Showing papers in "Discrete and Continuous Dynamical Systems - Series S in 2022"

Data-driven control of hydraulic servo actuator based on adaptive dynamic programming

Abstract: <p style='text-indent:20px;'>The hydraulic servo actuators (HSA) are often used in the industry in tasks that request great powers, high accuracy and dynamic motion. It is well known that HSA is a highly complex nonlinear system, and that the system parameters cannot be accurately determined due to various uncertainties, inability to measure some parameters, and disturbances. This paper considers control problem of the HSA with unknown dynamics, based on adaptive dynamic programming via output feedback. Due to increasing practical application of the control algorithm, a linear discrete model of HSA is considered and an online learning data-driven controller is used, which is based on measured input and output data instead of unmeasurable states and unknown system parameters. Hence, the ADP based data-driven controller in this paper requires neither the knowledge of the HSA dynamics nor exosystem dynamics. The convergence of the ADP based control algorithm is also theoretically shown. Simulation results verify the feasibility and effectiveness of the proposed approach in solving the optimal control problem of HSA.</p>

A novel bond stress-slip model for 3-D printed concretes

Abstract: <p style='text-indent:20px;'>This paper considers the 3D printing process as a discontinuous control system and gives a simple and coherent bond stress-slip model for a new and intelligent building 3-D printed concrete. The previous models focused on either the maximal stress or the maximal slip, however, the novel model uses an energy approach by the dimension analysis, so that the main factors affecting the bond stress-slip relationship can be clearly revealed, mainly including the concrete's properties (its porous structure and its strength), the steel bar's properties (its printing direction, its strength, its surface roughness and its geometrical property) and the printing process. It is confirmed that the proposed model, similar to the constitutive relationship in elasticity, plays a key role in concrete mechanics, and it can conveniently explain the observed phenomena from the experiment.</p>

A geometric multiscale model for the numerical simulation of blood flow in the human left heart

Abstract: We present a new computational model for the numerical simulation of blood flow in the human left heart. To this aim, we use the Navier-Stokes equations in an Arbitrary Lagrangian Eulerian formulation to account for the endocardium motion and we model the cardiac valves by means of the Resistive Immersed Implicit Surface method. To impose a physiological displacement of the domain boundary, we use a 3D cardiac electromechanical model of the left ventricle coupled to a lumped-parameter (0D) closed-loop model of the remaining circulation. We thus obtain a one-way coupled electromechanics-fluid dynamics model in the left ventricle. To extend the left ventricle motion to the endocardium of the left atrium and to that of the ascending aorta, we introduce a preprocessing procedure according to which an harmonic extension of the left ventricle displacement is combined with the motion of the left atrium based on the 0D model. To better match the 3D cardiac fluid flow with the external blood circulation, we couple the 3D Navier-Stokes equations to the 0D circulation model, obtaining a multiscale coupled 3D-0D fluid dynamics model that we solve via a segregated numerical scheme. We carry out numerical simulations for a healthy left heart and we validate our model by showing that meaningful hemodynamic indicators are correctly reproduced.

Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species

Abstract: <p style='text-indent:20px;'>Environmental factors and random variation have strong effects on the dynamics of biological and ecological systems. In this paper, we propose a stochastic delay differential model of two-prey, one-predator system with cooperation among prey species against predator. The model has a global positive solution. Sufficient conditions of existence and uniqueness of an ergodic stationary distribution of the positive solution are provided, by constructing suitable Lyapunov functionals. Sufficient conditions for possible extinction of the predator populations are also obtained. The conditions are expressed in terms of a threshold parameter <inline-formula><tex-math id="M1">\begin{document}$ {\mathcal R}_0^s $\end{document}</tex-math></inline-formula> that relies on the environmental noise. Illustrative examples and numerical simulations, using Milstein's scheme, are carried out to illustrate the theoretical results. A small scale of noise can promote survival of the species. While relative large noises can lead to possible extinction of the species in such an environment.</p>

Fractional Laplacians : A short survey

Abstract: <p style='text-indent:20px;'>This paper describes the state of the art and gives a survey of the wide literature published in the last years on the fractional Laplacian. We will first recall some definitions of this operator in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> and its main properties. Then, we will introduce the four main operators often used in the case of a bounded domain; and we will give several simple and significant examples to highlight the difference between these four operators. Also we give a rather long list of references : it is certainly not exhaustive but hopefully rich enough to track most connected results. We hope that this short survey will be useful for young researchers of all ages who wish to have a quick idea of the fractional Laplacian(s).</p>

Hyers-Ulam-Rassias stability of high-dimensional quaternion impulsive fuzzy dynamic equations on time scales

Abstract: <p style='text-indent:20px;'>In this paper, the Hyers-Ulam-Rassias stability of high-dimensional quaternion fuzzy dynamic equations with impulses is first considered on time scales. Some fundamental calculus results of the high-dimensional fuzzy quaternion functions in fuzzy quaternion space are established. Based on it, some sufficient conditions are obtained to guarantee the Hyers-Ulam-Rassias stability of the quaternion impulsive fuzzy dynamic equations in high-dimensional case. Moreover, several examples are provided to show the feasibility of our main results on various types of time scales.</p>

Dynamics of delayed cellular neural networks in the Stepanov pseudo almost automorphic space

Abstract: Pseudo almost automorphy (PAA) is a natural generalization of Bochner almost automorphy and Stepanov almost automorphy. Therefore, the results of the existence of PAA solutions of differential equations are few, and the results of the existence of pseudo almost automorphic solutions of difference equations are rare. In this work, we are concerned with a model of delayed cellular neural networks (CNNs). The delays are considered in varying-time form. By the Banach's fixed point theorem, Stepanov like PAA, and constructing a novel Lyapunov functional, we fixed a sufficient criteria that agreement the existence and the Stepanov-exponential stability of Stepanov-like PAA solution of this model of CNNs are obtained. In addition, a numerical example and simulations are performed to verify our theoretical results.

Existence and regularity results for stochastic fractional pseudo-parabolic equations driven by white noise

Abstract: <p style='text-indent:20px;'>Solutions of a direct problem for a stochastic pseudo-parabolic equation with fractional Caputo derivative are investigated, in which the non-linear space-time-noise is assumed to satisfy distinct Lipshitz conditions including globally and locally assumptions. The main aim of this work is to establish some existence, uniqueness, regularity, and continuity results for mild solutions.</p>

Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity

Abstract: <p style='text-indent:20px;'>The Jordan–Moore–Gibson–Thompson (JMGT) equation is a well-established and recently widely studied model for nonlinear acoustics (NLA). It is a third–order (in time) semilinear Partial Differential Equation (PDE) with a distinctive feature of predicting the propagation of ultrasound waves at <i>finite</i> speed. This is due to the heat phenomenon known as <i>second sound</i> which leads to hyperbolic heat-wave propagation. In this paper, we consider the problem in the so called "critical" case, where free dynamics is unstable. In order to stabilize, we shall use boundary feedback controls supported on a portion of the boundary only. Since the remaining part of the boundary is not "controlled", and the imposed boundary conditions of Neumann type fail to saitsfy Lopatinski condition, several mathematical issues typical for mixed problems within the context o boundary stabilizability arise. To resolve these, special geometric constructs along with sharp trace estimates will be developed. The imposed geometric conditions are motivated by the geometry that is suitable for modeling the problem of controlling (from the boundary) the acoustic pressure involved in medical treatments such as lithotripsy, thermotherapy, sonochemistry, or any other procedure involving High Intensity Focused Ultrasound (HIFU).</p>

Shape optimization method for an inverse geometric source problem and stability at critical shape

Abstract: <p style='text-indent:20px;'>This work deals with a geometric inverse source problem. It consists in recovering inclusion in a fixed domain based on boundary measurements. The inverse problem is solved via a shape optimization formulation. Two cost functions are investigated, namely, the least squares fitting, and the Kohn-Vogelius function. In this case, the existence of the shape derivative is given via the first order material derivative of the state problems. Furthermore, using the adjoint approach, the shape gradient of both cost functions is characterized. Then, the stability is investigated by proving the compactness of the Hessian at the critical shape for both considered cases. Finally, based on the gradient method, a steepest descent algorithm is developed, and some numerical experiments for non-parametric shapes are discussed.</p>

A backward SDE method for uncertainty quantification in deep learning

Abstract: We develop a backward stochastic differential equation based probabilistic machine learning method, which formulates a class of stochastic neural networks as a stochastic optimal control problem. An efficient stochastic gradient descent algorithm is introduced with the gradient computed through a backward stochastic differential equation. Convergence analysis for stochastic gradient descent optimization and numerical experiments for applications of stochastic neural networks are carried out to validate our methodology in both theory and performance.

Hadamard Itô-Doob Stochastic Fractional Order Systems

Abstract: In this paper, we study the existence and uniqueness of Hadamard Itô-Doob Stochastic Fractional Order Systems (HIDSFOS) using the Picard iteration method. Different from the previous works, our paper presents a new theory using the Hadamard fractional integral. We have proved the convergence of the solution of the averaged HIDSFOS to that of the standard HIDSFOS in the sense of the mean square and also in probability. Some examples are given at the end of this paper to illustrate our theoretical results.

Bounded consensus of double-integrator stochastic multi-agent systems

Abstract: In the framework of fixed topology and stochastic switching topologies, we study the mean-square bounded consensus(MSBC) of double-integrator stochastic multi-agent systems(SMASs) including additive system noises and communication noises. Combining algebra, graph theory and random analysis, we obtain several equivalent conditions for double-integrator SMASs to reach MSBC. In addition, the simulation examples also verify the correctness of the theoretical results.

Existence and regularity results for a singular parabolic equations with degenerate coercivity

Abstract: <p style='text-indent:20px;'>The aim of this paper is to prove existence and regularity of solutions for the following nonlinear singular parabolic problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{array}{lll} \dfrac{\partial u}{\partial t}-\mbox{div}\left( \dfrac{a(x,t,u, abla u)}{(1+|u|)^{\theta(p-1)}}\right) +g(x,t,u) = \dfrac{f}{u^{\gamma}} &amp;\mbox{in}&amp;\,\, Q,\\ u(x,0) = 0 &amp;\mbox{on} &amp; \Omega,\\ u = 0 &amp;\mbox{on} &amp;\,\, \Gamma. \end{array} \right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Here <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded open subset of <inline-formula><tex-math id="M2">\begin{document}$ I\!\!R^{N} (N&gt;p\geq 2), T&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ f $\end{document}</tex-math></inline-formula> is a non-negative function that belong to some Lebesgue space, <inline-formula><tex-math id="M4">\begin{document}$ f\in L^{m}(Q) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ Q = \Omega \times(0,T) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \Gamma = \partial\Omega\times(0,T) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ g(x,t,u) = |u|^{s-1}u $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ s\geq 1, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M9">\begin{document}$ 0\leq\theta&lt; 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ 0&lt;\gamma&lt;1. $\end{document}</tex-math></inline-formula></p>

An efficient D-N alternating algorithm for solving an inverse problem for Helmholtz equation

Abstract: <p style='text-indent:20px;'>Data completion known as Cauchy problem is one most investigated inverse problems. In this work we consider a Cauchy problem associated with Helmholtz equation. Our concerned is the convergence of the well-known alternating iterative method [<xref ref-type="bibr" rid="b25">25</xref>]. Our main result is to restore the convergence for the classical iterative algorithm (KMF) when the wave numbers are considerable. This is achieved by, some simple modification for the Neumann condition on the under-specified boundary and replacement by relaxed Neumann ones. Moreover, for the small wave number <inline-formula><tex-math id="M111111">\begin{document}$ k $\end{document}</tex-math></inline-formula>, when the convergence of KMF algorithm's [<xref ref-type="bibr" rid="b25">25</xref>] is ensured, our algorithm can be used as an acceleration of convergence.</p><p style='text-indent:20px;'>In this case, we present theoretical results of the convergence of this relaxed algorithm. Meanwhile it, we can deduce the convergence intervals related to the relaxation parameters in different situations. In contrast to the existing results, the proposed algorithm is simple to implement converges for all choice of wave number.</p><p style='text-indent:20px;'>We approach our algorithm using finite element method to obtain an accurate numerical results, to affirm theoretical results and to prove it's effectiveness.</p>

An algorithm for solving linear nonhomogeneous quaternion-valued differential equations and some open problems

Abstract: <p style='text-indent:20px;'>Quaternion-valued differential equations (QDEs) is a new kind of differential equations. In this paper, an algorithm was presented for solving linear nonhomogeneous quaternionic-valued differential equations. The variation of constants formula was established for the nonhomogeneous quaternionic-valued differential equations. Moreover, several examples showed the feasibility of our algorithm. Finally, some open problems end this paper.</p>

A novel semi-analytical method for solutions of two dimensional fuzzy fractional wave equation using natural transform

Abstract: <p style='text-indent:20px;'>In this research article, the techniques for computing an analytical solution of 2D fuzzy wave equation with some affecting term of force has been provided. Such type of achievement for the aforesaid solution is obtained by applying the notions of a Caputo non-integer derivative in the vague or uncertainty form. At the first attempt the fuzzy natural transform is applied for obtaining the series solution. Secondly the homotopy perturbation (HPM) technique is used, for the analysis of the proposed result by comparing the co-efficient of homotopy parameter <inline-formula><tex-math id="M1">\begin{document}$ q $\end{document}</tex-math></inline-formula> to get hierarchy of equation of different order for <inline-formula><tex-math id="M2">\begin{document}$ q $\end{document}</tex-math></inline-formula>. For this purpose, some new results about Natural transform of an arbitrary derivative under uncertainty are established, for the first time in the literature. The solution has been assumed in term of infinite series, which break the problem to a small number of equations, for the respective investigation. The required results are then determined in a series solution form which goes rapidly towards the analytical result. The solution has two parts or branches in fuzzy form, one is lower branch and the other is upper branch. To illustrate the ability of the considered approach, we have proved some test problems.</p>

Solving the linear transport equation by a deep neural network approach

Abstract: <p style='text-indent:20px;'>In this paper, we study linear transport model by adopting <i>deep learning method</i>, in particular deep neural network (DNN) approach. While the interest of using DNN to study partial differential equations is arising, here we adapt it to study kinetic models, in particular the linear transport model. Moreover, theoretical analysis on the convergence of neural network and its approximated solution towards analytic solution is shown. We demonstrate the accuracy and effectiveness of the proposed DNN method in numerical experiments.</p>

Link theorem and distributions of solutions to uncertain Liouville-Caputo difference equations

Abstract: <p style='text-indent:20px;'>We consider a class of initial fractional Liouville-Caputo difference equations (IFLCDEs) and its corresponding initial uncertain fractional Liouville-Caputo difference equations (IUFLCDEs). Next, we make comparisons between two unique solutions of the IFLCDEs by deriving an important theorem, namely the main theorem. Besides, we make comparisons between IUFLCDEs and their <inline-formula><tex-math id="M1">\begin{document}$ \varrho $\end{document}</tex-math></inline-formula>-paths by deriving another important theorem, namely the link theorem, which is obtained by the help of the main theorem. We consider a special case of the IUFLCDEs and its solution involving the discrete Mittag-Leffler. Also, we present the solution of its <inline-formula><tex-math id="M2">\begin{document}$ \varrho $\end{document}</tex-math></inline-formula>-paths via the solution of the special linear IUFLCDE. Furthermore, we derive the uniqueness of IUFLCDEs. Finally, we present some test examples of IUFLCDEs by using the uniqueness theorem and the link theorem to find a relation between the solutions for the IUFLCDEs of symmetrical uncertain variables and their <inline-formula><tex-math id="M3">\begin{document}$ \varrho $\end{document}</tex-math></inline-formula>-paths.</p>

An out-of-distribution-aware autoencoder model for reduced chemical kinetics

Abstract: <p style='text-indent:20px;'>While detailed chemical kinetic models have been successful in representing rates of chemical reactions in continuum scale computational fluid dynamics (CFD) simulations, applying the models in simulations for engineering device conditions is computationally prohibitive. To reduce the cost, data-driven methods, e.g., autoencoders, have been used to construct reduced chemical kinetic models for CFD simulations. Despite their success, data-driven methods rely heavily on training data sets and can be unreliable when used in out-of-distribution (OOD) regions (i.e., when extrapolating outside of the training set). In this paper, we present an enhanced autoencoder model for combustion chemical kinetics with uncertainty quantification to enable the detection of model usage in OOD regions, and thereby creating an OOD-aware autoencoder model that contributes to more robust CFD simulations of reacting flows. We first demonstrate the effectiveness of the method in OOD detection in two well-known datasets, MNIST and Fashion-MNIST, in comparison with the deep ensemble method, and then present the OOD-aware autoencoder for reduced chemistry model in syngas combustion.</p>

Sharper and finer energy decay rate for an elastic string with localized Kelvin-Voigt damping

Abstract:

This paper is on the asymptotic behavior of the elastic string equation with localized Kelvin-Voigt damping

\begin{document}$ u_{tt}(x, t)-[u_{x}(x, t)+b(x)u_{x, t}(x, t)]_{x} = 0, \; x\in(-1, 1), \; t>0, $\end{document}

where \begin{document}$ b(x) = 0 $\end{document} on \begin{document}$ x\in (-1, 0] $\end{document}, and \begin{document}$ b(x) = a(x)>0 $\end{document} on \begin{document}$ x\in (0, 1) $\end{document}. It is known that the Geometric Optics Condition for exponential stability does not apply to Kelvin-Voigt damping. Under the assumption that \begin{document}$ a'(x) $\end{document} has a singularity at \begin{document}$ x = 0 $\end{document}, we investigate the decay rate of the solution which depends on the order of the singularity.

When \begin{document}$ a(x) $\end{document} behaves like \begin{document}$ x^{\alpha}(-\log x)^{-\beta} $\end{document} near \begin{document}$ x = 0 $\end{document} for \begin{document}$ 0\le{\alpha}<1, \;0\le\beta $\end{document} or \begin{document}$ 0<{\alpha}<1, \;\beta<0 $\end{document}, we show that the system can achieve a mixed polynomial-logarithmic decay rate.

As a byproduct, when \begin{document}$ \beta = 0 $\end{document}, we obtain the decay rate \begin{document}$ t^{-\frac{ 3-\alpha-\varepsilon}{2(1-{\alpha})}} $\end{document} of solution for arbitrarily small \begin{document}$ \varepsilon>0 $\end{document}, which improves the rate \begin{document}$ t^{-\frac{1}{1-{\alpha}}} $\end{document} obtained in [14]. The new rate is again consistent with the exponential decay rate in the limit case \begin{document}$ \alpha\to 1^- $\end{document}. This is a step toward the goal of obtaining the optimal decay rate eventually.

Weak solvability of nonlinear elliptic equations involving variable exponents

Abstract:

We are concerned with the study of the existence and multiplicity of solutions for Dirichlet boundary value problems, involving the \begin{document}$ ( p( m ), \, q( m ) )- $\end{document} equation and the nonlinearity is superlinear but does not fulfil the Ambrossetti-Rabinowitz condition in the framework of Sobolev spaces with variable exponents in a complete manifold. The main results are proved using the mountain pass theorem and Fountain theorem with Cerami sequences. Moreover, an example of a \begin{document}$ ( p( m ), \, q( m ) ) $\end{document} equation that highlights the applicability of our theoretical results is also provided.

A stochastic collocation method based on sparse grids for a stochastic Stokes-Darcy model

Abstract: <p style='text-indent:20px;'>In this paper, we develop a sparse grid stochastic collocation method to improve the computational efficiency in handling the steady Stokes-Darcy model with random hydraulic conductivity. To represent the random hydraulic conductivity, the truncated Karhunen-Loève expansion is used. For the discrete form in probability space, we adopt the stochastic collocation method and then use the Smolyak sparse grid method to improve the efficiency. For the uncoupled deterministic subproblems at collocation nodes, we apply the general coupled finite element method. Numerical experiment results are presented to illustrate the features of this method, such as the sample size, convergence, and randomness transmission through the interface.</p>

Synchronization for singularity-perturbed complex networks via event-triggered impulsive control

Abstract: This paper studies synchronization of singularity-perturbed complex networks (SPCNs) with a small singular perturbation parameter (SPP) via event-triggered impulsive control (ETIC). A novel dynamic event-triggered mechanism is proposed where an auxiliary impulse parameter is introduced to regulate the triggering threshold dynamically for saving the network resource. Based on SPP-dependent Lyapunov function, some sufficient conditions involving the impulsive gain, triggering parameters and singular perturbation parameter (SPP) are obtained to synchronize the SPCNs, and the upper bound of SPP is also determined. Moreover, it proves that the Zeno behavior can be excluded. Finally, two simulations are provided to demonstrate the validity of the obtained results.

Null controllability for semilinear heat equation with dynamic boundary conditions

Abstract:

This paper deals with the null controllability of the semilinear heat equation with dynamic boundary conditions of surface diffusion type, with nonlinearities involving drift terms. First, we prove a negative result for some function \begin{document}$ F $\end{document} that behaves at infinity like \begin{document}$ |s| \ln ^{p}(1+|s|), $\end{document} with \begin{document}$ p > 2 $\end{document}. Then, by a careful analysis of the linearized system and a fixed point method, a null controllability result is proved for nonlinearties \begin{document}$ F(s, \xi) $\end{document} and \begin{document}$ G(s, \xi) $\end{document} growing slower than \begin{document}$ |s| \ln ^{3 / 2}(1+|s|+\|\xi\|)+\|\xi\| \ln^{1 / 2}(1+|s|+\|\xi\|) $\end{document} at infinity.

Novel criteria for robust stability of Cohen-Grossberg neural networks with multiple time delays

Abstract: This research paper deals with the investigation of global robust stability results for Cohen-Grossberg neural networks involving the multiple constant time delays. The activation functions in this neural network model are supposed to be in the set of non-decreasing slope-bounded nonlinear functions and the uncertainties in the constant network parameters are considered to have bounded upper norms. By employing a proper positive definite Lyapunov-type functional and using homeomorphism mapping theory, we propose some novel sets of novel conditions that assure both existence, uniqueness and global robust asymptotic stability of equilibrium points of this nonlinear Cohen-Grossberg-type neural network model involving the multiple time delays. The derived robustly stable conditions mainly rely on examining some proper relationships that are imposed on constant valued interconnection matrices of this delayed neural network. These stability conditions can be certainly verified by employing various simple and useful properties of real interval matrices. Some comparisons are made to address the key advantages of these novel criteria over previously reported corresponding results. An instructive example is also examined to observe novelty of these proposed criteria.

Qualitative structure of a discrete predator-prey model with nonmonotonic functional response

Abstract:

In this paper, we study the qualitative structure of a discrete predator-prey model with nonmonotonic functional response near a degenerate fixed point whose eigenvalues are \begin{document}$ \pm1 $\end{document}. Firstly, the model is transformed into an ordinary differential system by using the normal form theory and the Takens's theorem. Then, the qualitative properties of this ordinary differential system near the degenerate equilibrium are analyzed with the blowing-up method. Finally, according to the conjugacy between the discrete model and the time-one mapping of the vector field, the qualitative structure of this discrete model is obtained. Numerical simulations are also given.

Probabilistic analysis of a class of impulsive linear random differential equations forced by stochastic processes admitting Karhunen-Loève expansions

Abstract: We study a full randomization of the complete linear differential equation subject to an infinite train of Dirac's delta functions applied at different time instants. The initial condition and coefficients of the differential equation are assumed to be absolutely continuous random variables, while the external or forcing term is a stochastic process. We first approximate the forcing term using the Karhunen-Loève expansion, and then we take advantage of the Random Variable Transformation method to construct a formal approximation of the first probability density function (1-p.d.f.) of the solution. By imposing mild conditions on the model parameters, we prove the convergence of the aforementioned approximation to the exact 1-p.d.f. of the solution. All the theoretical findings are illustrated by means of two examples, where different types of probability distributions are assumed to model parameters.

Positive solutions for the $ p(x)- $Laplacian : Application of the Nehari method

Abstract: <p style='text-indent:20px;'>In this paper, we study the existence of positive solutions of the following equation</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE10000">\begin{document}$\begin{equation} (P_{\lambda}) \left\{ \begin{array}{rclll} - \Delta_{p(x)} u+V(x)\vert u\vert^{p(x)-2}u &amp; = &amp; \lambda k(x) \vert u\vert^{\alpha(x)-2}u\\ &amp;+&amp; h(x) \vert u\vert^{\beta(x)-2}u&amp;\mbox{ in }&amp;\Omega\\ u&amp; = &amp;0 &amp;\mbox{ on }&amp; \partial \Omega. \end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right) \end{equation}$\end{document}</tex-math></disp-formula></p> <p style='text-indent:20px;'>The study of the problem <inline-formula><tex-math id="M2">\begin{document}$ (P_{\lambda}) $\end{document}</tex-math></inline-formula> needs generalized Lebesgue and Sobolev spaces. In this work, under suitable assumptions, we prove that some variational methods still work. We use them to prove the existence of positive solutions to the problem <inline-formula><tex-math id="M3">\begin{document}$ (P_{\lambda}) $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M4">\begin{document}$ W_{0}^{1,p(x)}(\Omega) $\end{document}</tex-math></inline-formula>.</p>

Analytic continuation of noisy data using Adams Bashforth residual neural network

Abstract: <p style='text-indent:20px;'>We propose a data-driven learning framework for the analytic continuation problem in numerical quantum many-body physics. Designing an accurate and efficient framework for the analytic continuation of imaginary time using computational data is a grand challenge that has hindered meaningful links with experimental data. The standard Maximum Entropy (MaxEnt)-based method is limited by the quality of the computational data and the availability of prior information. Also, the MaxEnt is not able to solve the inversion problem under high level of noise in the data. Here we introduce a novel learning model for the analytic continuation problem using a Adams-Bashforth residual neural network (AB-ResNet). The advantage of this deep learning network is that it is model independent and, therefore, does not require prior information concerning the quantity of interest given by the spectral function. More importantly, the ResNet-based model achieves higher accuracy than MaxEnt for data with higher level of noise. Finally, numerical examples show that the developed AB-ResNet is able to recover the spectral function with accuracy comparable to MaxEnt where the noise level is relatively small.</p>