*Bijan Mohammadi and Olivier Pironneau*

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199546909
- eISBN:
- 9780191720482
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199546909.003.0009
- Subject:
- Mathematics, Mathematical Physics

This chapter puts forward a general argument to support the use of approximate gradients within optimization loops integrated with mesh refinements. However, this does not justify all the procedures ...
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This chapter puts forward a general argument to support the use of approximate gradients within optimization loops integrated with mesh refinements. However, this does not justify all the procedures that are presented in the previous chapter. The chapter proves also that smoothers are essential. This part was done in collaboration with E. Polak and N. Dicesare.Less

This chapter puts forward a general argument to support the use of approximate gradients within optimization loops integrated with mesh refinements. However, this does not justify all the procedures that are presented in the previous chapter. The chapter proves also that smoothers are essential. This part was done in collaboration with E. Polak and N. Dicesare.

*Éric Blayo, Marc Bocquet, Emmanuel Cosme, and Leticia F. Cugliandolo (eds)*

- Published in print:
- 2014
- Published Online:
- March 2015
- ISBN:
- 9780198723844
- eISBN:
- 9780191791185
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198723844.001.0001
- Subject:
- Physics, Geophysics, Atmospheric and Environmental Physics

This book gathers notes from lectures and seminars given during a three-week school on theoretical and applied data assimilation held in Les Houches in 2012. Data assimilation aims at determining as ...
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This book gathers notes from lectures and seminars given during a three-week school on theoretical and applied data assimilation held in Les Houches in 2012. Data assimilation aims at determining as accurately as possible the state of a dynamical system by combining heterogeneous sources of information in an optimal way. Generally speaking, the mathematical methods of data assimilation describe algorithms for forming optimal combinations of observations of a system, a numerical model that describes its evolution, and appropriate prior information. Data assimilation has a long history of application to high-dimensional geophysical systems dating back to the 1960s, with application to the estimation of initial conditions for weather forecasts. It has become a major component of numerical forecasting systems in geophysics, and an intensive field of research, with numerous additional applications in oceanography and atmospheric chemistry, with extensions to other geophysical sciences. The physical complexity and the high dimensionality of geophysical systems have led the community of geophysics to make significant contributions to the fundamental theory of data assimilation. This book is composed of a series of main lectures, presenting the fundamentals of four-dimensional variational data assimilation, the Kalman filter, smoothers, and the information theory background required to understand and evaluate the role of observations; a series of specialized lectures, addressing various aspects of data assimilation in detail, from the most recent developments in the theory to the specificities of various thematic applications.Less

This book gathers notes from lectures and seminars given during a three-week school on theoretical and applied data assimilation held in Les Houches in 2012. Data assimilation aims at determining as accurately as possible the state of a dynamical system by combining heterogeneous sources of information in an optimal way. Generally speaking, the mathematical methods of data assimilation describe algorithms for forming optimal combinations of observations of a system, a numerical model that describes its evolution, and appropriate prior information. Data assimilation has a long history of application to high-dimensional geophysical systems dating back to the 1960s, with application to the estimation of initial conditions for weather forecasts. It has become a major component of numerical forecasting systems in geophysics, and an intensive field of research, with numerous additional applications in oceanography and atmospheric chemistry, with extensions to other geophysical sciences. The physical complexity and the high dimensionality of geophysical systems have led the community of geophysics to make significant contributions to the fundamental theory of data assimilation. This book is composed of a series of main lectures, presenting the fundamentals of four-dimensional variational data assimilation, the Kalman filter, smoothers, and the information theory background required to understand and evaluate the role of observations; a series of specialized lectures, addressing various aspects of data assimilation in detail, from the most recent developments in the theory to the specificities of various thematic applications.

*Ronald K. Pearson*

- Published in print:
- 1999
- Published Online:
- November 2020
- ISBN:
- 9780195121988
- eISBN:
- 9780197561294
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195121988.003.0005
- Subject:
- Computer Science, Mathematical Theory of Computation

The review of linear models presented in Chapter 2 was intended to provide a baseline, establishing notation and reviewing some important aspects of this reference class against which nonlinear ...
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The review of linear models presented in Chapter 2 was intended to provide a baseline, establishing notation and reviewing some important aspects of this reference class against which nonlinear models are necessarily judged. This chapter demonstrates that nonlinearity can be approached from at least two fundamentally different directions. The first of these directions is structural-— by far the most common approach—in which a class of nonlinear models is defined by specifying the basic structure of all elements of that class. The Hammerstein, Wiener, and Uryson model classes discussed in Chapter 1 illustrate this approach. The structurally defined model class considered in this chapter is the class of bilinear models discussed in Sec. 3.1, which may be viewed as “almost linear” for reasons that will become apparent. Alternatively, it is possible to adopt behavioral definitions of nonlinear model classes, although this approach is generally more difficult. There, a particular type of input/output behavior is specified and subsequent analysis seeks model structures that can exhibit this behavior. In practice, this approach tends to be difficult because it is often not clear how to constuct explicit examples that exhibit specified qualitative behavior. Of necessity, then, the primary focus of this book is structurally defined model classes like the bilinear models discussed in Sec. 3.1, although three behaviorally defined model classes are considered in some detail in Secs. 3.2 through 3.4. The first of these is the class of homogeneous models, obtained by relaxing one of the two defining conditions for linearity: homogeneous models do not obey the superposition principle of linear systems, but they are invariant under scaling of the input sequence by arbitrary real constants. Relaxing these conditions further and requiring only that this scaling hold for positive constants leads to the class of positive-homogeneous models, described in Sec. 3.3. Alternatively, requiring linearity to hold but only for constant input sequences leads to the class of static-linear models, described in Sec. 3.4. As these and subsequent discussions illustrate, some remarkably general results may be obtained concerning the structure of these three model classes.
Less

The review of linear models presented in Chapter 2 was intended to provide a baseline, establishing notation and reviewing some important aspects of this reference class against which nonlinear models are necessarily judged. This chapter demonstrates that nonlinearity can be approached from at least two fundamentally different directions. The first of these directions is structural-— by far the most common approach—in which a class of nonlinear models is defined by specifying the basic structure of all elements of that class. The Hammerstein, Wiener, and Uryson model classes discussed in Chapter 1 illustrate this approach. The structurally defined model class considered in this chapter is the class of bilinear models discussed in Sec. 3.1, which may be viewed as “almost linear” for reasons that will become apparent. Alternatively, it is possible to adopt behavioral definitions of nonlinear model classes, although this approach is generally more difficult. There, a particular type of input/output behavior is specified and subsequent analysis seeks model structures that can exhibit this behavior. In practice, this approach tends to be difficult because it is often not clear how to constuct explicit examples that exhibit specified qualitative behavior. Of necessity, then, the primary focus of this book is structurally defined model classes like the bilinear models discussed in Sec. 3.1, although three behaviorally defined model classes are considered in some detail in Secs. 3.2 through 3.4. The first of these is the class of homogeneous models, obtained by relaxing one of the two defining conditions for linearity: homogeneous models do not obey the superposition principle of linear systems, but they are invariant under scaling of the input sequence by arbitrary real constants. Relaxing these conditions further and requiring only that this scaling hold for positive constants leads to the class of positive-homogeneous models, described in Sec. 3.3. Alternatively, requiring linearity to hold but only for constant input sequences leads to the class of static-linear models, described in Sec. 3.4. As these and subsequent discussions illustrate, some remarkably general results may be obtained concerning the structure of these three model classes.