Extending bluff-and-fix estimates for polynomial chaos expansions
Introduction
Uncertainty quantification (UQ) in physical models, including those governed by systems of partial differential equations, is important for building confidence in the resulting predictions. Uncertainties can originate from imperfect knowledge of boundary or initial conditions in the system of interest or variability in the operating conditions. A common approach is to represent the sources of uncertainty as stochastic variables; in this context the solution to the original differential equation becomes a random quantity. Stochastic Galerkin schemes (SGS) are used to approximate the solution to parametrized differential equations [2], [3], [4]. In particular, they utilize a functional basis in the parameter space to express the solution and then derive and solve a deterministic system of PDEs with standard numerical techniques [5], [6]. A Galerkin method projects the randomness in a solution onto a finite-dimensional basis, making deterministic computations possible. SGS are part of a broader class known as spectral methods [7], [8], [9].
The most common UQ strategies involve Monte Carlo (MC) algorithms, which suffer from a slow convergence rate proportional to the inverse square root of the number of samples [10]. If each sample evaluation is expensive — as is often the case for the solution of a PDE — this slow convergence can make obtaining tens of thousands of evaluations computationally infeasible [8]. Initial spectral method applications to UQ problems showed orders-of-magnitude reductions in the cost needed to estimate statistics with comparable accuracy [11].
In the conventional approach for SGS, the unknown quantities are expressed as infinite series of orthogonal polynomials in the space of the random input variables. This representation has its roots in the work of Wiener [12], who expressed a Gaussian process as a countably infinite series of Hermite polynomials. Ghanem and Spanos [7] truncated Wiener’s representation and used the resulting finite series as a key ingredient in a stochastic finite element method. SGS based on polynomial expansions are often referred to as polynomial chaos approaches. In contrast to sampling methods (e.g. Monte Carlo simulations), polynomial chaos approaches are intrusive methods i.e. they require the solution of a mathematical problem that is different from the one originally considered. Uncertain quantities are represented as expansions that separate deterministic coefficients from a chosen random orthogonal basis [13].
Let be a subset of the spatial and time domain .1 Then let be continuous and differential in its space and time components; further, let . This represents the solution to a differential equation, Here is a general differential operator that contains both spatial and temporal derivatives. Often is assumed to be nonlinear.
Let be a zero mean, square-integrable, real random variable. We assume uncertainty is present in the initial condition and represent it by setting where is a known function of and . Accordingly, the solution to is now a random variable indexed by , meaning is a stochastic process.
As statistics of , we require that both and have existing second moments. Note that these are the only restrictions; namely, even though is chosen as sinusoidal in the example of Section 2, we do not require to be periodic, bounded over the real line, zero on the whole boundary , etc.
We consider a polynomial chaos expansion (PCE) and separate the deterministic and random components of by writing The are deterministic coefficients, while the are orthogonal polynomials with respect to the measure induced by . Let denote the inner product mapping . The triple-product notation is understood as and the singleton as . Then we require the to satisfy the properties [5] where the are nonzero and is the Kronecker delta. Note that the Hermite are necessarily independent as random variables in from their orthogonality.
Let be a stochastic process indexed by . By the Cameron–Martin theorem2 the truncated PCE denoted by converges in mean square,3 at every fixed in the domain . This justifies the PCE and its truncation to a finite number of terms for the sake of computation.
Stochastic properties of the solution can be readily computed [8], [15], [17], [18] given the attributes in (2). Namely, Going forward, the symbol denotes a deterministic function in and in addition to the usual expectation operator.
Similarly, the variance of the solution is Note that variance is defined, because we assume has existing second moments over all indexes. Using the truncated expansion , we can approximate the solution’s true variance as4 As mentioned in the abstract, this paper is an extended version of [1] that considers a PDE system obtained by explicitly representing the PDEs in terms of and . In Section 3.4, our objective will not be to compute but instead to estimate and . The beauty is that, unlike the case of approximating by , not all of the coefficients functions need to be determined perfectly in order to approximate and accurately. In this sense, the algorithm we introduce (bluff-and-fix) is preferable to conventional methods that do not have the flexibility to choose which are more important to estimate well.
Substituting the truncation into Eq. (1), we have Furthermore, we can determine the initial conditions for the deterministic component functions. Multiplying by any and integrating with respect to the -measure yields The scalars of course are dependent on the choice of the orthogonal polynomials. Consequently, the initial conditions for and are In a similar manner, we can “integrate away” the randomness in Eq. (7) by projecting onto each basis polynomial. This is discussed in detail in the next section.
Section snippets
Inviscid Burgers’ Equation
Our choice of orthogonal polynomials relies on the distribution of the random variable. Throughout this paper, we will choose and the to be Hermite polynomials; however, many of the results apply almost identically to other distributions and their corresponding polynomials, with some caveats around convergence [15], [20].
Note that Hermite polynomials satisfy and by [21], where . Now
Bluff-and-fix (BNF) Algorithm
Recall that we are solving inviscid Burgers’ equation with the uncertain initial condition for , as given by Eq. (10). In Sections 3.3 Iterative bluff-and-fix, 3.1 Solving the, 3.2 One step bluff-and-fix, the goal is to solve the system in Eq. (11) for the coefficient functions numerically, therefore giving an approximation of by the partial summation . In Section 3.4, the objective instead is to estimate the mean and variance of the
Numerical Results
We report solutions for Burgers’ equation with an uncertain initial condition, namely for . The equation is solved for on a uniform grid with . Time integration is based on the Runge–Kutta 4-step (RK4) scheme with and .
Throughout this discussion, we will define error as the deviation of the solution approximation produced by bluff-and-fix from the solution yielded from solving the full system via RK4. That is, our computations
Conclusion
Polynomial chaos (PC) methods are effective for incorporating and quantifying uncertainties in problems governed by partial differential equations. In this paper, which is an extended version of [1], we present a promising algorithm (one step bluff-and-fix) for utilizing the solution to a polynomial chaos system arising from inviscid Burgers’ equation to approximate the solution to the corresponding system. Bluff-and-fix is considered in the context of inviscid Burgers’ equation to
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References (22)
- et al.
Solution of stochastic partial differential equations using galerkin finite element techniques
Computer Methods in Applied Mechanics and Engineering
(2001) - et al.
Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations
Computer Methods in Applied Mechanics and Engineering
(2005) A generalized spectral decomposition technique to solve stochastic partial differential equations
Computer Methods in Applied Mechanics and Engineering
(2007)- et al.
A bluff-and-fix algorithm for polynomial chaos methods
- et al.
Stochastic Galerkin methods and model order reduction for linear dynamical systems
International Journal for Uncertainty Quantification
(2015) A primer on stochastic Galerkin methods
(2007)- et al.
Stochastic Finite Elements: A Spectral Approach
(1991) Spectral Methods for Parametrized Matrix Equations
(2009)- et al.
Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics
(2010) Monte Carlo Theory, Methods and Examples
(2013)
The Wiener–Askey polynomial chaos for stochastic differential equations
SIAM J. Sci. Comput.
Cited by (2)
20 years of computational science: Selected papers from 2020 International Conference on Computational Science
2021, Journal of Computational ScienceSecond Order Moments of Multivariate Hermite Polynomials in Correlated Random Variables
2021, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)