Generalized polynomial chaos expansion by reanalysis using static condensation based on substructuring

This paper presents a new computational method for forward uncertainty quantification(UQ)analyses on large-scale structural systems in the presence of arbitrary and dependent random inputs.The method consists of a generalized polynomial chaos expansion(GPCE)for statistical moment and reliability analyses associated with the stochastic output and a static reanalysis method to generate the input-output data set.In the reanalysis,we employ substructuring for a structure to isolate its local regions that vary due to random inputs.This allows for avoiding repeated computations of invariant substructures while generating the input-output data set.Combining substructuring with static condensation further improves the computational efficiency of the reanalysis without losing accuracy.Consequently,the GPCE with the static reanalysis method can achieve significant computational saving,thus mitigating the curse of dimensionality to some degree for UQ under high-dimensional inputs.The numerical results obtained from a simple structure indicate that the proposed method for UQ produces accurate solutions more efficiently than the GPCE using full finite element analyses(FEAs).We also demonstrate the efficiency and scalability of the proposed method by executing UQ for a large-scale wing-box structure under ten-dimensional(all-dependent)random inputs.

A Non-symmetric Digital Image Secure Communication Scheme Based on Generalized Chaos Synchronization System

Based on a generalized chaos synchronization system and a discrete Sinai map, a non-symmetric true color （RGB） digital image secure communication scheme is proposed. The scheme first changes an ordinary RGB digital image with 8 bits into unrecognizable disorder codes and then transforms the disorder codes into an RGB digital image with 16 bits for transmitting. A receiver uses a non-symmetric key to verify the authentication of the received data origin, and decrypts the ciphertext. The scheme can encrypt and decz：Fpt most formatted digital RGB images recognized by computers, and recover the plaintext almost without any errors. The scheme is suitable to be applied in network image communications. The analysis of the key space, sensitivity of key parameters, and correlation of encrypted images imply that this scheme has sound security.

A voltage-controlled chaotic oscillator based on carbon nanotube field-effect transistor for low-power embedded systems

This paper presents a compact and low-power-based discrete-time chaotic oscillator based on a carbon nanotube field-effect transistor implemented using Wong and Deng＇s well-known model. The chaotic circuit is composed of a nonlinear circuit that creates an adjustable chaos map, two sample and hold cells for capture and delay functions, and a voltage shifter that works as a buffer and adjusts the output voltage for feedback. The operation of the chaotic circuit is verified with the SPICE software package, which uses a supply voltage of 0.9 V at a frequency of 20 kHz. The time series, frequency spectra, transitions in phase space, sensitivity with the initial condition diagrams, and bifurcation phenomena are presented. The main advantage of this circuit is that its chaotic signal can be generated while dissipating approximately 7.8 μW of power, making it suitable for embedded systems where many chaos-signal generators are required on a single chip.

Generalized projective synchronization of the fractional-order chaotic system using adaptive fuzzy sliding mode control

An adaptive fuzzy sliding mode strategy is developed for the generalized projective synchronization of a fractional- order chaotic system, where the slave system is not necessarily known in advance. Based on the designed adaptive update laws and the linear feedback method, the adaptive fuzzy sliding controllers are proposed via the fuzzy design, and the strength of the designed controllers can he adaptively adjusted according to the external disturbances. Based on the Lya- punov stability theorem, the stability and the robustness of the controlled system are proved theoretically. Numerical simu- lations further support the theoretical results of the paper and demonstrate the efficiency of the proposed method. Moreover, it is revealed that the proposed method allows us to manipulate arbitrarily the response dynamics of the slave system by adjusting the desired scaling factor λi and the desired translating factor ηi, which may be used in a channel-independent chaotic secure communication.

Theoretical study of electric energy consumption for self-powered chaos signal generator

Self-powered chaos signal generator is potentially useful in future medical system,such as low cost portable human healthy monitor and treatment without external power source.For both functional and power unit,the power level of electric energy generator and consumption is a key factor for self-powered system.In this paper,we have investigated the power consumption of three typical output modes of a simple chaos circuit.Analytical analysis for power consumption of fixed output mode is obtained for evaluating the power characteristics of chaos signal generator.Numerical calculations are given for predicting the power characteristics of periodical and chaotic output modes.This study is important for not only understanding the power consumption of chaos signal generator,but also guiding new self-powered chaos signal generator design.

Generalized reduced-order synchronization of chaotic system based on fast slide mode

A new kind of generalized reduced-order synchronization of different chaotic systems is proposed in this paper. It is shown that dynamical evolution of third-order oscillator can be synchronized with the canonical projection of a fourth-order chaotic system generated through nonsingular states transformation from a cell neural net chaotic system. In this sense, it is said that generalized synchronization is achieved in reduced-order. The synchronization discussed here expands the scope of reduced-order synchronization studied in relevant literatures. In this way, we can achieve generalized reduced-order synchronization between many famous chaotic systems such as the second-order Drifting system and the third-order Lorenz system by designing a fast slide mode controller. Simulation results are provided to verify the operation of the designed synchronization.

Entropy‑Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty

In this paper,we develop an entropy-conservative discontinuous Galerkin(DG)method for the shallow water(SW)equation with random inputs.One of the most popular methods for uncertainty quantifcation is the generalized Polynomial Chaos(gPC)approach which we consider in the following manuscript.We apply the stochastic Galerkin(SG)method to the stochastic SW equations.Using the SG approach in the stochastic hyperbolic SW system yields a purely deterministic system that is not necessarily hyperbolic anymore.The lack of the hyperbolicity leads to ill-posedness and stability issues in numerical simulations.By transforming the system using Roe variables,the hyperbolicity can be ensured and an entropy-entropy fux pair is known from a recent investigation by Gerster and Herty(Commun.Comput.Phys.27(3):639–671,2020).We use this pair and determine a corresponding entropy fux potential.Then,we construct entropy conservative numerical twopoint fuxes for this augmented system.By applying these new numerical fuxes in a nodal DG spectral element method(DGSEM)with fux diferencing ansatz,we obtain a provable entropy conservative(dissipative)scheme.In numerical experiments,we validate our theoretical fndings.

Generalized Polynomial Chaos for Nonlinear Random Pantograph Equations

This paper is concerned with the application of generalized polynomial chaos （gPC） method to nonlinear random pantograph equations. An error estimation of gPC method is derived. The global error analysis is given for the error arising from finite-dimensional noise （FDN） assumption, projection error, aliasing error and discretization error. In the end, with several numerical experiments, the theoretical results are further illustrated.

Fast Numerical Methods for Stochastic Computations:A Review

This paper presents a review of the current state-of-the-art of numerical methods for stochastic computations.The focus is on efficient high-order methods suitable for practical applications,with a particular emphasis on those based on generalized polynomial chaos(gPC)methodology.The framework of gPC is reviewed,along with its Galerkin and collocation approaches for solving stochastic equations.Properties of these methods are summarized by using results from literature.This paper also attempts to present the gPC based methods in a unified framework based on an extension of the classical spectral methods into multi-dimensional random spaces.

A Stochastic Collocation Approach to Bayesian Inference in Inverse Problems

We present an efficient numerical strategy for the Bayesian solution of inverse problems.Stochastic collocation methods,based on generalized polynomial chaos(gPC),are used to construct a polynomial approximation of the forward solution over the support of the prior distribution.This approximation then defines a surrogate posterior probability density that can be evaluated repeatedly at minimal computational cost.The ability to simulate a large number of samples from the posterior distribution results in very accurate estimates of the inverse solution and its associated uncertainty.Combined with high accuracy of the gPC-based forward solver,the new algorithm can provide great efficiency in practical applications.A rigorous error analysis of the algorithm is conducted,where we establish convergence of the approximate posterior to the true posterior and obtain an estimate of the convergence rate.It is proved that fast(exponential)convergence of the gPC forward solution yields similarly fast(exponential)convergence of the posterior.The numerical strategy and the predicted convergence rates are then demonstrated on nonlinear inverse problems of varying smoothness and dimension.

Galerkin Method for Wave Equations with Uncertain Coefficients

Polynomial chaos methods(and generalized polynomial chaos methods)have been extensively applied to analyze PDEs that contain uncertainties.However this approach is rarely applied to hyperbolic systems.In this paper we analyze the properties of the resulting deterministic system of equations obtained by stochastic Galerkin projection.We consider a simple model of a scalar wave equation with random wave speed.We show that when uncertainty causes the change of characteristic directions,the resulting deterministic system of equations is a symmetric hyperbolic system with both positive and negative eigenvalues.A consistent method of imposing the boundary conditions is proposed and its convergence is established.Numerical examples are presented to support the analysis.

Distributional Sensitivity for Uncertainty Quantification

In this work we consider a general notion of distributional sensitivity,which measures the variation in solutions of a given physical/mathematical system with respect to the variation of probability distribution of the inputs.This is distinctively different from the classical sensitivity analysis,which studies the changes of solutions with respect to the values of the inputs.The general idea is measurement of sensitivity of outputs with respect to probability distributions,which is a well-studied concept in related disciplines.We adapt these ideas to present a quantitative framework in the context of uncertainty quantification for measuring such a kind of sensitivity and a set of efficient algorithms to approximate the distributional sensitivity numerically.A remarkable feature of the algorithms is that they do not incur additional computational effort in addition to a one-time stochastic solver.Therefore,an accurate stochastic computation with respect to a prior input distribution is needed only once,and the ensuing distributional sensitivity computation for different input distributions is a post-processing step.We prove that an accurate numerical model leads to accurate calculations of this sensitivity,which applies not just to slowly-converging Monte-Carlo estimates,but also to exponentially convergent spectral approximations.We provide computational examples to demonstrate the ease of applicability and verify the convergence claims.

Adaptive Bayesian Inference for Discontinuous Inverse Problems,Application to Hyperbolic Conservation Laws

Various works from the literature aimed at accelerating Bayesian inference in inverse problems.Stochastic spectral methods have been recently proposed as surrogate approximations of the forward uncertainty propagation model over the support of the prior distribution.These representations are efficient because they allow affordable simulation of a large number of samples from the posterior distribution.Unfortunately,they do not perform well when the forward model exhibits strong nonlinear behavior with respect to its input.In this work,we first relate the fast(exponential)L2-convergence of the forward approximation to the fast(exponential)convergence(in terms of Kullback-Leibler divergence)of the approximate posterior.In particular,we prove that in case the prior distribution is uniform,the posterior is at least twice as fast as the convergence rate of the forward model in those norms.The Bayesian inference strategy is developed in the framework of a stochastic spectral projection method.The predicted convergence rates are then demonstrated for simple nonlinear inverse problems of varying smoothness.We then propose an efficient numerical approach for the Bayesian solution of inverse problems presenting strongly nonlinear or discontinuous system responses.This comes with the improvement of the forward model that is adaptively approximated by an iterative generalized Polynomial Chaos-based representation.The numerical approximations and predicted convergence rates of the former approach are compared to the new iterative numerical method for nonlinear time-dependent test cases of varying dimension and complexity,which are relevant regarding our hydrodynamics motivations and therefore regarding hyperbolic conservation laws and the apparition of discontinuities in finite time.