scholarly journals Structured coagulation-fragmentation equation in the space of radon measures: Unifying discrete and continuous models

2021 ◽  
Author(s):  
Azmy S. Ackleh ◽  
Rainey Lyons ◽  
Nicolas Saintier
Keyword(s):  
Regularity Result ◽  
Stationary Solutions ◽  
Fragmentation Model ◽  
Model Parameters ◽  
Physical Processes ◽  
Absolutely Continuous ◽  
Radon Measures ◽  
Continuous Models ◽  
Fragmentation Equation ◽  
Well Posed

We present a structured coagulation-fragmentation model which describes the population dynamics of oceanic phytoplankton. This model is formulated on the space of Radon measures equipped with the bounded Lipschitz norm and unifies the study of the discrete and continuous coagulation-fragmentation models. We prove that the model is well-posed and show it can reduce down to the classic discrete and continuous coagulation-fragmentation models. To understand the interplay between the physical processes of coagulation and fragmentation and the biological processes of growth, reproduction, and death, we establish a regularity result for the solutions and use it to show that stationary solutions are absolutely continuous under some conditions on model parameters. We develop a semi-discrete approximation scheme which conserves mass and prove its convergence to the unique weak solution. We then use the scheme to perform numerical simulations for the model.

scholarly journals The confrontation of the shock-powered synchrotron maser model with the Galactic FRB 200428

2020 ◽  
Vol 500 (2) ◽  
pp. 2704-2710 ◽  
Author(s):  
Yun-Wei Yu ◽  
Yuan-Chuan Zou ◽  
Zi-Gao Dai ◽  
Wen-Fei Yu
Keyword(s):  
Line Of Sight ◽  
Model Parameters ◽  
Observational Constraints ◽  
Synchrotron Emission ◽  
Physical Processes ◽  
X Ray ◽  
Emission Efficiency ◽  
Free Parameters ◽  
Fast Radio Burst ◽  
Parameter Constraints

ABSTRACT The association of FRB 200428 with an X-ray burst (XRB) from the Galactic magnetar SGR 1935+2154 offers important implications for the physical processes responsible for the fast radio burst (FRB) phenomena. By assuming that the XRB emission is produced in the magnetosphere, we investigate the possibility that the FRB emission is produced by shock-powered synchrotron maser (SM), which is phenomenologically described with a number of free parameters. The observational constraints on the model parameters indicate that the model can in principle be consistent with the FRB 200428 observations, if the ejecta lunched by magnetar activities can have appropriate ingredients and structures and the shock processes occur on the line of sight. To be specific, a complete burst ejecta should consist of an ultra-relativistic and extremely highly collimated e± component and a sub-relativistic and wide-spreading baryonic component. The internal shocks producing the FRB emission arise from a collision between the e± ejecta and the remnant of a previous baryonic ejecta at the same direction. The parameter constraints depend on the uncertain spectrum and efficiency of the SM emission. While the spectrum is tentatively described by a spectral index of −2, we estimate the emission efficiency to be around 10−4 by requiring that the synchrotron emission of the shocked material cannot be much brighter than the magnetosphere XRB emission.

scholarly journals Compressed sensing with continuous parametric reconstruction

2021 ◽  
Vol 11 (1) ◽  
pp. 851
Author(s):  
Imrich Andras ◽  
Linus Michaeli ◽  
Jan Saliga
Keyword(s):  
Signal Reconstruction ◽  
Sampling Rate ◽  
Quality Monitoring ◽  
Wireless Sensor ◽  
Model Parameters ◽  
Sampling Process ◽  
Continuous Approach ◽  
Continuous Models ◽  
Stable Performance ◽  
Unconventional Method

This work presents a novel unconventional method of signal reconstruction after compressive sensing. Instead of usual matrices, continuous models are used to describe both the sampling process and acquired signal. Reconstruction is performed by finding suitable values of model parameters in order to obtain the most probable fit. A continuous approach allows more precise modelling of physical sampling circuitry and signal reconstruction at arbitrary sampling rate. Application of this method is demonstrated using a wireless sensor network used for freshwater quality monitoring. Results show that the proposed method is more robust and offers stable performance when the samples are noisy or otherwise distorted.

scholarly journals Integro-differential equations for the self-organisation of liver zones by competitive exclusion of cell-types

1987 ◽  
Vol 29 (2) ◽  
pp. 156-194 ◽  
Author(s):  
L. Bass ◽  
A. J. Bracken ◽  
K. Holmåker ◽  
B. R. F. Jefferies
Keyword(s):  
Differential Equations ◽  
Enzymatic Activity ◽  
Cell Types ◽  
Stationary Solutions ◽  
Stable Set ◽  
Model Parameters ◽  
Asymptotically Stable ◽  
Hepatic Sinusoid ◽  
Jump Discontinuities ◽  
Cell Densities

AbstractA model is developed for the seif-organisation of zones of enzymatic activity along a liver capillary (hepatic sinusoid) lined with cells of two types, which contain different enzymes and compete for sites on the wall of the sinusoid. An effectively non-local interaction between the cells arises from local consumption of oxygen from blood flowing throug1 the sinusoid, which gives rise to gradients of oxygen concentration in turn influencing rates of division and of death of the two cell-types. The process is modelled by a pair of coupled non-linear integro-differential equations for the cell-densities as functions of time and position along the sinusoid. Existence of a unique, bounded, non-negative solution of the equations is proved, for prescribed initial values. The equations admit infinitely many stationary solutions, but it is shown that all except one are unstable, for any given set of the model parameters. The remaining solution is shown to be asymptotically stable against a large class of perturbations. For certain ranges of the model parameters, the asymptotically stable stationaxy solution has a zonal structure, with cells of one type located entirely upstream of cells of the other type, and with jump discontinuities in the cell densities at a certain distance along the sinusoid. Such sinusoidal zones can account for zones of enzymatic activity observed in the intact liver. Exceptional cases are found for singular choices of model parameters, such that stationary cell-densities cannot be asymptotically stable individually, but together form an asymptotically stable set. Certain mathematical questions are left open, notably the behaviour of large deviations from stationary solutions, and the global stability of such solutions. Possible generalisations of the model are described.

Inverse Modeling Techniques for Parameter Estimation in Engineering Simulation Models

2006 ◽  
Author(s):  
Srikanth Akkaram ◽  
Don Beeson ◽  
Harish Agarwal ◽  
Gene Wiggs
Keyword(s):  
Heat Transfer ◽  
Parameter Estimation ◽  
Inverse Modeling ◽  
Simulation Models ◽  
Posteriori Probability ◽  
Design System ◽  
Model Parameters ◽  
Input And Output ◽  
Ill Posed ◽  
Well Posed

Computational simulation models are extensively used in the development, design and analysis of an aircraft engine and its components to represent the physics of an underlying phenomenon. The use of such a model-based simulation in engineering often necessitates the need to estimate model parameters based on physical experiments or field data. This class of problems, referred to as inverse problems [1] in the literature can be classified as well-posed or ill-posed dependent on the quality (uncertainty) and quantity (amount) of data that is available to the engineer. The development of a generic inverse modeling solver in a probabilistic design system [2] requires the ability to handle diverse characteristics in various models. These characteristics include (a) varying fidelity in model accuracy with simulation times from a couple of seconds to many hours (b) models being black-box with the engineer having access to only the input and output (c) non-linearity in the model (d) time-dependent model input and output. The paper demonstrates methods that have been implemented to handle these features with emphasis on applications in heat transfer and applied mechanics. A practical issue faced in the application of inverse modeling for parameter estimation is ill-posedness that is characterized by instability and non-uniqueness in the solution. Generic methods to deal with ill-posedness include (a) model development, (b) optimal experimental design and (c) regularization methods. The purpose of this paper is to communicate the development and implementation of an inverse method that provides a solution for both well-posed as well as ill-posed problems using regularization based on the prior values of the parameters. In the case of an ill-posed problem, the method provides two solution schemes — a most probable solution closest to the prior, based on the singular value decomposition (SVD) and a maximum a-posteriori probability solution (MAP). The inverse problem is solved as a finite dimensional non-linear optimization problem using the SVD and/or MAP techniques tailored to the specifics of the application. The paper concludes with numerical examples and applications demonstrating the scope as well as validating the developed method. Engineering applications include heat transfer coefficient estimation for disk quenching in process modeling, material model parameter estimation, sparse clearance data modeling, steady state and transient engine high-pressure compressor heat transfer estimation.

scholarly journals A finite volume scheme for the solution of a mixed discrete-continuous fragmentation model

2021 ◽  
Author(s):  
Graham Baird ◽  
Endre Suli
Keyword(s):  
Finite Volume ◽  
Numerical Scheme ◽  
Truncation Error ◽  
Approximate Solutions ◽  
Fragmentation Model ◽  
Finite Volume Scheme ◽  
Finite Volume Schemes ◽  
Operator Semigroups ◽  
Volume Scheme ◽  
Fragmentation Equation

This paper concerns the construction and analysis of a numerical scheme for a mixed discrete-continuous fragmentation equation. A finite volume scheme is developed, based on a conservative formulation of a truncated version of the equations. The approximate solutions provided by this scheme are first shown to display conservation of mass and preservation of nonnegativity. Then, by utilising a Dunford-Pettis style argument, the sequence of approximate solutions generated is shown, under given restrictions on the model and the mesh, to converge (weakly) in an appropriate L1 space to a weak solution to the problem. By applying the methods and theory of operator semigroups, we are able to show that these weak solutions are unique and necessarily classical (differentiable) solutions, a degree of regularity not generally established when finite volume schemes are applied to such problems. Furthermore, this approach enabled us to derive a bound for the error induced by the truncation of the mass domain, and also establish the convergence of the truncated solutions as the truncation point is increased without bound. Finally, numerical simulations are performed to investigate the performance of the scheme and assess its rate of convergence as the mesh is refined, whilst also verifying the bound on the truncation error.

scholarly journals Ordering results between the largest claims arising from two general heterogeneous portfolios

Filomat ◽  
2021 ◽  
Vol 35 (4) ◽  
pp. 1315-1332
Author(s):  
Sangita Das ◽  
Suchandan Kayal
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Model Parameters ◽  
Absolutely Continuous ◽  
Stochastic Orderings ◽  
General Family ◽  
Insurance Portfolio ◽  
Family Of Distributions

This work is entirely devoted to compare the largest claims from two heterogeneous portfolios. It is assumed that the claim amounts in an insurance portfolio are nonnegative absolutely continuous random variables and belong to a general family of distributions. The largest claims have been compared based on various stochastic orderings. The established sufficient conditions are associated with the matrices and vectors of model parameters. Applications of the results are provided for the purpose of illustration.

Parameter estimation and output feedback stabilization for the linear Korteweg-de Vries equation with disturbed boundary measurement

2019 ◽  
Vol 25 ◽  
pp. 76
Author(s):  
Chaohua Jia ◽  
Wei Guo ◽  
Diao Luo
Keyword(s):  
Parameter Estimation ◽  
Vries Equation ◽  
Regularity Result ◽  
Feedback Stabilization ◽  
Adaptive Observer ◽  
De Vries ◽  
Adaptive Laws ◽  
Unknown Disturbance ◽  
The Right ◽  
Well Posed

This paper is concerned with the parameter estimation and boundary feedback stabilization for the linear Korteweg-de Vries equation posed on a finite interval with the boundary observation at the right end and the non-collocated control at the left end. The boundary observation suffers from some unknown disturbance. An adaptive observer is designed and the adaptive laws of the parameters are obtained by the Lyapunov method. The resulted closed-loop system is proved to be well-posed and asymptotically stable in case that the length of the interval is not critical. Moreover, it is shown that the estimated parameter converges to the unknown parameter. As a by-product, a hidden regularity result is proved.

scholarly journals Flow Properties Inferred from Generalized Maxwell Models

2009 ◽  
Vol 64 (1-2) ◽  
pp. 81-95 ◽  
Author(s):  
Siegfried Hess ◽  
Bastian Arlt ◽  
Sebastian eidenreich ◽  
Patrick Ilg ◽  
Chris Goddard ◽  
...  
Keyword(s):  
Stress Tensor ◽  
Maxwell Model ◽  
Stationary Solutions ◽  
Model Parameters ◽  
Relaxation Equation ◽  
Newtonian Viscosity ◽  
Stick Slip ◽  
First Case ◽  
Scalar Invariant ◽  
Special Cases

The generalized Maxwell model is formulated as a nonlinear relaxation equation for the symmetric traceless stress tensor. The relaxation term of the equation involves the derivative of a potential function with respect to the stress tensor. Two special cases for this potential referred to as “isotropic” and “anisotropic” are considered. In the first case, the potential solely depends on the second scalar invariant, viz. the norm of the tensor. In the second case, also a dependence on the third scalar invariant, essentially the determinant, is taken into account in analogy to the Landau-de Gennes potential of nematic liquid crystals. Rheological consequences of the model are presented for a plane Couette flow with an imposed shear rate. The non-Newtonian viscosity and the normal stress differences are analyzed for stationary solutions. The dependence on the model parameters is discussed in detail. In particular, the occurrence of a shear-thickening behaviour is studied. The possibility to describe substances with yield stress and the existence of non-stationary, stick-slip-like solutions are pointed out. The extension of the model to magneto-rheological fluids is indicated.

scholarly journals Assessment of the Tumbling-Snake Model against Linear and Nonlinear Rheological Data of Bidisperse Polymer Blends

Polymers ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 376 ◽  
Author(s):  
Pavlos Stephanou ◽  
Martin Kröger
Keyword(s):  
Brownian Dynamics ◽  
Stationary Solutions ◽  
Model Parameters ◽  
Snake Model ◽  
Shear Rates ◽  
Rheological Data ◽  
Wide Range ◽  
Dynamics Simulations ◽  
Brownian Dynamics Simulations ◽  
Entangled Melts

We have recently solved the tumbling-snake model for concentrated polymer solutions and entangled melts in the academic case of a monodisperse sample. Here, we extend these studies and provide the stationary solutions of the tumbling-snake model both analytically, for small shear rates, and via Brownian dynamics simulations, for a bidisperse sample over a wide range of shear rates and model parameters. We further show that the tumbling-snake model bears the necessary capacity to compare well with available linear and non-linear rheological data for bidisperse systems. This capacity is added to the already documented ability of the model to accurately predict the shear rheology of monodisperse systems.

A New Methodology for Observation-Based Parameterization Development

2020 ◽  
Vol 148 (10) ◽  
pp. 4159-4184
Author(s):  
Kay Suselj ◽  
Derek Posselt ◽  
Mark Smalley ◽  
Matthew D. Lebsock ◽  
Joao Teixeira
Keyword(s):  
Numerical Models ◽  
Eddy Diffusivity ◽  
Cloud Fraction ◽  
Model Parameters ◽  
Vertical Resolution ◽  
Shallow Convection ◽  
Physical Processes ◽  
Marine Stratocumulus ◽  
Objective Basis ◽  
Observing Systems

AbstractWe develop a methodology for identification of candidate observables that best constrain the parameterization of physical processes in numerical models. This methodology consists of three steps: (i) identifying processes that significantly impact model results, (ii) identifying observables that best constrain the influential processes, and (iii) investigating the sensitivity of the model results to the measurement error and vertical resolution of the constraining observables. This new methodology is applied to the Jet Propulsion Laboratory stochastic multiplume Eddy-Diffusivity/Mass-Flux (JPL-EDMF) model for two case studies representing nonprecipitating marine stratocumulus and marine shallow convection. The uncertainty of physical processes is characterized with uncertainty of model parameters. We find that the most uncertain processes in the JPL-EDMF model are related to the representation of lateral entrainment for convective plumes and parameterization of mixing length scale for the eddy-diffusivity part of the model. The results show a strong interaction between these uncertain processes. Measurements of the water vapor profile for shallow convection and of the cloud fraction profile for the stratocumulus case are among those measurements that best constrain the uncertain JPL-EDMF processes. The interdependence of the required vertical resolution and error characteristics of the observational system are shown. If the observations are associated with larger error, their vertical resolution has to be finer and vice versa. We suggest that the methodology and results presented here provide an objective basis for defining requirements for future observing systems such as future satellite missions to observe clouds and the planetary boundary layer.

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