The computer revolution alluded to above and the
development of new codes allow one to solve rather routinely
certain estimation problems that would have been out of the
question ten years ago.
|
You may like...
Tenet
John David Washington, Robert Pattinson
Blu-ray disc
(1)
R52
Discovery Miles 520
|