scholarly journals Scaling effects on the periodic homogenization of a reaction-diffusion-convection problem posed in homogeneous domains connected by a thin composite layer

2021 ◽  
Vol 80 (1) ◽  
pp. 157-200
Author(s):  
Vishnu Raveendran ◽  
Emilio Cirillo ◽  
Ida de Bonis ◽  
Adrian Muntean
Keyword(s):  
Thin Layer ◽  
Composite Layer ◽  
Exclusion Process ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Model Parameters ◽  
Periodic Homogenization ◽  
Scaling Effects ◽  
Material Interfaces ◽  
Non Linear

We study the question of periodic homogenization of a variably scaled reaction-diffusion problem with non-linear drift posed for a domain crossed by a flat composite thin layer. The structure of the non-linearity in the drift was obtained in earlier works as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) for a population of interacting particles crossing a domain with obstacle. Using energy-type estimates as well as concepts like thin-layer convergence and two-scale convergence, we derive the homogenized evolution equation and the corresponding effective model parameters for a regularized problem. Special attention is paid to the derivation of the effective transmission conditions across the separating limit interface in essentially two different situations: (i) finitely thin layer and (ii) infinitely thin layer. This study should be seen as a preliminary step needed for the investigation of averaging fast non-linear drifts across material interfaces—a topic with direct applications in the design of thin composite materials meant to be impenetrable to high-velocity impacts.

scholarly journals Singular limit for reactive transport through a thin heterogeneous layer including a nonlinear diffusion coefficient

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Markus Gahn
Keyword(s):  
Thin Layer ◽  
Nonlinear Diffusion ◽  
Reactive Transport ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Singular Limit ◽  
Transport Processes ◽  
Homogenization Theory ◽  
Macroscopic Model ◽  
Heterogeneous Structure

<p style='text-indent:20px;'>Reactive transport processes in porous media including thin heterogeneous layers play an important role in many applications. In this paper, we investigate a reaction-diffusion problem with nonlinear diffusion in a domain consisting of two bulk domains which are separated by a thin layer with a periodic heterogeneous structure. The thickness of the layer, as well as the periodicity within the layer are of order <inline-formula><tex-math id="M1">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> is much smaller than the size of the bulk domains. For the singular limit <inline-formula><tex-math id="M3">\begin{document}$ \epsilon \to 0 $\end{document}</tex-math></inline-formula>, when the thin layer reduces to an interface, we rigorously derive a macroscopic model with effective interface conditions between the two bulk domains. Due to the oscillations within the layer, we have the combine dimension reduction techniques with methods from the homogenization theory. To cope with these difficulties, we make use of the two-scale convergence in thin heterogeneous layers. However, in our case the diffusion in the thin layer is low and depends nonlinearly on the concentration itself. The low diffusion leads to a two-scale limit depending on a macroscopic and a microscopic variable. Hence, weak compactness results based on standard <i>a priori</i> estimates are not enough to pass to the limit <inline-formula><tex-math id="M4">\begin{document}$ \epsilon \to 0 $\end{document}</tex-math></inline-formula> in the nonlinear terms. Therefore, we derive strong two-scale compactness results based on a variational principle. Further, we establish uniqueness for the microscopic and the macroscopic model.</p>

Adaptive control of time-varying non-linear plants by speed-gradient algorithms

2019 ◽  
pp. 37-44 ◽  
Author(s):  
O. P. Tomchina ◽  
D. N. Polyakhov ◽  
O. I. Tokareva ◽  
A. L. Fradkov
Keyword(s):  
Adaptive Control ◽  
Adaptive Systems ◽  
Reference Model ◽  
Maximum Deviation ◽  
Model Parameters ◽  
Time Varying ◽  
System Solution ◽  
Non Linear ◽  
Initial Uncertainty ◽  
Speed Gradient

Introduction: The motion of many real world systems is described by essentially non-linear and non-stationary models. A number of approaches to the control of such plants are based on constructing an internal model of non-stationarity. However, the non-stationarity model parameters can vary widely, leading to more errors. It is only assumed in this paper that the change rate of the object parameters is limited, while the initial uncertainty can be quite large.Purpose: Analysis of adaptive control algorithms for non-linear and time-varying systems with an explicit reference model, synthesized by the speed gradient method.Results: An estimate was obtained for the maximum deviation of a closed-loop system solution from the reference model solution. It is shown that with sufficiently slow changes in the parameters and a small initial uncertainty, the limit error in the system can be made arbitrarily small. Systems designed by the direct approach and systems based on the identification approach are both considered. The procedures for the synthesis of an adaptive regulator and analysis of the synthesized system are illustrated by an example.Practical relevance: The obtained results allow us to build and analyze a broad class of adaptive systems with reference models under non-stationary conditions.

Nearly Exact Bayesian Estimation of Non-linear No-Arbitrage Term-Structure Models*

2021 ◽  
Author(s):  
Marcello Pericoli ◽  
Marco Taboga
Keyword(s):  
Bayesian Estimation ◽  
Term Structure ◽  
Broad Class ◽  
Model Parameters ◽  
State Variables ◽  
Computationally Efficient ◽  
Macroeconomic Factors ◽  
No Arbitrage ◽  
Term Structure Models ◽  
Non Linear

Abstract We propose a general method for the Bayesian estimation of a very broad class of non-linear no-arbitrage term-structure models. The main innovation we introduce is a computationally efficient method, based on deep learning techniques, for approximating no-arbitrage model-implied bond yields to any desired degree of accuracy. Once the pricing function is approximated, the posterior distribution of model parameters and unobservable state variables can be estimated by standard Markov Chain Monte Carlo methods. As an illustrative example, we apply the proposed techniques to the estimation of a shadow-rate model with a time-varying lower bound and unspanned macroeconomic factors.

scholarly journals Reduced basis approximations of the solutions to spectral fractional diffusion problems

2020 ◽  
Vol 28 (3) ◽  
pp. 147-160
Author(s):  
Andrea Bonito ◽  
Diane Guignard ◽  
Ashley R. Zhang
Keyword(s):  
Computational Cost ◽  
Fractional Power ◽  
Reaction Diffusion ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Quadrature Point ◽  
Reduced Basis ◽  
Fast Evaluation ◽  
Bottle Neck ◽  
Diffusion Problems

AbstractWe consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction–diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction–diffusion problems. The reduced basis does not depend on the fractional power s for 0 < smin ⩽ s ⩽ smax < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.

scholarly journals Identification of mechanical properties of arteries with certification of global optimality

2021 ◽  
Author(s):  
Jan-Lucas Gade ◽  
Carl-Johan Thore ◽  
Jonas Stålhand
Keyword(s):  
Global Solution ◽  
Branch And Bound ◽  
Clinical Data ◽  
Minimization Problem ◽  
Branch And Bound Algorithm ◽  
Stationary Points ◽  
Model Parameters ◽  
Global Optimality ◽  
Clinical Value ◽  
Non Linear

AbstractIn this study, we consider identification of parameters in a non-linear continuum-mechanical model of arteries by fitting the models response to clinical data. The fitting of the model is formulated as a constrained non-linear, non-convex least-squares minimization problem. The model parameters are directly related to the underlying physiology of arteries, and correctly identified they can be of great clinical value. The non-convexity of the minimization problem implies that incorrect parameter values, corresponding to local minima or stationary points may be found, however. Therefore, we investigate the feasibility of using a branch-and-bound algorithm to identify the parameters to global optimality. The algorithm is tested on three clinical data sets, in each case using four increasingly larger regions around a candidate global solution in the parameter space. In all cases, the candidate global solution is found already in the initialization phase when solving the original non-convex minimization problem from multiple starting points, and the remaining time is spent on increasing the lower bound on the optimal value. Although the branch-and-bound algorithm is parallelized, the overall procedure is in general very time-consuming.

scholarly journals Functional A Posteriori Error Control for Conforming Mixed Approximations of Coercive Problems with Lower Order Terms

2016 ◽  
Vol 16 (4) ◽  
pp. 609-631 ◽  
Author(s):  
Immanuel Anjam ◽  
Dirk Pauly
Keyword(s):  
Error Control ◽  
A Posteriori Error Estimates ◽  
Reaction Diffusion ◽  
Posteriori Error ◽  
Diffusion Problem ◽  
Mixed Formulation ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Linear Partial Differential Equations ◽  
Primal And Dual

AbstractThe results of this contribution are derived in the framework of functional type a posteriori error estimates. The error is measured in a combined norm which takes into account both the primal and dual variables denoted by x and y, respectively. Our first main result is an error equality for all equations of the class ${\mathrm{A}^{*}\mathrm{A}x+x=f}$ or in mixed formulation ${\mathrm{A}^{*}y+x=f}$, ${\mathrm{A}x=y}$, where the exact solution $(x,y)$ is in $D(\mathrm{A})\times D(\mathrm{A}^{*})$. Here ${\mathrm{A}}$ is a linear, densely defined and closed (usually a differential) operator and ${\mathrm{A}^{*}}$ its adjoint. In this paper we deal with very conforming mixed approximations, i.e., we assume that the approximation ${(\tilde{x},\tilde{y})}$ belongs to ${D(\mathrm{A})\times D(\mathrm{A}^{*})}$. In order to obtain the exact global error value of this approximation one only needs the problem data and the mixed approximation itself, i.e., we have the equality$\lvert x-\tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-% \tilde{y}\rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}=\mathcal{M}(% \tilde{x},\tilde{y}),$where ${\mathcal{M}(\tilde{x},\tilde{y}):=\lvert f-\tilde{x}-\mathrm{A}^{*}\tilde{y}% \rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$ contains only known data. Our second main result is an error estimate for all equations of the class ${\mathrm{A}^{*}\mathrm{A}x+ix=f}$ or in mixed formulation ${\mathrm{A}^{*}y+ix=f}$, ${\mathrm{A}x=y}$, where i is the imaginary unit. For this problem we have the two-sided estimate$\frac{\sqrt{2}}{\sqrt{2}+1}\mathcal{M}_{i}(\tilde{x},\tilde{y})\leq\lvert x-% \tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-\tilde{y}% \rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}\leq\frac{\sqrt{2}}{% \sqrt{2}-1}\mathcal{M}_{i}(\tilde{x},\tilde{y}),$where ${\mathcal{M}_{i}(\tilde{x},\tilde{y}):=\lvert f-i\tilde{x}-\mathrm{A}^{*}% \tilde{y}\rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$ contains only known data. We will point out a motivation for the study of the latter problems by time discretizations or time-harmonic ansatz of linear partial differential equations and we will present an extensive list of applications including the reaction-diffusion problem and the eddy current problem.

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