Adaptive Filter for High Dimensional Inverse Engineering Problems: From Theory to Practical Implementation

In this contribution, the problem of data assimilation as state estimation for dynamical systems under uncertainties is addressed. This emphasize is put on high-dimensional systems context. Major difficulties in the design of data assimilation algorithms is a concern for computational resources (computational power and memory) and uncertainties (system parameters, statistics of model, and observational errors). The idea of the adaptive filter will be given in detail to see how it is possible to overcome uncertainties as well as to explain the main principle and tools for implementation of the adaptive filter for complex dynamical systems. Simple numerical examples are given to illustrate the principal differences of the AF with the Kalman filter and other methods. The simulation results are presented to compare the performance of the adaptive filter with the Kalman filter.

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MODELLING OF HIGH-DIMENSIONAL DIFFUSION STOCHASTIC PROCESS WITH NONLINEAR COEFFICIENTS FOR ENGINEERING APPLICATIONS — PART I: APPROXIMATIONS FOR EXPECTATION AND VARIANCE OF NONSTATIONARY PROCESS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202599000543 ◽

1999 ◽

Vol 09 (08) ◽

pp. 1201-1246 ◽

Cited By ~ 5

Author(s):

Y. V. MAMONTOV ◽

M. WILLANDER ◽

T. LEWIN

Keyword(s):

Combined Treatment ◽

Numerical Approach ◽

Nonstationary Process ◽

High Dimensional ◽

Dimensional Euclidean Space ◽

Practical Implementation ◽

Dimensional Diffusion ◽

Analytical Expressions ◽

Ode Systems ◽

Real World Problems

This work is devoted to diffusion stochastic processes (DSPs) with nonlinear coefficients in n-dimensional Euclidean space at high n (n is much greater than a few units). It deals with expectation and variance of a nonstationary process whereas our accompanying work deals with covariance and spectral density of a stationary process. Combined, analytical-numerical approach is a reasonable and perhaps the only way to treat high-dimensional DSPs in practice. Each of the above works develops the corresponding parts of the analytical basis for this combined treatment. The present work proposes approximate analytical expressions for the expectation and variance in the form of two ordinary differential equation (ODE) systems. They are derived within DSP theory without any techniques directly related to stochastic differential equations. Both ODE systems allow for space nonhomogeneities of the diffusion and damping matrixes and thereby do take nonlinearities of the DSP coefficients into account. Some related topics like invariant processes and the aspects of practical implementation of the above expressions are discussed as well. A proper attention is paid to formulation of some features important in applications of DSPs to the real-world problems. The results of this work can equally be used in various engineering fields.

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A Comparison Study on Performance of an Adaptive Filter with Other Estimation Methods for State Estimation in High-Dimensional System

Advances in Statistical Methodologies and Their Application to Real Problems ◽

10.5772/67005 ◽

2017 ◽

Author(s):

Hong Son Hoang ◽

Remy Baraille

Keyword(s):

State Estimation ◽

Adaptive Filter ◽

Estimation Methods ◽

High Dimensional ◽

Dimensional System ◽

Comparison Study

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The role of optimal perturbations in improvement of stabil- ity and forecasting capacity of adaptive filter for high di- mensional systems

MATEC Web of Conferences ◽

10.1051/matecconf/201821002033 ◽

2018 ◽

Vol 210 ◽

pp. 02033

Author(s):

Hong Son Hoang ◽

Rémy Baraille

Keyword(s):

Dynamical System ◽

Invariant Subspace ◽

Adaptive Filter ◽

Numerical Algorithms ◽

High Dimensional ◽

Optimal Perturbation ◽

Different Types ◽

Theoretical Results ◽

Very High

This paper is devoted to an optimal perturbation (OP) which allows to qualify a capacity of the dynamical system to predict more or less correctly the system behaviour in the future. The different types of OPs, deterministic, stochastic, OP in an invariant subspace, are introduced. The theoretical results on the optimality of the introduced OPs are presented. The different simple and efficient numerical algorithms for computation of the OPs are outlined which constitute a basis for implementation of a stable adaptive filter in a very high dimensional environment.

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A new reduced-order adaptive filter for state estimation in high-dimensional systems

Automatica ◽

10.1016/s0005-1098(97)00069-1 ◽

1997 ◽

Vol 33 (8) ◽

pp. 1475-1498 ◽

Cited By ~ 15

Author(s):

S. Hoang ◽

P. De Mey ◽

O. Talagrand ◽

R. Baraille

Keyword(s):

State Estimation ◽

Adaptive Filter ◽

High Dimensional ◽

Reduced Order

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Towards a Vector Field Based Approach to the Proper Generalized Decomposition (PGD)

Mathematics ◽

10.3390/math9010034 ◽

2020 ◽

Vol 9 (1) ◽

pp. 34

Author(s):

Antonio Falcó ◽

Lucía Hilario ◽

Nicolás Montés ◽

Marta C. Mora ◽

Enrique Nadal

Keyword(s):

Vector Field ◽

Optimization Problem ◽

Stationary Points ◽

High Dimensional ◽

Practical Implementation ◽

Approximation Procedure ◽

Proper Generalized Decomposition ◽

Rank One ◽

Fixed Rank ◽

Low Dimensional

A novel algorithm called the Proper Generalized Decomposition (PGD) is widely used by the engineering community to compute the solution of high dimensional problems. However, it is well-known that the bottleneck of its practical implementation focuses on the computation of the so-called best rank-one approximation. Motivated by this fact, we are going to discuss some of the geometrical aspects of the best rank-one approximation procedure. More precisely, our main result is to construct explicitly a vector field over a low-dimensional vector space and to prove that we can identify its stationary points with the critical points of the best rank-one optimization problem. To obtain this result, we endow the set of tensors with fixed rank-one with an explicit geometric structure.

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