scholarly journals Multigrid methods and data assimilation ― Convergence study and first experiments on non-linear equations

2011 ◽  
Vol Volume 14 - 2011 - Special... ◽  
Author(s):  
Emilie Neveu ◽  
Laurent Debreu ◽  
François-Xavier Le Dimet
Keyword(s):  
Optimal Control ◽  
Data Assimilation ◽  
Linear Equations ◽  
Computational Cost ◽  
Multigrid Methods ◽  
Advection Equation ◽  
Non Linear ◽  
Discretization Errors ◽  
Correction Step ◽  
The Impact

International audience In order to limit the computational cost of the variational data assimilation process, we investigate the use of multigrid methods to solve the associated optimal control system. On a linear advection equation, we study the impact of the regularization term and the discretization errors on the efficiency of the coarse grid correction step introduced by the multigrid method. We show that even if for a perfect numerical model the optimal control problem leads to the solution of an elliptic system, discretization errors introduce implicit diffusion that can alter the success of the multigrid methods. Then we test the multigrids configuration and the influence of the algorithmic parameters on a non-linear Burgers equation to show that the algorithm is robust and converges much faster than the monogrid one. Afin de limiter le coût de calcul lié aux méthodes variationnelles d’assimilation de données, nous nous intéressons ici à l’utilisation de méthodes multigrilles pour la résolution de systèmes de contrôle optimal. Sur un modèle simple d’advection linéaire, nous étudions l’impact du terme de régularisation du contrôle optimal ainsi que l’impact des erreurs de discrétisation sur l’efficacité de la correction grille grossière introduite par cette méthode. En particulier, nous montrons que pour un modèle numérique parfait, le problème de contrôle optimal est elliptique mais que les erreurs de discrétisation introduisant une diffusion implicite peuvent altérer les performances de la méthode multigrille. Enfin, sur une équation de Burgers, non linéaire, nous étudions l’influence des différents paramètres inhérents aux méthodes multigrilles et montrons que ces méthodes sont robustes et convergent beaucoup plus rapidement que les méthodes monogrilles.

Meteoroid trajectories from BRAMS data

2021 ◽  
Author(s):  
Hervé Lamy
Keyword(s):  
Time Delays ◽  
Specular Reflection ◽  
Linear Equations ◽  
Radio Interferometer ◽  
Additional Information ◽  
Sensitivity Pattern ◽  
Non Linear ◽  
Using Data ◽  
The Impact ◽  
Non Linear Equations

<p>BRAMS (Belgian RAdio Meteor Stations) is a Belgian radio network using forward scatter observations to detect and characterize meteoroids. A dedicated transmitter located in south of Belgium emits a CW signal with no modulation at a frequency of 49.97 MHz and with a power of 130 W. The network comprises currently 35 similar receiving stations located in Belgium and neighboring countries. They use Yagi antennas with a wide sensitivity pattern which therefore provide no information about the directivity of the meteor echoes. One of these stations is however a radio interferometer using the classical Jones configuration and is able to retrieve the direction of the meteor echoes.</p><p>We discuss here a general method to retrieve meteoroid trajectories based solely on time delays measured between meteor echoes recorded at multiple receiving stations. It is based on solving at least 6 non-linear equations to solve for the position of one specular reflection point (3 unknowns) and the 3 components of the speed. This method has also been described recently in Mazur et al (2020) and applied to CMOR data. However, specificities of the CMOR configuration has allowed simplifications that cannot be made with the BRAMS network. In order to maximize the number of meteoroid trajectories with at least 6 stations detecting meteor echoes, a number of additional stations geographically close to each other have been installed in the Limburg province in 2020. Another method to retrieve meteoroid trajectories using data from the radio interferometer and from 3 other stations is also presented.</p><p>We show preliminary results from both methods using also complementary data from the optical CAMS Benelux network.  The CAMS trajectories are used to select specific meteor echoes in the BRAMS data. The time delays between them are computed and used to solve the set of non-linear equations to retrieve the meteoroid trajectory and speed, which are then compared to the CAMS values. This allows us to assess the accuracy of both methods.</p><p>Finally we simulate the impact of using additional information, not currently available but that might become in a near future. This includes data from a monostatic system (a radar nearby our BRAMS transmitter is currently built), from a second radio interferometer (to be located in Limburg and/or near the transmitter), or the total range traveled by the radio wave if a coded CW transmitter such as in Vierinen et al (2016) is used.</p>

Computationally Efficient Combined Plant Design and Controller Optimization Using a Coupling Measure

2012 ◽  
Vol 134 (7) ◽  
Author(s):  
Rakesh Patil ◽  
Zoran Filipi ◽  
Hosam Fathy
Keyword(s):  
Optimal Control ◽  
Computational Cost ◽  
Systems Design ◽  
Plant Design ◽  
Coupling Term ◽  
Control Optimization ◽  
Novel Approach ◽  
Design Variables ◽  
The Impact ◽  
Plant Control

This paper presents a novel approach to the optimization of a dynamic systems design and control. Traditionally, these problems have been solved either sequentially or in a combined manner. We propose a novel approach that uses a previously derived coupling measure to quantify the impact of plant design variables on optimal control cost. This proposed approach has two key advantages. First, because the coupling term quantifies the gradient of the control optimization objective with respect to plant design variables, the approach ensures combined plant/control optimality. Second, because the coupling term equals the integral of optimal control co-states multiplied by static gradient terms that can be computed a priori, the proposed approach is computationally attractive. We illustrate this approach using an example cantilever beam structural design and vibration control problem. The results show significant computational cost improvements compared to traditional combined plant/control optimization. This reduction in computational cost becomes more pronounced as the number of plant design variables increases.

scholarly journals On Efficient Iterative Numerical Methods for Simultaneous Determination of all Roots of Non-Linear Function

2020 ◽  
Author(s):  
Mudassir Shams ◽  
Nazir Mir ◽  
Naila Rafiq
Keyword(s):  
Simultaneous Determination ◽  
Iterative Methods ◽  
Linear Equations ◽  
Basin Of Attraction ◽  
Numerical Test ◽  
Computational Cost ◽  
Single Root ◽  
Non Linear ◽  
Non Linear Equations

We construct a family of two-step optimal fourth order iterative methods for finding single root of non-linear equations. We generalize these methods to simultaneous iterative methods for determining all the distinct as well as multiple roots of single variable non-linear equations. Convergence analysis is present for both cases to show that the order of convergence is four in case of single root finding method and is twelve for simultaneous determination of all roots of non-linear equation. The computational cost, Basin of attraction, efficiency, log of residual and numerical test examples shows, the newly constructed methods are more efficient as compared to the existing methods in literature.

scholarly journals Impact of a Multiple Pendulum with a Non-Linear Contact Force

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1202
Author(s):  
Dan B. Marghitu ◽  
Jing Zhao
Keyword(s):  
Kinetic Energy ◽  
Contact Force ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Kinematic Chain ◽  
Double Pendulum ◽  
Lagrange Equations ◽  
Non Linear ◽  
Impact Kinetic Energy ◽  
The Impact

This article presents a method to solve the impact of a kinematic chain in terms of a non-linear contact force. The nonlinear contact force has different expressions for elastic compression, elasto-plastic compression, and elastic restitution. Lagrange equations of motion are used to obtain the non-linear equations of motion with friction for the collision period. The kinetic energy during the impact is compared with the pre-impact kinetic energy. During the impact of a double pendulum the kinetic energy of the non-impacting link is increasing and the total kinetic energy of the impacting link is decreasing.

Slip Effects on a Ree-Eyring Liquid Peristaltic Flow Towards an Inclined Channel and Variable Liquid Properties

2021 ◽  
Vol 10 (2) ◽  
pp. 246-258
Author(s):  
H. Balachandra ◽  
C. Rajashekhar ◽  
F. Mebarek-Oudina ◽  
G. Manjunatha ◽  
H. Vaidya ◽  
...  
Keyword(s):  
Perturbation Technique ◽  
Linear Equations ◽  
Velocity Slip ◽  
Peripheral Region ◽  
Pictorial Representation ◽  
Linear Partial Differential Equations ◽  
Non Linear ◽  
Flow Phenomena ◽  
The Impact ◽  
Matlab Programming

The current paper aims to model the flow of blood in narrow arteries by taking non-Newtonian Ree-Eyring fluid. The impact of wall properties, slip, mass, and heat transport are considered. In earlier studies, the viscosity of blood was considered a constant parameter. However, many researchers have claimed that the blood’s viscosity in the peripheral region is less than that of the core region. Thus, the present model incorporates the variation in thermal conductivity and viscosity, which takes a vital character in scrutinizing blood flow in narrow arteries. The mathematical model is developed on low Reynold’s number approximation and long wavelength, leading to the governing non-linear equations representing the flow phenomena. The governing non-linear partial differential equations are resolved by utilizing the series solution method (perturbation technique). MATLAB programming has been used to plot the pictorial representation of all the parameters through the flow quantities. The investigation shows that the variable velocity slip parameter and viscosity develops the velocity profiles. Further, the liquid parameter can be used to control the dimension of the trapped bolus. Moreover, the obtained outcomes aid in recognizing the flow of blood in micro arteries.

scholarly journals Impact from a Volumetric Radar-Sampling Operator for Radial Velocity Observations within EnKF Supercell Assimilation

2012 ◽  
Vol 29 (10) ◽  
pp. 1417-1427 ◽  
Author(s):  
Therese E. Thompson ◽  
Louis J. Wicker ◽  
Xuguang Wang
Keyword(s):  
Data Assimilation ◽  
Computational Cost ◽  
Radar Data ◽  
Volume Averaging ◽  
Doppler Velocity ◽  
Observation Operator ◽  
Model State ◽  
Point Operator ◽  
The Impact ◽  
Ensemble Kalman Filtering

Abstract Maximizing the accuracy of ensemble Kalman filtering (EnKF) radar data assimilation requires that the observation operator sample the model state in the same manner that the radar sampled the atmosphere. It may therefore be desirable to include volume averaging and power weighting in the observation operator. This study examines the impact of including radar-sampling effects in the Doppler velocity observation operator on EnKF analyses and forecasts. Locally substantial differences are found between a simple point operator and a realistic radar-sampling operator when they are applied to the model state at a single time. However, assimilation results indicate that the radar-sampling operator does not substantially improve the EnKF analyses or forecasts, and it greatly increases the computational cost of the data assimilation.

scholarly journals Assimilating compact phase space retrievals (CPSRs): comparison with independent observations (MOZAIC in situ and IASI retrievals) and extension to assimilation of truncated retrieval profiles

2018 ◽  
Vol 11 (9) ◽  
pp. 3727-3745 ◽  
Author(s):  
Arthur P. Mizzi ◽  
David P. Edwards ◽  
Jeffrey L. Anderson
Keyword(s):  
Carbon Monoxide ◽  
Phase Space ◽  
Data Assimilation ◽  
Computational Cost ◽  
Lower Troposphere ◽  
Forecast Skill ◽  
Promising Alternative ◽  
Independent Observations ◽  
The Impact

Abstract. Assimilation of atmospheric composition retrievals presents computational challenges due to their high data volume and often sparse information density. Assimilation of compact phase space retrievals (CPSRs) meets those challenges and offers a promising alternative to assimilation of raw retrievals at reduced computational cost (Mizzi et al., 2016). This paper compares analysis and forecast results from assimilation of Terra/Measurement of Pollution in the Troposphere (MOPITT) carbon monoxide (CO) CPSRs with independent observations. We use MetOp-A/Infrared Atmospheric Sounding Interferometer (IASI) CO retrievals and Measurement of OZone, water vapor, carbon monoxide, and nitrogen oxides by in-service AIrbus airCraft (MOZAIC) in situ CO profiles for our independent observation comparisons. Generally, the results confirm that assimilation of MOPITT CPSRs improves the Weather Research and Forecasting model with chemistry coupled to the ensemble Kalman filter data assimilation from the Data Assimilation Research Testbed (WRF-Chem/DART) analysis fit and forecast skill at a reduced computational cost compared to assimilation of raw retrievals. Comparison with the independent observations shows that assimilation of MOPITT CO generally improved the analysis fit and forecast skill in the lower troposphere but degraded it in the upper troposphere. We attribute that degradation to assimilation of MOPITT CO retrievals with a possible bias of  ∼ 14 % above 300 hPa. To discard the biased retrievals, in this paper, we also extend CPSRs to assimilation of truncated retrieval profiles (as opposed to assimilation of full retrieval profiles). Those results show that not assimilating the biased retrievals (i) resolves the upper tropospheric analysis fit degradation issue and (ii) reduces the impact of assimilating the remaining unbiased retrievals because the total information content and vertical sensitivities are changed.

scholarly journals A master exceptional field theory

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Guillaume Bossard ◽  
Axel Kleinschmidt ◽  
Ergin Sezgin
Keyword(s):  
Field Theory ◽  
Gauge Invariance ◽  
Sigma Model ◽  
Linear Equations ◽  
Field Theories ◽  
Gauge Invariant ◽  
Non Linear ◽  
Non Linear Equations

Abstract We construct a pseudo-Lagrangian that is invariant under rigid E11 and transforms as a density under E11 generalised diffeomorphisms. The gauge-invariance requires the use of a section condition studied in previous work on E11 exceptional field theory and the inclusion of constrained fields that transform in an indecomposable E11-representation together with the E11 coset fields. We show that, in combination with gauge-invariant and E11-invariant duality equations, this pseudo-Lagrangian reduces to the bosonic sector of non-linear eleven-dimensional supergravity for one choice of solution to the section condi- tion. For another choice, we reobtain the E8 exceptional field theory and conjecture that our pseudo-Lagrangian and duality equations produce all exceptional field theories with maximal supersymmetry in any dimension. We also describe how the theory entails non-linear equations for higher dual fields, including the dual graviton in eleven dimensions. Furthermore, we speculate on the relation to the E10 sigma model.

Sign in / Sign up

Export Citation Format

Share Document