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Domain Decomposition: A Unified Approach for Solving Fluid Mechanics Problems on Parallel Computers
Published in Hojjat Adeli, Parallel Processing in Computational Mechanics, 2020
A uniform grid is placed on Ω with Δx = Δy = 1/16. Algorithm (21) is used to solve Eq. (22), where the Δ operators are replaced by central difference operators. The domain decomposition of Ω is given by Figure 4, where xk =i Δx, xi = j Δx for some integers j < i. The individual subproblems in algorithm (21) are solved to an error tolerance of 10− 11 using the preconditioned conjugate gradient method with a point-Jacobi preconditioner. In Table 3 we compare the number of iterations for algorithms DD (β1= β2 = 0, α1 = α2 = 1) and DN (β1 = α2 = 0, α1 = β2 = 1) to achieve . In all cases .
Bayesian Classification of Genomic Big Data
Published in Ervin Sejdić, Tiago H. Falk, Signal Processing and Machine Learning for Biomedical Big Data, 2018
Ulisses M. Braga-Neto, Emre Arslan, Upamanyu Banerjee, Arghavan Bahadorinejad
First, we calibrate k, θ, and φ using the control sample only, since these parameters are common across control and treatment populations and f has not been calibrated yet. The procedure used is displayed in Algorithm 1. In this algorithm, ϵ is the error tolerance. It has been proved [53] that smaller ϵ gives better approximation of posterior p(k|Sn). However, this must be balanced against the possibility that P(||T(S0(t)),T(S0)||<ϵ)≈0, which would prevent convergence to the posterior.
Continuum and Atomic-Scale Finite Element Modeling of Multilayer Self-Positioning Nanostructures
Published in Sarhan M. Musa, Computational Finite Element Methods in Nanotechnology, 2013
At each updated configuration, the tangent stiffness matrix and the load vector should be calculated using Equations 6.79 and 6.80. In the iteration algorithm (6.81), e is an error tolerance and g is a function for estimating the load relaxation factor a. In general, initial configuration of atoms leads to large forces due to steep energy descent, and thus a may be small at the first step and gets closer to 1 as current position gets closer to the equilibrium atomic configuration where energy descent is small (Figure 6.8). Constant α factor throughout overall solution can cause unnecessary iterations or can lead to divergence, because derivatives (slope) of the interaction potential energy usually become smaller as current configuration gets closer to the equilibrium where loading becomes zero.
Scrutinisation of Chebyshev collocation method for mass transfer on a continuous flat plate moving in parallel to a free stream in the presence of a chemical reaction
Published in International Journal of Ambient Energy, 2023
Vishwanath B. Awati, Akash Goravar, Mahesh Kumar N, Ali J. Chamkha
To obtain CCM results in the present study, minimum 32 terms are required in TSCSE (i.e. N = 31). As N = 34, the time taken by algorithm to obtain the required values of skin-friction for upper branch solution is approximately 45.57 s and for lower branch solution it is approximately 23.8 s. To analyse the convergence of CCM results, attempt is made to increase the degree of polynomial, i.e. by taking sufficiently large number of terms in Equation (21) (i.e. approximately N = 45). In all the computations is the default error tolerance set for this algorithm. The accurate solutions of are obtained for different values of and are presented in Table 1 along with the number of iterations. For this, the algorithm performs approximately same number of iterations, viz. N = 34, but time consumed for upper branch solution is approximately 147.07 s and lower branch solution it requires 71.75 s. This increase in time is acceptable as the number of unknown Chebyshev coefficients is increased from 35 to 46. Also, the number of equations in the system (31) and (37) increases accordingly.
Mathematical programs with equilibrium constraints: a sequential optimality condition, new constraint qualifications and algorithmic consequences
Published in Optimization Methods and Software, 2021
The complementary-penalty method consists in solving where is a given penalty parameter. In [40], the authors used an interior-point method for solving (60). That algorithm is called Algorithm I. For an error tolerance and a barrier parameter , Algorithm I tries to find a penalty parameter and vectors satisfying , , , , such that where , , . Theorem 3.4 of [40] guarantees convergence to a C-stationary point if MPEC-LICQ holds at the limit point and to S-stationary point if additionally we require that , . Here, we show convergence to M-stationary point using MPEC-CCP (strictly weaker than MPEC-LICQ) under some assumptions.
Localization strategies for robotic endoscopic capsules: a review
Published in Expert Review of Medical Devices, 2019
Federico Bianchi, Antonino Masaracchia, Erfan Shojaei Barjuei, Arianna Menciassi, Alberto Arezzo, Anastasios Koulaouzidis, Danail Stoyanov, Paolo Dario, Gastone Ciuti
In 2009, Yang et al. proposed a method able to estimate the six DoFs of the capsule, that is, the 3D position and the 3D orientation, by using a rectangular IPM and a Particle Swarm Optimizer (PSO) [19]. Using a dipole-dipole magnetic model, the authors derived the mathematical model of the magnetic flux generated by a rectangular magnet. Then, by using an array of 16 magneto-resistive sensors, the capsule pose was obtained by solving the nonlinear relative objective function. Also, in this case, the Levenberg-Marquardt optimizer was confirmed to be the most proper algorithm in terms of computational cost and error tolerance. Simulation results showed an average execution time of 0.17 s in case of an average estimation error of 3.9 mm and 5.06°, while an average execution time of 0.63 s in case of an average estimation error of 0.59 mm and 0.66° in position and orientation, respectively.