From 14c7137ada5001014a3d6b68f9d9275ee73ca1bb Mon Sep 17 00:00:00 2001 From: Jean-Philippe ARGAUD Date: Wed, 12 May 2021 21:03:49 +0200 Subject: [PATCH] Minor documentation review corrections (2) --- doc/en/images/meth_ad_and_opt.png | Bin 0 -> 202237 bytes doc/en/images/meth_steps_in_study.png | Bin 111010 -> 238767 bytes doc/en/snippets/BackgroundError.rst | 2 + doc/en/snippets/EvolutionError.rst | 2 + doc/en/snippets/ObservationError.rst | 2 + doc/en/theory.rst | 152 ++++++++++++++++++++++++ doc/fr/images/meth_ad_and_opt.png | Bin 0 -> 222990 bytes doc/fr/images/meth_steps_in_study.png | Bin 117046 -> 244200 bytes doc/fr/snippets/BackgroundError.rst | 2 + doc/fr/snippets/EvolutionError.rst | 2 + doc/fr/snippets/ObservationError.rst | 2 + doc/fr/theory.rst | 162 ++++++++++++++++++++++++++ 12 files changed, 326 insertions(+) create mode 100644 doc/en/images/meth_ad_and_opt.png create mode 100644 doc/fr/images/meth_ad_and_opt.png diff --git a/doc/en/images/meth_ad_and_opt.png 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x8<`r}5dK7$XNoB(m2S^Z{EsZno<8gT37!o%P)$hIG8_V4FXdI`N@a`!{x8ntdZPdU diff --git a/doc/en/snippets/BackgroundError.rst b/doc/en/snippets/BackgroundError.rst index a694d64..df771e5 100644 --- a/doc/en/snippets/BackgroundError.rst +++ b/doc/en/snippets/BackgroundError.rst @@ -1,4 +1,6 @@ .. index:: single: BackgroundError +.. index:: single: ScalarSparseMatrix +.. index:: single: DiagonalSparseMatrix BackgroundError *Matrix*. This indicates the background error covariance matrix, previously diff --git a/doc/en/snippets/EvolutionError.rst b/doc/en/snippets/EvolutionError.rst index 6721362..3b7729c 100644 --- a/doc/en/snippets/EvolutionError.rst +++ b/doc/en/snippets/EvolutionError.rst @@ -1,4 +1,6 @@ .. index:: single: EvolutionError +.. index:: single: ScalarSparseMatrix +.. index:: single: DiagonalSparseMatrix EvolutionError *Matrix*. The variable indicates the evolution error covariance matrix, diff --git a/doc/en/snippets/ObservationError.rst b/doc/en/snippets/ObservationError.rst index 058aa4e..46d964e 100644 --- a/doc/en/snippets/ObservationError.rst +++ b/doc/en/snippets/ObservationError.rst @@ -1,4 +1,6 @@ .. index:: single: ObservationError +.. index:: single: ScalarSparseMatrix +.. index:: single: DiagonalSparseMatrix ObservationError *Matrix*. The variable indicates the observation error covariance matrix, diff --git a/doc/en/theory.rst b/doc/en/theory.rst index a319ac3..328dde9 100644 --- a/doc/en/theory.rst +++ b/doc/en/theory.rst @@ -143,6 +143,35 @@ consideration is greatly facilitated by the data assimilation framework, which makes it possible to objectively process a heterogeneous set of available information. +Joint estimation of states and parameters +----------------------------------------- + +.. index:: single: joint estimation of states and parameters + +It is sometimes necessary, when considering the two previous types of +applications, to need to simultaneously estimate states (fields) and parameters +characterising a physical phenomenon. This is known as **joint estimation of +states and parameters**. + +Without going into the advanced methods to solve this problem, we can mention +the conceptually very simple approach of considering the vector of states to be +interpolated as *augmented* by the vector of parameters to be calibrated. The +assimilation or optimization algorithms can then be applied to the augmented +vector. Valid for moderate nonlinearities in the simulation, this simple method +extends the optimization space, and thus leads to larger problems, but it is +often possible to reduce the representation to numerically computable cases. +Without exhaustiveness, the separated variables optimization, the reduced rank +filtering, or the specific treatment of covariance matrices, are common +techniques to avoid this dimension problem. We note that, in the temporal case, +the evolution of the parameters to be estimated is simply the identity. + +To go further, we refer to the mathematical methods of optimization and +augmentation developed in many books or specialized articles, finding their +origin for example in [Lions68]_, [Jazwinski70]_ or [Dautray85]_. In particular +in the case of more marked nonlinearities during the numerical simulation of +the states, it is advisable to treat in a more complete but also more complex +way the problem of joint estimation of states and parameters. + Simple description of the data assimilation methodological framework -------------------------------------------------------------------- @@ -262,6 +291,30 @@ operators. Many other variants have been developed to improve the numerical quality of the methods or to take into account computer requirements such as calculation size and time. +A schematic view of Data Assimilation and Optimization approaches +----------------------------------------------------------------- + +To help the reader get an idea of the approaches that can be used with ADAO in +Data Assimilation and Optimization, we propose here a simplified scheme +describing an arbitrary classification of methods. It is partially and freely +inspired by [Asch16]_ (Figure 1.5). + + .. _meth_steps_in_study: + .. image:: images/meth_ad_and_opt.png + :align: center + :width: 75% + .. centered:: + **A simplified classification of methods that can be used with ADAO in Data Assimilation and Optimization** + +It is deliberately simple to remain readable, the dashed lines showing some of +the simplifications. For example, it does not specifically mention the methods +with reductions, some of which were variations of the basic methods shown here, +nor does it mention the more detailed extensions. It also omits the test +methods available in ADAO and useful for the study. + +Each method mentioned in this diagram is the subject of a specific descriptive +section in the chapter on :ref:`section_reference_assimilation`. + Going further in the data assimilation framework ------------------------------------------------ @@ -411,3 +464,102 @@ it is available in the ADAO module: The reader interested in the subject of optimization can look at [WikipediaMO]_ as a general entry point. + +Reduction methods and reduced optimization +------------------------------------------ + +.. index:: single: reduction +.. index:: single: reduction methods +.. index:: single: reduced methods +.. index:: single: reduced space +.. index:: single: neutral sub-space +.. index:: single: SVD +.. index:: single: POD +.. index:: single: PCA +.. index:: single: Kahrunen-Loeve +.. index:: single: RBM +.. index:: single: EIM +.. index:: single: Fourier +.. index:: single: wavelets +.. index:: single: EOF +.. index:: single: sparse + +Data assimilation and optimization approaches always imply a certain amount of +reiteration of a unitary numerical simulation representing the physics that is +to be treated. In order to handle this physics as well as possible, this +elementary numerical simulation is often of large size, even huge, and leads to +an extremely high computational cost when it is repeated. The complete physical +simulation is often called "*high fidelity simulation*" (or "*full scale +simulation*"). + +In a generic way, **reduction methods thus aim at reducing the computational +cost of the optimization while controlling as much as possible the numerical +error implied by this reduction**. + +To establish this, one seeks to reduce at least one of the ingredients that +make up the data assimilation or optimization problem. One can thus classify +the reduction methods according to the ingredient on which they operate, +knowing that some methods deal with several of them. A rough classification is +provided here, which the reader can complete by reading general mathematical +books or articles, or those specialized in his physics. + +Reduction of data assimilation or optimization algorithms: + the optimization algorithms themselves can generate significant + computational costs to process numerical information. Various methods can + be used to reduce their algorithmic cost, for example by working in the + most suitable reduced space for optimization, or by using multi-level + optimization techniques. ADAO has such techniques that are included in + variants of classical algorithms, leading to exact or approximate but + numerically more efficient resolutions. By default, the algorithmic options + chosen in ADAO are always the most efficient when they do not impact the + quality of the optimization. + +Reduction of the representation of covariances: + in data assimilation algorithms, covariances are the most expensive + quantities to handle or to store, often becoming the limiting quantities + from the point of view of the computational cost. Many methods try to use a + reduced representation of these matrices (leading sometimes but not + necessarily to reduce the dimension of the optimization space). + Classically, factorization, decomposition (spectral, Fourier, wavelets...) + or ensemble estimation (EOF...) techniques, or combinations, are used to + reduce the numerical load of these covariances in the computations. ADAO + uses some of these techniques, in combination with sparse computation + techniques, to make the handling of covariance matrices more efficient. + +Reduction of the physical model: + the simplest way to reduce the cost of the unit calculation consists in + reducing the simulation model itself, by representing it in a more economic + way. Numerous methods allow this reduction of models by ensuring a more or + less rigorous control of the approximation error generated by the + reduction. The use of simplified models of the physics allows a reduction + but without always producing an error control. On the contrary, all + decomposition methods (Fourier, wavelets, SVD, POD, PCA, Kahrunen-Loeve, + RBM, EIM, etc.) aim at a reduction of the representation space with an + explicit error control. Although they are very frequently used, they must + nevertheless be completed by a fine analysis of the interaction with the + optimization algorithm in which the reduced computation is inserted, in + order to avoid instabilities, discrepancies or inconsistencies that are + notoriously harmful. ADAO fully supports the use of this type of reduction + method, even if it is often necessary to establish this generic independent + reduction prior to the optimization. + +Reduction of the data assimilation or optimization space: + the size of the optimization space depends greatly on the type of problem + treated (estimation of states or parameters) but also on the number of + observations available to conduct the data assimilation. It is therefore + sometimes possible to conduct the optimization in the smallest space by + adapting the internal formulation of the optimization algorithms. When it + is possible and judicious, ADAO integrates this kind of reduced formulation + to improve the numerical performance without reducing the quality of the + optimization. + +Combining multiple reductions: + many advanced algorithms seek to combine multiple reduction techniques + simultaneously. However, it is difficult to have both generic and robust + methods, and to use several very efficient reduction techniques at the same + time. ADAO integrates some of the most robust methods, but this aspect is + still largely the subject of research and development. + +One can end this quick overview of reduction methods by pointing out that their +use is ubiquitous in real applications and in numerical tools, and that ADAO +allows to use proven methods without even knowing it. diff --git a/doc/fr/images/meth_ad_and_opt.png b/doc/fr/images/meth_ad_and_opt.png new file mode 100644 index 0000000000000000000000000000000000000000..104d0bde73504fd3e770458efea19d237c879ea2 GIT binary patch literal 222990 zcmd?Rg;&(u8$LRU$T=Wl5E6nSok~iJpmcXjcXx+}?(UMIOS(fT>1JqlG(J8)qBzDQk($txsCjtb9+pQ&TN4U3vt!gB=N8Qi z)x#9mLFZ>ajE5^J>yu>pZ5xb9D;!5<5+<)8**>Gx{$}UYbd-i;BT_;J=bR*9su0M) zemt3f+i~^E_TrH}8-%-yLZqID{H;|JRHRt~xpIhC%OrXCYr~gXl=oZ3QkM4Ek#A-w zmtAn)L;ePVsB2#K`$#$VD@F%7E3V&fMQNTS%0RskzcE2U{bu{RgM6^>exv)uk31Ieiu?8o_3k$@ znTY$hzj2zGDU%6IO zBp?6Gg_4O|FzccHxPX9bo=dh3s%FJpfYOQ`C&uOcg0Fv!Oj|;aiDqdMXH{Nd+73a! 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