Strategies for solving index one DAE with non-negative constraints: Application to liquid–liquid extraction

Métivier, Ludovic; Montarnal, Philippe

doi:10.1016/j.jcp.2011.12.039

Cited by 7 publications

(2 citation statements)

References 18 publications

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“…where A, B and C denote the concentrations of [A], [B] and [C], respectively. In our simulations, we set A(0) = 1, B(0) = 0 as initial conditions of the concentrations and a very large time interval [0 4 • 10 11 ] as proposed in [54]. Table 1 summarizes the l 2 , l ∞ and mean absolute (MAE) approximation errors with respect to the reference solution ].…”

Section: Case Study 1: the Robertson Index-1 Daesmentioning

confidence: 99%

Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEs

Fabiani¹,

Galaris²,

Russo³

et al. 2022

Preprint

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We address a physics-informed neural network based on the concept of random projections for the numerical solution of initial value problems of nonlinear ODEs in linear-implicit form and index-1 DAEs, which may also arise from the spatial discretization of PDEs. The proposed scheme has a single hidden layer with appropriately randomly parametrized Gaussian kernels and a linear output layer, while the internal weights are fixed to ones. The unknown weights between the hidden and output layer are computed by Newton's iterations, using the Moore-Penrose pseudoinverse for low to medium scale, and sparse QR decomposition with L 2 regularization for medium to large scale systems. To deal with stiffness and sharp gradients, we thus propose an variable step size scheme based on the elementary local error control algorithm for adjusting the step size of integration and address a natural continuation method for providing good initial guesses for the Newton iterations. Building on previous works on random projections, we prove the approximation capability of the scheme for ODEs in the canonical form and index-1 DAEs in the semiexplicit form. The "optimal" bounds of the uniform distribution from which the values of the shape parameters of the Gaussian kernels are sampled are "parsimoniously" chosen based on the bias-variance trade-off decomposition, thus using the stiff van der Pol model as the reference solution for this task. The optimal bounds are fixed once and for all the problems studied here. In particular, the performance of the scheme is assessed through seven benchmark problems. Namely, we considered four index-1 DAEs, the Robertson model, a no autonomous model of five DAEs describing the motion of a bead on a rotating needle, a non autonomous model of six DAEs describing a power discharge control problem, the chemical so-called Akzo Nobel problem and three stiff problems, the Belousov-Zhabotinsky model, the Allen-Cahn PDE phase-field model and the Kuramoto-Sivashinsky PDE giving rise to chaotic dynamics. The efficiency of the scheme in terms of both numerical accuracy and computational cost is compared with three stiff/DAE solvers (ode23t, ode23s, ode15s) of the MATLAB ODE suite. Our results show that, the proposed scheme outperforms the aforementioned stiff solvers in several * Corresponding author A PREPRINT -MARCH 14, 2022 cases, especially in regimes where high stiffness and/or sharp gradients arise, in terms of numerical accuracy, while the computational costs are for any practical purposes comparable.

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Section: Case Study 1: the Robertson Index-1 Daesmentioning

confidence: 99%

Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEs

Fabiani¹,

Galaris²,

Russo³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Theory, practice of applications, and R programs for solution of SDAEs are provided in [10] and earlier publications in this series, and publications [11] and [12]. Algorithms aimed at the inclusion of the non-negativity constraints are considered in [13].…”

Section: Differential-algebraic Equations Of Locally Optimal Trajectomentioning

confidence: 99%

Environmental protection via optimal global economic restructuring

Vaninsky

2019

MethodsX

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Multiobjective restructuring aimed at green economic growth

Vaninsky

2021

Environ Syst Decis

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Strategies for solving index one DAE with non-negative constraints: Application to liquid–liquid extraction

Abstract: To cite this version:Ludovic Métivier, Philippe Montarnal. Strategies for solving index one DAE with non-negative constraints: Application to liquid-liquid extraction. Journal of Computational Physics, Elsevier, 2012, 231 (7)

Cited by 7 publications

References 18 publications

Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEs

Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEs

Environmental protection via optimal global economic restructuring

Multiobjective restructuring aimed at green economic growth

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