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∞ Filtering
Published in Jitendra R. Raol, Girija Gopalratnam, Bhekisipho Twala, Nonlinear Filtering, 2017
Jitendra R. Raol, Girija Gopalratnam, Bhekisipho Twala
These equations are similar to that of the KF given in Chapter 2 for both the a priori and a posteriori parts that led to the extensions of KF to Krein space. In essence, H∞ filters are KFs in a Krein space. Although, the similarity of these equations with the KF equations is evident, the following differences between the two are observed [39,40]:
Rocking of Non-symmetric Rigid Blocks in Buildings Considering Effects Associated with Dynamic Soil-Structure Interaction
Published in Journal of Earthquake Engineering, 2018
Miguel A. Jaimes, Cesar Arredondo, Luciano Fernández-Sola
Equation (7) is solved through the average central difference method, which is based on an approximation in finite differences of displacement derivatives (velocity and acceleration) with respect to time. The and responses at a time step are calculated from the equation of dynamic motion, its derivatives, and the known response in previous time steps; in our case from the and values that correspond to half-time steps a priori and a posteriori to the time of interest s. It is possible to use the above-mentioned method only if the time increments are modified in such a way that , and the half-time steps are redefined for the angular velocity and rotation at time as follows:
Identifying an unknown source term of a parabolic equation in Banach spaces
Published in Applicable Analysis, 2022
Nguyen Van Duc, Nguyen Van Thang, Luong Duy Nhat Minh, Nguyen Trung Thành
We proposed a regularization method for an inverse source problem in the Banach space setting and proved the Hölder-type error estimates for the regularized solution using both a priori and a posteriori parameter choice rules. Numerical examples showed good reconstruction results for a simple case. Numerical realization in more general cases is under consideration and will be reported in a future work.
An Efficient Machine-Learning Approach for PDF Tabulation in Turbulent Combustion Closure
Published in Combustion Science and Technology, 2021
Rishikesh Ranade, Genong Li, Shaoping Li, Tarek Echekki
The main objective of this work is to introduce an adaptive, computationally efficient MLP-SOM framework to train multi-dimensional PDF tables and explore its feasibility in realistic scenarios where an acceptable solution accuracy is required without expensive computations. Several a priori and a posteriori tests are carried out to analyze this framework in terms of solution accuracy, computational cost, training cost, memory storage, and long-time error propagation during numerical simulations. The following conclusions are drawn from this work: The adaptive MLP-SOM algorithm identifies the optimum number of neurons required to fit a thermo-chemical quantity table for a specific network architecture. The simpler networks obtained to ensure that the training and interpolation time are significantly reduced and a need for restarting the training is eliminated.The training time using SOM-based clustering and scalar grouping is around 3 times faster than using a conventional MLP or ANN training methodology. The capability of parallelization over clusters or species groups provides further speed-up and makes training negligible in comparison to the total simulation time. Moreover, the interpolation time using MLP-SOM based functional evaluation matches well with the linear interpolation method which is a tremendous improvement over previous ANN models in this context (Ihme, Schmitt, Pitsch 2009).The RANS and LES studies of flame DLR-A show great agreement between the linear PDF interpolation and MLP-SOM based functional evaluation of thermo-chemical quantities including several minor species. The results at multiple radial and axial locations match well in both cases. Moreover, the CFD solution times are comparable for both methods.Finally, the memory requirements of the MLP-SOM method are about 1000 times smaller than those of a PDF table and therefore can easily accommodate a larger number of parameterized quantities.