Polynomial Chaos Expansion Approach Research Articles

The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach

We consider the stochastic linear quadratic optimal control problem for state equations of the Ito-Skorokhod type, where the dynamics are driven by strongly continuous semigroup. We provide a numerical framework for solving the control problem using a polynomial chaos expansion approach in white noise setting. After applying polynomial chaos expansion to the state equation, we obtain a system of infinitely many deterministic partial differential equations in terms of the coefficients of the state and the control variables. We set up a control problem for each equation, which results in a set of deterministic linear quadratic regulator problems. Solving these control problems, we find optimal coefficients for the state and the control. We prove the optimality of the solution expressed in terms of the expansion of these coefficients compared to a direct approach. Moreover, we apply our result to a fully stochastic problem, in which the state, control and observation operators can be random, and we also consider an extension to state equations with memory noise.

Prediction of the intramembranous tissue formation during perisprosthetic healing with uncertainties. Part 2. Global clinical healing due to combination of random sources

This work proposes to examine the variability of the bone tissue healing process in the early period after the implantation surgery. The first part took into account the effect of variability of individual biochemical factors on the solid phase fraction, which is an indicator of the quality of the primary fixation and condition of its long-term behaviour. The next issue, addressed in this second part, is the effect of cumulative sources of uncertainties on the same problem of a canine implant. This paper is concerned with the ability to increase the number of random parameters to assess the coupled influence of those variabilities on the tissue healing. To avoid an excessive increase in the complexity of the numerical modelling and further, to maintain efficiency in computational cost, a collocation-based polynomial chaos expansion approach is implemented. A progressive set of simulations with an increasing number of sources of uncertainty is performed. This information is helpful for future implant design and decision process for the implantation surgical act.

Uncertainty Quantification of Subcritical Nonlinear Aeroelastic System Using Integrated Interpolation Method and Polynomial Chaos Expansion

The present paper is about analyzing and quantifying parametric uncertainties in a nonlinear aeroelastic system. The response surface of a subcritical system has discontinuities which makes it difficult to analyze using spectral approaches. A quadrature based spectral parametric uncertainty tool called nonintrusive spectral projection (NISP) can capture discontinuities in the response surface only by infinite number of points. As the number of random variable increases, the order of expansion has to increase drastically in order to capture the response accurately. Hence an alternate approach using integrated interpolation scheme which was earlier applied for a thermoacoustic problem is being applied to a classical flutter problem. It is found to be accurate and computationally cheaper than traditional Monte Carlo simulations (MCS) in capturing the response surface. This paper intends to apply the interpolation scheme to the nonlinear aeroelastic model and compares it with quadrature based polynomial chaos expansion (PCE) approach and MCS.

Polynomial chaos expansion with random and fuzzy variables

A dynamical uncertain system is studied in this paper. Two kinds of uncertainties are addressed, where the uncertain parameters are described through random variables and/or fuzzy variables. A general framework is proposed to deal with both kinds of uncertainty using a polynomial chaos expansion (PCE). It is shown that fuzzy variables may be expanded in terms of polynomial chaos when Legendre polynomials are used. The components of the PCE are a solution of an equation that does not depend on the nature of uncertainty. Once this equation is solved, the post-processing of the data gives the moments of the random response when the uncertainties are random or gives the response interval when the variables are fuzzy. With the PCE approach, it is also possible to deal with mixed uncertainty, when some parameters are random and others are fuzzy. The results provide a fuzzy description of the response statistical moments.

A Computational Spectral Approach to Interest Rate Models

The Polynomial Chaos Expansion (PCE) technique recovers a finite second order random variable exploiting suitable linear combinations of orthogonal polynomials which are functions of a given stochastic quantity $\xi$, hence acting as a kind of random basis. The PCE methodology has been developed as a mathematically rigorous Uncertainty Quantification (UQ) method which aims at providing reliable numerical estimates for some uncertain physical quantities defining the dynamic of certain engineering models and their related simulations. In the present paper we exploit the PCE approach to analyze some equity and interest rate models considering, without loss of generality, the one dimensional case. In particular we will take into account those models which are based on the Geometric Brownian Motion (gBm), e.g. the Vasicek model, the CIR model, etc. We also provide several numerical applications and results which are discussed for a set of volatility values. The latter allows us to test the PCE technique on a quite large set of different scenarios, hence providing a rather complete and detailed investigation on PCE-approximation's features and properties, such as the convergence of statistics, distribution and quantiles. Moreover we give results concerning both an efficiency and an accuracy study of our approach by comparing our outputs with the ones obtained adopting the Monte Carlo approach in its standard form as well as in its enhanced version.

A stochastically and spatially adaptive parallel scheme for uncertain and nonlinear two-phase flow problems

Simulating in flow problems in porous media often requires techniques for uncertainty quantification in order to represent parameter values that are not given exactly. Straightforward Monte-Carlo (MC) methods have a limited efficiency due to slow convergence. Better convergence in low-parametric problems can be achieved with polynomial chaos expansion (PCE) techniques. The PCE approach yields a coupled deterministic system to be solved. The degree of coupling increases with the non-linearity of the considered equations and with the order of polynomial expansion. This fact increases the computational effort of PCE and significantly reduces the scalability in parallelization. We present an application of the hybrid stochastic Galerkin finite volume (HSG-FV) method to a two-phase flow problem in two space dimensions. The method extends the classical polynomial chaos expansion by a multi-element discretization in the probability space of the parameters. It leads to a deterministic system that is coupled to a lesser degree than in element-free PCE versions, respectively, fully decoupled in stochastic elements (SE). Therefore, the HSG-FV method allows for more efficient parallelization. For the further reduction of complexity, we present a new stochastic adaptivity method. We present numerical examples in two spatial dimensions with linear and nonlinear fractional flow functions in the two-phase flow problem. The flow functions might depend in a discontinuous manner on the unknown spatial position of porous-medium heterogeneities. Finally, we discuss the interplay of the new method with spatial adaptivity per SE for these problems.

Polynomial Chaos Expansion Approach to Interest Rate Models

The Polynomial Chaos Expansion (PCE) technique allows us to recover a finite second-order random variable exploiting suitable linear combinations of orthogonal polynomials which are functions of a given stochastic quantityξ, hence acting as a kind of random basis. The PCE methodology has been developed as a mathematically rigorous Uncertainty Quantification (UQ) method which aims at providing reliable numerical estimates for some uncertain physical quantities defining the dynamic of certain engineering models and their related simulations. In the present paper, we use the PCE approach in order to analyze some equity and interest rate models. In particular, we take into consideration those models which are based on, for example, the Geometric Brownian Motion, the Vasicek model, and the CIR model. We present theoretical as well as related concrete numerical approximation results considering, without loss of generality, the one-dimensional case. We also provide both an efficiency study and an accuracy study of our approach by comparing its outputs with the ones obtained adopting the Monte Carlo approach, both in its standard and its enhanced version.

Relating in situ gas measurements to the surface outgassing properties of cometary nuclei

The sensitivity of the gas flow field to changes in different initial conditions has been studied for the case of a highly simplified cometary nucleus model. The nucleus model simulated a homogeneously outgassing sphere with a more active ring around an axis of symmetry. The varied initial conditions were the number density of the homogeneous region, the surface temperature, and the composition of the flow (varying amounts of H2O and CO2) from the active ring.The sensitivity analysis was performed using the Polynomial Chaos Expansion (PCE) method. Direct Simulation Monte Carlo (DSMC) was used for the flow, thereby allowing strong deviations from local thermal equilibrium. The PCE approach can be used to produce a sensitivity analysis with only four runs per modified input parameter and allows one to study and quantify non-linear responses of measurable parameters to linear changes in the input over a wide range. Hence the PCE allows one to obtain a functional relationship between the flow field properties at every point in the inner coma and the input conditions. It is for example shown that the velocity and the temperature of the background gas are not simply linear functions of the initial number density at the source.As probably expected, the main influence on the resulting flow field parameter is the corresponding initial parameter (i.e. the initial number density determines the background number density, the temperature of the surface determines the flow field temperature, etc.). However, the velocity of the flow field is also influenced by the surface temperature while the number density is not sensitive to the surface temperature at all in our model set-up. Another example is the change in the composition of the flow over the active area. Such changes can be seen in the velocity but again not in the number density.Although this study uses only a simple test case, we suggest that the approach, when applied to a real case in 3D, should assist in identifying the sensitivity of gas parameters measured in situ by, for example, the Rosetta spacecraft to the surface boundary conditions and vice versa.

Scattering From Two-Dimensional Objects of Varying Shape Combining the Multilevel Fast Multipole Method (MLFMM) With the Stochastic Galerkin Method (SGM)

In this letter, the Multilevel Fast Multipole Method (MLFMM) is combined with the polynomial chaos expansion (PCE) approach to model the stochastic variations of a scatterer. In particular, it is demonstrated how the Stochastic Galerkin Method (SGM) can be combined with an MLFMM accelerated method of moments (MoM) and how the beneficial effects of the MLFMM for electromagnetically large scatterers are retained in the stochastic case.

Global Sensitivity Analysis in a Multi-State Physics Model of Component Degradation Based on a Hybrid State-Space Enrichment and Polynomial Chaos Expansion Approach

This paper extends previous works related to the assessment of component degradation, through Markov multi-state physic models. The extension includes the evaluation of the effects of uncertain parameters in the model, and the definition of their importance with respect to their influence on the output of the model. Global Sensitivity Analysis (GSA) is selected as the technique because it enables us to 1) consider the simultaneous effects of parameters variations, and 2) to define importance indexes that allow a ranking of the components. GSA requires a large number of evaluations for specific points, identified by an appropriate design of experiment. To avoid the many costly evaluations, a meta-model is built based on polynomial chaos expansion (PCE). A PCE is a multi-dimensional polynomial approximation of the model with coefficients determined by evaluating the model in a reduced set of predetermined points. Importance index values are then derived directly from the PCE. Because, in the problem considered, the model provides the time-dependent behavior of the state probabilities, the importance indexes are also functions of time. An application is presented, related to the cracking process in an Alloy 82/182 dissimilar metal weld in the primary coolant system of a nuclear power plant.

Generalised polynomial chaos expansion approaches to approximate stochastic model predictive control†

This paper considers the model predictive control of dynamic systems subject to stochastic uncertainties due to parametric uncertainties and exogenous disturbance. The effects of uncertainties are quantified using generalised polynomial chaos expansions with an additive Gaussian random process as the exogenous disturbance. With Gaussian approximation of the resulting solution trajectory of a stochastic differential equation using generalised polynomial chaos expansion, convex finite-horizon model predictive control problems are solved that are amenable to online computation of a stochastically robust control policy over the time horizon. Using generalised polynomial chaos expansions combined with convex relaxation methods, the probabilistic constraints are replaced by convex deterministic constraints that approximate the probabilistic violations. This approach to chance-constrained model predictive control provides an explicit way to handle a stochastic system model in the presence of both model uncertainty and exogenous disturbances.

Transient Response of Structural Dynamic Systems with Parametric Uncertainty

The time-domain response of a randomly parameterized structural dynamic system is investigated with a polynomial chaos expansion approach and a stochastic Krylov subspace projection, which has been proposed here. The latter uses time-adaptive stochastic spectral functions as weighting functions of the deterministic orthogonal basis onto which the solution is projected. The spectral functions are rational functions of the input random variables and depend on the spectral properties of the unperturbed system. The stochastic system response can be accurately resolved even when using low-order spectral functions, which are computationally advantageous. The time integration required for the resolution of the transient stochastic response has been performed with the unconditionally stable single-step implicit Newmark scheme using a stochastic integration operator. A semistatistical hybrid analytical and simulation-based computational approach has been utilized to obtain the moments and probability density functions of the solution. The simulations have been performed for different degrees of variability of the input randomness and different dimensions of the input stochastic space and compared with the direct Monte-Carlo simulations for accuracy and computational efficiency.

Stochastic Analysis of the Eigenvalue Problem for Mechanical Systems Using Polynomial Chaos Expansion— Application to a Finite Element Rotor

This paper proposes to use a polynomial chaos expansion approach to compute stochastic complex eigenvalues and eigenvectors of structures including damping or gyroscopic effects. Its application to a finite element rotor model is compared to Monte Carlo simulations. This lets us validate the method and emphasize its advantages. Three different uncertain configurations are studied. For each, a stochastic Campbell diagram is proposed and interpreted and critical speeds dispersion is evaluated. Furthermore, an adaptation of the Modal Accordance Criterion (MAC) is proposed in order to monitor the eigenvectors dispersion.

Hybrid perturbation-Polynomial Chaos approaches to the random algebraic eigenvalue problem

The analysis of structures is affected by uncertainty in the structure’s material properties, geometric parameters, boundary conditions and applied loads. These uncertainties can be modelled by random variables and random fields. Amongst the various problems affected by uncertainty, the random eigenvalue problem is specially important when analyzing the dynamic behavior or the buckling of a structure. The methods that stand out in dealing with the random eigenvalue problem are the perturbation method and methods based on Monte Carlo Simulation. In the past few years, methods based on Polynomial Chaos (PC) have been developed for this problem, where each eigenvalue and eigenvector are represented by a PC expansion. In this paper four variants of a method hybridizing perturbation and PC expansion approaches are proposed and compared. The methods use Rayleigh quotient, the power method, the inverse power method and the eigenvalue equation. PC expansions of eigenvalues and eigenvectors are obtained with the proposed methods. The new methods are applied to the problem of an Euler Bernoulli beam and a thin plate with stochastic properties.

Robust Hierarchical Multiscale Design Using PATC and PCE

Multiscale design for dealing with 2-scale material and product system is implemented by employing the probabilistic analytical target cascading (PATC) and polynomial chaos expansion (PCE) approaches in this paper. PATC allows design autonomy at each scale subsystem by formulating the multiscale design problem as a hierarchical structure. PCE ensures uncertainties to be propagated within and across each scale accurately and efficiently. In addition, correlation between the random inputs is also considered during uncertainty propagation. Comparative study on a multiscale bracket design problem shows that the results obtained by our strategy are very close to the reference values. It is demonstrated that PATC and PCE are effective and applicable on multiscale design.

Stochastic Modeling of the Pull-In Voltage in a MEMS Beam Structure

This paper deals with a stochastic approach to model the pull-in voltage in a microelectromechanical system (MEMS) beam structure. Two nonintrusive stochastic techniques are used: a sampling method, based on Monte-Carlo simulations and a polynomial chaos expansion (PCE) approach. The deterministic model solves an electro-mechanical problem using a coupled 2-D finite element (FE) electrostatic and Euler-Bernoulli beam equation mechanical model. The influence on the pull-in voltage of three parameters subjected to uncertainty (material properties and shape) was investigated. Also, the advantage of the PCE method, compared to Monte-Carlo sampling method, was emphasized.

Boundary procedures for the time-dependent Burgers' equation under uncertainty

The Burgers' equation with uncertain initial and boundary conditions is approximated using a Polynomial Chaos Expansion (PCE) approach where the solution is represented as a series of stochastic, orthogonal polynomials. The resulting truncated PCE system is solved using a novel numerical discretization method based on spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability. The resulting PCE solution yields an accurate quantitative description of the stochastic evolution of the system, provided that appropriate boundary conditions are available. The specification of the boundary data is shown to inuence the solution; we will discuss the problematic implications of the lack of precisely characterized boundary data and possible ways of imposing stable and accurate boundary conditions.

Numerical analysis of the Burgers’ equation in the presence of uncertainty

The Burgers’ equation with uncertain initial and boundary conditions is investigated using a polynomial chaos (PC) expansion approach where the solution is represented as a truncated series of stochastic, orthogonal polynomials. The analysis of well-posedness for the system resulting after Galerkin projection is presented and follows the pattern of the corresponding deterministic Burgers equation. The numerical discretization is based on spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability. Similarly to the deterministic case, the explicit time step for the hyperbolic stochastic problem is proportional to the inverse of the largest eigenvalue of the system matrix. The time step naturally decreases compared to the deterministic case since the spectral radius of the continuous problem grows with the number of polynomial chaos coefficients. An estimate of the eigenvalues is provided. A characteristic analysis of the truncated PC system is presented and gives a qualitative description of the development of the system over time for different initial and boundary conditions. It is shown that a precise statistical characterization of the input uncertainty is required and partial information, e.g. the expected values and the variance, are not sufficient to obtain a solution. An analytical solution is derived and the coefficients of the infinite PC expansion are shown to be smooth, while the corresponding coefficients of the truncated expansion are discontinuous.

Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods

An efficient method for uncertainty analysis of flow in random porous media is explored in this study, on the basis of combination of Karhunen‐Loeve expansion and probabilistic collocation method (PCM). The random log transformed hydraulic conductivity field is represented by the Karhunen‐Loeve expansion and the hydraulic head is expressed by the polynomial chaos expansion. Probabilistic collocation method is used to determine the coefficients of the polynomial chaos expansion by solving for the hydraulic head fields for different sets of collocation points. The procedure is straightforward and analogous to the Monte Carlo method, but the number of simulations required in PCM is significantly reduced. Steady state flows in saturated random porous media are simulated with the probabilistic collocation method, and comparisons are made with other stochastic methods: Monte Carlo method, the traditional polynomial chaos expansion (PCE) approach based on Galerkin scheme, and the moment‐equation approach based on Karhunen‐Loeve expansion (KLME). This study reveals that PCM and KLME are more efficient than the Galerkin PCE approach. While the computational efforts are greatly reduced compared to the direct sampling Monte Carlo method, the PCM and KLME approaches are able to accurately estimate the statistical moments and probability density function of the hydraulic head.

Using stochastic analysis to capture unstable equilibrium in natural convection

A stabilized stochastic finite element implementation for the natural convection system of equations under Boussinesq assumptions with uncertainty in inputs is considered. The stabilized formulations are derived using the variational multiscale framework assuming a one-step trapezoidal time integration rule. The stabilization parameters are shown to be functions of the time-step size. Provision is made for explicit tracking of the subgrid-scale solution through time. A support-space/stochastic Galerkin approach and the generalized polynomial chaos expansion (GPCE) approach are considered for input–output uncertainty representation. Stochastic versions of standard Rayleigh–Bènard convection problems are used to evaluate the approach. It is shown that for simulations around critical points, the GPCE approach fails to capture the highly non-linear input uncertainty propagation whereas the support-space approach gives fairly accurate results. A summary of the results and findings is provided.