Stochastic Moments Research Articles

A framework for robust quadratic optimal control with parametric dynamic model uncertainty using polynomial chaos

AbstractWe propose a framework tailored to robust optimal control (OC) problems subject to parametric model uncertainty of system dynamics. First, we identify a generic class of robust objective kernels that are based on the class of deterministic quadratic objectives. It is demonstrated how such kernels can be expressed as a function of the stochastic moments of the state and how the objective terms relate to the robustness and performance of the optimal solution. Second, we engage the generalized polynomial chaos (gPC) framework to propagate uncertainty along the state trajectory. Integrating both frameworks makes way to reformulate the problem as a deterministic OC problem in function of the gPC expansion coefficients that can be solved using existing methods. We apply the framework to solve the problem of robust optimal startup behavior of a nonlinear mechanical drivetrain that is subject to a bifurcation in its dynamics.

PoCET: a Polynomial Chaos Expansion Toolbox for Matlab

We introduce PoCET: a free and open-scource Polynomial Chaos Expansion Toolbox for Matlab, featuring the automatic generation of polynomial chaos expansion (PCE) for linear and nonlinear dynamic systems with time-invariant stochastic parameters or initial conditions, as well as several simulation tools. It offers a built-in handling of Gaussian, uniform, and beta probability density functions, projection and collocation-based calculation of PCE coefficients, and the calculation of stochastic moments from a PCE. Efficient algorithms for the calculation of the involved integrals have been designed in order to increase its applicability. PoCET comes with a variety of introductory and instructive examples. Throughout the paper we show how to perform a polynomial chaos expansion on a simple ordinary differential equation using PoCET, as well as how it can be used to solve the more complex task of optimal experimental design.

Uncertainty Quantification for the Transient Response of Human Equivalent Antenna Using the Stochastic Collocation Approach

The paper deals with the uncertainty quantification of the transient axial current induced along the human body exposed to electromagnetic pulse radiation. The body is modeled as a straight wire antenna whose length and radius exhibit random nature. The uncertainty is propagated to the output transient current by means of the stochastic collocation method. The stochastic approach is entirely nonintrusive and serves as a wrapper around the deterministic code. The numerical deterministic model is based on the time domain Hallen integral equation solved by means of the Galerkin-Bubnov indirect boundary element method (GB-IBEM). The stochastic moments, i.e., the mean and the variance of the transient current, are calculated. Confidence margins are obtained for the whole duration of the transient response as well as for the maximal current value. The presented approach enables the estimation of the probability for the induced current to exceed the basic restrictions prescribed by regulatory bodies. The sensitivity analysis of the input parameters indicates to which extent the variation of the input parameter set influences the output values which is particularly interesting for the design of the human equivalent antenna.

Response probability density of a system with cross-correlated parametric and additive input noises

Data processing and subsequent mining is a widely followed task. Employment of suitable evaluation and interpretation procedures can significantly improve the effective resolution of measuring facility using an identical hardware equipment. Recording of time variable processes is accompanied by various internal disturbing effects as a rule. They influence parameters of the measuring facility, transducer-device transmitting, etc. These parasitic processes are usually of the random character and, consequently, they exercise as parametric noises. Moreover, the input signal mostly consists of a useful signal, which can be taken for deterministic, and of a random additive part. Due to interaction of additive noises with the device itself, the cross-correlation of both additive and multiplicative noises cannot be neglected as a rule. Various combinations of noises are the origin of random and also systematic measuring errors which can have under certain circumstances a cumulative character. Their influence deteriorates the output signal quality and can lead finally to the stochastic stability loss. These effects can be theoretically described using differential systems with stochastic coefficients and a stochastic right hand side considering all input and output processes to be of the Markov type. A direct investigation of the relevant Fokker-Planck equation is employed as the main tool. Two first stochastic moments (mathematical mean value and variance) as evolutionary processes are investigated for a general deterministic useful signal and subsequently for two special cases of this one. Both types of input random noises are considered. Conditions of stochastic stability with respect to intensities of input random processes are formulated. The probability density function is deduced as well, in order to illustrate the probabilistic character of the system response as a whole. The stochastic asymmetry of the output signal is identified. Limitation procedures show a smooth transition from a general stochastic problem to deterministic noise free input signal and its processing.

Scenario-based planning of active distribution systems under uncertainties of renewable generation and electricity demand

The rising penetration of intermittent renewable distributed generation leads to uncertainties in the planning of electric distribution networks. Fully considering the uncertainties pertinent to wind power generation, photovoltaic power generation and load demand, this paper proposes a scenario-based model for the planning of active distribution systems. The solution obtains the optimal capacities and locations of wind and photovoltaic based distributed generators in the distribution system, whilst minimizing the active and reactive power losses as well as voltage deviation. A scenario matrix is generated using the heuristic moment matching technique that captures the stochastic moments and correlation among historical wind and photovoltaic power, and electricity demand. The scenario matrix is then incorporated to propose a stochastic planning model that considers a multi-objective index for minimizing power losses and voltage deviation. Finally, the effectiveness of the proposed planning model is confirmed using case-studies in 53-bus and IEEE 123-bus distribution systems.

Correlation imaging in inverse scattering is tomography on probability distributions

Scattering from a non-smooth random field on the time domain is studied for plane waves that propagate simultaneously through the potential in variable angles. We first derive sufficient conditions for stochastic moments of the field to be recovered from empirical correlations between amplitude measurements of the leading singularities, detected in the exterior of a region where the potential is almost surely supported. The result is then applied to show that if two sufficiently regular random fields yield the same correlations, they have identical laws as function-valued random variables.

Scenario-based investment planning of isolated multi-energy microgrids considering electricity, heating and cooling demand

Multi-energy microgrids provide a flexible solution for the utilization of the distributed energy resources in order to meet the electrical, heating and cooling energy demands in the off-grid communities. However, the planning of the multi-energy microgrids is a non-trivial problem due to the complex energy flows between the sources and the loads pertaining to the electrical, heating and cooling energy, along with the intermittency of the renewable distributed generation. This work proposes a scenario-based stochastic multi-energy microgrid investment planning model that aims to minimize the investment and operation costs as well as the Carbon dioxide emissions by determining the optimal distributed energy resource mix, siting and sizing in the isolated microgrids. The proposed planning model employs the power flow and heat transfer equations to explicitly model the energy flows between electrical, heating and cooling energy sources and loads. Moreover, an uncertainty matrix is employed to tackle the operational uncertainties associated with the wind and photovoltaic generation, and the electrical, heating and cooling loads. The uncertainty matrix is modeled using the heuristic moment matching method that effectively captures the stochastic moments and correlation among the historical scenarios. The numerical results obtained from the case-study in the 19-bus microgrid test system confirm that the proposed methodology provides significant reductions in the investment and operation costs as well as the Carbon dioxide emissions. Finally, the superiority of the proposed planning solution is also validated using the deterministic planning solution as the comparison benchmark.

Maximum Entropy Probability Density Principle in Probabilistic Investigations of Dynamic Systems.

In this study, we consider a method for investigating the stochastic response of a nonlinear dynamical system affected by a random seismic process. We present the solution of the probability density of a single/multiple-degree of freedom (SDOF/MDOF) system with several statically stable equilibrium states and with possible jumps of the snap-through type. The system is a Hamiltonian system with weak damping excited by a system of non-stationary Gaussian white noise. The solution based on the Gibbs principle of the maximum entropy of probability could potentially be implemented in various branches of engineering. The search for the extreme of the Gibbs entropy functional is formulated as a constrained optimization problem. The secondary constraints follow from the Fokker–Planck equation (FPE) for the system considered or from the system of ordinary differential equations for the stochastic moments of the response derived from the relevant FPE. In terms of the application type, this strategy is most suitable for SDOF/MDOF systems containing polynomial type nonlinearities. Thus, the solution links up with the customary formulation of the finite elements discretization for strongly nonlinear continuous systems.

Stochastic process design kits for photonic circuits based on polynomial chaos augmented macro-modelling

Fabrication tolerances can significantly degrade the performance of fabricated photonic circuits and process yield. It is essential to include these stochastic uncertainties in the design phase in order to predict the statistical behaviour of a device before the final fabrication. This paper presents a method to build a novel class of stochastic-based building blocks for the preparation of Process Design Kits for the analysis and design of photonic circuits. The proposed design kits directly store the information on the stochastic behaviour of each building block in the form of a generalized-polynomial-chaos-based augmented macro-model obtained by properly exploiting stochastic collocation and Galerkin methods. Using these macro-models, only a single deterministic simulation is required to compute the stochastic moments of any arbitrary photonic circuit, without the need of running a large number of time-consuming circuit simulations thereby dramatically improving simulation efficiency. The effectiveness of the proposed approach is verified by means of classical photonic circuit examples with multiple uncertain variables.

A Stochastic Dynamical Model of Arctic Sea Ice

The noise forcing underlying the variability in the Arctic ice cover has a wide range of principally unknown origins. For this reason, the analytical and numerical solutions of a stochastic Arctic sea ice model are analyzed with both additive and multiplicative noise over a wide range of external heat fluxes Δ F0, corresponding to greenhouse gas forcing. The stochastic variability fundamentally influences the nature of the deterministic steady-state solutions corresponding to perennial and seasonal ice and ice-free states. Thus, the results are particularly relevant for the interpretation of the state of the system as the ice cover thins with Δ F0, allowing a thorough examination of the differing effects of additive versus multiplicative noise. In the perennial ice regime, the principal stochastic moments are calculated and compared to those determined from a stochastic perturbation theory described previously. As Δ F0increases, the competing contributions to the variability of the destabilizing sea ice–albedo feedback and the stabilizing longwave radiative loss are examined in detail. At the end of summer the variability of the stochastic paths shows a clear maximum, which is due to the combination of the increasing influence of the albedo feedback and an associated “memory effect,” in which fluctuations accumulate from early spring to late summer. This is counterbalanced by the stabilization of the ice cover resulting from the longwave loss of energy from the ice surface, which is enhanced during winter, thereby focusing the stochastic paths and decreasing the variability. Finally, common examples in stochastic dynamics with multiplicative noise are discussed wherein the choice of the stochastic calculus (Itô or Stratonovich) is not necessarily determinable a priori from observations alone, which is why both calculi are treated on equal footing herein.

Sublinear growth of the corrector in stochastic homogenization: optimal stochastic estimates for slowly decaying correlations

We establish sublinear growth of correctors in the context of stochastic homogenization of linear elliptic PDEs. In case of weak decorrelation and “essentially Gaussian” coefficient fields, we obtain optimal (stretched exponential) stochastic moments for the minimal radius above which the corrector is sublinear. Our estimates also capture the quantitative sublinearity of the corrector (caused by the quantitative decorrelation on larger scales) correctly. The result is based on estimates on the Malliavin derivative for certain functionals which are basically averages of the gradient of the corrector, on concentration of measure, and on a mean value property for a-harmonic functions.

Correlation Risk and International Portfolio Choice

We study the optimal portfolio choice of international investors when variances and correlations are stochastic. We assume that the returns from the perspective of the domestic investor are driven by a Wishart Affine Stochastic Correlation (WASC) model. We show that this also holds from the perspective of the foreign investor and give the relations between the variance-covariance matrices and the parameters of its dynamics in both currencies. Stochastic second moments have an impact on risk and returns that characterize the domestic and the foreign investment opportunity sets. Optimal portfolios and hedging demands of international investors differ due to their dependence on exchange rate variances and correlations. The benefits from investing can be different for domestic and foreign investors, and can also react differently to changes in second moments. These findings hold both in complete and incomplete markets.

Определение вероятностных моментов фазовых координат нелинейной модели конструкции

Определение вероятностных моментов фазовых координат нелинейной модели конструкции

Dynamics of a Fractional Derivative Type of a Viscoelastic Rod With Random Excitation

Abstract The axial vibrations of a viscoelastic rod with a body attached to its end are investigated. The problem is modelled by the constitutive equations with fractional derivatives as well as with the perturbations involving a bounded noise and a white noise process. The weak solutions for the equations given below in two cases of constitutive equations as well as their stochastic moments are determined.

Robust topology optimization of structures with imperfect geometry based on geometric nonlinear analysis

Topology optimization often leads to structures consisting of slender elements which are particularly sensitive to geometric imperfections. Such imperfections might affect the structural stability and induce large displacement effects in these slender structures. This paper therefore presents a robust approach to topology optimization which accounts for geometric imperfections and their potentially detrimental influence on the structural stability. Geometric nonlinear effects are incorporated in the optimization by means of a Total Lagrangian finite element formulation in the minimization of end-compliance. Geometric imperfections are modeled as a vector-valued random field in the design domain. The resulting uncertain performance of the design is taken into account by minimizing a weighted sum of the mean and standard deviation of the compliance in the robust optimization problem. These stochastic moments are typically estimated by means of sampling methods such as Monte Carlo simulation. However, these methods require multiple independent nonlinear finite element analyses in each design iteration of the optimization algorithm. An efficient solution algorithm which uses adjoint differentiation in a second-order perturbation method is therefore developed to estimate the stochastic moments during the optimization. Two applications with structures that exhibit different types of structural instabilities are examined. In both cases, it is demonstrated by means of an extensive Monte Carlo simulation that the deterministic design is very sensitive to imperfections, while the design obtained by means of the proposed method is much more robust.

Annealed estimates on the Green function

We consider a random, uniformly elliptic coefficient field $$a(x)$$ on the $$d$$ -dimensional integer lattice $$\mathbb {Z}^d$$ . We are interested in the spatial decay of the quenched elliptic Green function $$G(a;x,y)$$ . Next to stationarity, we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic Sobolev inequality holds for the ensemble $$\langle \cdot \rangle $$ . We prove that all stochastic moments of the first and second mixed derivatives of the Green function, that is, $$\langle |\nabla _x G(x,y)|^p\rangle $$ and $$\langle |\nabla _x\nabla _y G(x,y)|^p\rangle $$ , have the same decay rates in $$|x-y|\gg 1$$ as for the constant coefficient Green function, respectively. This result relies on and substantially extends the one by Delmotte and Deuschel (Probab Theory Relat Fields 133:358–390, 2005), which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of $$G$$ , that is, $$\langle |\nabla _x G(x,y)|^2\rangle $$ and $$\langle |\nabla _x\nabla _y G(x,y)|\rangle $$ . As an application, we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast.

Optimal design of fractional order linear system with stochastic inputs/parametric uncertainties by hybrid spectral method

This paper reports the design of a fractional linear system under stochastic inputs/uncertainties. The design methods were based on the hybrid spectral method for expanding the system signals over orthogonal functions. The use of the hybrid spectral method led to algebraic relationships between the first and second order stochastic moments of the input and output of a system. The spectral method could obtain a highly accurate solution with less computational demand than the traditional Monte Carlo method. Based on the hybrid spectral framework, the optimal design was elaborated by minimizing the suitably defined constrained-optimization problem.

Application of the generalized Polynomial Chaos expansion to the simulation of an internal combustion engine with uncertainties

The study of combustion associated phenomena in internal combustion engines is important for reasons that include pollutant formation control, and the heat transfer between the hot gases, the piston and combustion chamber walls. Although there are several models for the chemically reacting turbulent flow inside the combustion chamber, such models always suffer from some level of uncertainties in their formulations or in the physical parameters used for their simulation. The main objective of the present paper consists in modeling the combustion phenomena in a combustion chamber of an internal combustion engine, using a model based on the First Law of Thermodynamics, considering uncertainties in the mass fraction of burned fuel. As a result, the curve describing the in-cylinder pressure as a function of the crank angle can be obtained for different uncertainties levels. For this purpose, the generalized Polynomial Chaos (gPC) and Monte Carlo techniques were applied to a set of ordinary differential equations obtained from a First Law analysis, and the stochastic moments of the solution were obtained. Uniform and Gaussian uncertainties distributions were considered in order to obtain the stochastic solution. Results show that the gPC is more computationally efficient than the Monte Carlo simulation for the considered application . Besides, experimental engine results were also used in order to validate the implemented numerical procedure.

Momentos estocásticos de orden superior y la estimación de la volatilidad implícita: aplicación de la expansión de Edgeworth en el modelo Black-Scholes

In this document the Edgeworth expansion is used in the Black-Scholes model for estimating the implicit volatility and the impact of the higher order stochastic moments on the option price, over Grupo Financiero Galicia (GGAL) stock options contracts trading in the Buenos Aires Stocks Exchange (Argentina). First, the underlying probability distribution of returns is analysed; then the model is subject to iteration to obtain implicit values for volatility, skewnness and kurtosis. The main conclusions are the flat shape of the volatility curve of the model, and the significant weight of the skewnnes and kurtosisof the «in the money, out of the money» prices.

Influence of second-order random wave kinematics on the design loads of offshore wind turbine support structures

The impact of wave model nonlinearities on the design loads of wind turbine monopile foundations is delineated based on a second-order nonlinear random wave model that satisfies the boundary conditions at the free surface and by including the effects of convective acceleration in the inertial loads. The second-order nonlinear water kinematics is developed based on a Gram Charlier series expansion using the first four stochastic moments of the wave process. The wave surface velocities and accelerations are expressed using a Taylor series expansion about the mean sea level, which satisfies to the second-order, the unsteady Bernoulli equation and normal flow condition at the free surface. The operating design loads on the monopile are computed using fully coupled and uncoupled methods. The computation of the mud level loads based on the inclusion of nonlinear wave surface kinematics is compared with those obtained when using linear waves and Wheeler stretching. The effect of the spatial derivatives of the wave velocity on the wave surface kinematics is quantified and shown to determine the wave spectral cut-off frequency limit. The spatial derivatives of wave velocity also participate in the expression for the wave convective acceleration, whose effect is demonstrated on the inertial loads on the foundation in the presence of ocean currents. The effect of nonlinear water kinematics on the monopile design load reveals the large frequency bandwidth of wave structure interaction, but the phase differences between the hydrodynamic loads with the rotor loads tend to lower the probability of joint simultaneous extreme peaks in hydrodynamics and rotor loads.