scholarly journals Behavior of the iterative ensemble-based variational method in nonlinear problems

2021 ◽  
Vol 28 (1) ◽  
pp. 93-109
Author(s):  
Shin'ya Nakano
Keyword(s):  
Data Assimilation ◽  
Variational Method ◽  
Local Maximum ◽  
Nonlinear Problems ◽  
Dimensional Subspace ◽  
High Dimensional ◽  
Dimensional Approximation ◽  
Monotonic Convergence ◽  
System Nonlinearity ◽  
Lower Dimensional

Abstract. The behavior of the iterative ensemble-based data assimilation algorithm is discussed. The ensemble-based method for variational data assimilation problems, referred to as the 4D ensemble variational method (4DEnVar), is a useful tool for data assimilation problems. Although the 4DEnVar is derived based on a linear approximation, highly uncertain problems, in which system nonlinearity is significant, are solved by applying this method iteratively. However, the ensemble-based methods basically seek the solution within a lower-dimensional subspace spanned by the ensemble members. It is not necessarily trivial how high-dimensional problems can be solved with the ensemble-based algorithm which employs the lower-dimensional approximation based on the ensemble. In the present study, an ensemble-based iterative algorithm is reformulated to allow us to analyze its behavior in high-dimensional nonlinear problems. The conditions for monotonic convergence to a local maximum of the objective function are discussed in a high-dimensional context. It is shown that the ensemble-based algorithm can solve high-dimensional problems by distributing the ensemble in different subspace at each iteration. The findings as the results of the present study were also experimentally supported.

scholarly journals Behavior of the iterative ensemble-based variational method in nonlinear problems

2020 ◽  
Author(s):  
Shin'ya Nakano
Keyword(s):  
Data Assimilation ◽  
Variational Method ◽  
Objective Function ◽  
Iterative Algorithm ◽  
Linear Approximation ◽  
Local Maximum ◽  
Nonlinear Problems ◽  
Variational Data Assimilation ◽  
Monotonic Convergence ◽  
System Nonlinearity

Abstract. The behavior of the iterative ensemble-based data assimilation algorithm is discussed. The ensemble-based method for variational data assimilation problems, referred to as the 4-dimensional ensemble variational method (4DEnVar), is a useful tool for data assimilation problems. Although the 4DEnVar is derived based on a linear approximation, highly uncertain problems, where system nonlinearity is significant, are solved by applying this method iteratively. However, it is not necessarily trivial how the algorithm works in highly uncertain problems where nonlinearity is not negligible. In the present study, an ensemble-based iterative algorithm is reformulated to allow us to analyze its behavior in nonlinear problems. The conditions for monotonic convergence to a local maximum of the objective function are discussed in nonlinear context. The findings as the results of the present study were also experimentally supported.

scholarly journals Discovering a sparse set of pairwise discriminating features in high-dimensional data

2020 ◽  
Author(s):  
Samuel Melton ◽  
Sharad Ramanathan
Keyword(s):  
Single Cell ◽  
Dimensional Space ◽  
Cell Types ◽  
Dimensional Subspace ◽  
Supplementary Information ◽  
High Dimensional ◽  
Technological Advances ◽  
Data Points ◽  
Low Dimensional ◽  
Sparse Set

Abstract Motivation Recent technological advances produce a wealth of high-dimensional descriptions of biological processes, yet extracting meaningful insight and mechanistic understanding from these data remains challenging. For example, in developmental biology, the dynamics of differentiation can now be mapped quantitatively using single-cell RNA sequencing, yet it is difficult to infer molecular regulators of developmental transitions. Here, we show that discovering informative features in the data is crucial for statistical analysis as well as making experimental predictions. Results We identify features based on their ability to discriminate between clusters of the data points. We define a class of problems in which linear separability of clusters is hidden in a low-dimensional space. We propose an unsupervised method to identify the subset of features that define a low-dimensional subspace in which clustering can be conducted. This is achieved by averaging over discriminators trained on an ensemble of proposed cluster configurations. We then apply our method to single-cell RNA-seq data from mouse gastrulation, and identify 27 key transcription factors (out of 409 total), 18 of which are known to define cell states through their expression levels. In this inferred subspace, we find clear signatures of known cell types that eluded classification prior to discovery of the correct low-dimensional subspace. Availability and implementation https://github.com/smelton/SMD. Supplementary information Supplementary data are available at Bioinformatics online.

scholarly journals A nudging-based data assimilation method: the Back and Forth Nudging (BFN) algorithm

2008 ◽  
Vol 15 (2) ◽  
pp. 305-319 ◽  
Author(s):  
D. Auroux ◽  
J. Blum
Keyword(s):  
Data Assimilation ◽  
Variational Method ◽  
Burgers Equation ◽  
Reconstruction Error ◽  
Sea Surface ◽  
Reference Trajectory ◽  
Discrete Observations ◽  
Lorenz Model ◽  
Time Period ◽  
The Difference

Abstract. This paper deals with a new data assimilation algorithm, called Back and Forth Nudging. The standard nudging technique consists in adding to the equations of the model a relaxation term that is supposed to force the observations to the model. The BFN algorithm consists in repeatedly performing forward and backward integrations of the model with relaxation (or nudging) terms, using opposite signs in the direct and inverse integrations, so as to make the backward evolution numerically stable. This algorithm has first been tested on the standard Lorenz model with discrete observations (perfect or noisy) and compared with the variational assimilation method. The same type of study has then been performed on the viscous Burgers equation, comparing again with the variational method and focusing on the time evolution of the reconstruction error, i.e. the difference between the reference trajectory and the identified one over a time period composed of an assimilation period followed by a prediction period. The possible use of the BFN algorithm as an initialization for the variational method has also been investigated. Finally the algorithm has been tested on a layered quasi-geostrophic model with sea-surface height observations. The behaviours of the two algorithms have been compared in the presence of perfect or noisy observations, and also for imperfect models. This has allowed us to reach a conclusion concerning the relative performances of the two algorithms.

scholarly journals Fast high-dimensional approximation with sparse occupancy trees

2011 ◽  
Vol 235 (8) ◽  
pp. 2063-2076 ◽  
Author(s):  
Peter Binev ◽  
Wolfgang Dahmen ◽  
Philipp Lamby
Keyword(s):  
High Dimensional ◽  
Dimensional Approximation

A steady vortex ring in Poiseuille flow and rearrangements of a function

2006 ◽  
Vol 462 (2068) ◽  
pp. 1235-1253 ◽  
Author(s):  
Dirar Rebah
Keyword(s):  
Kinetic Energy ◽  
Variational Method ◽  
Fluid Mechanics ◽  
Vortex Ring ◽  
Ideal Fluid ◽  
Poiseuille Flow ◽  
Nonlinear Problems ◽  
Insight Into

This study proves the existence of a steady vortex ring of an ideal fluid in Poiseuille flow. The method that was used is a variational method proposed by Benjamin (Benjamin 1976 The alliance of practical and analytical insight into the nonlinear problems of fluid mechanics , vol. 503, pp. 8–29), in which a steady vortex ring can be obtained as a maximizer of a functional that is related to kinetic energy and the impulse over the set of rearrangements of a prescribed function.

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