input-output data. Even if some knowledge regarding the parameters is available, the values of the model parameters need to be adjusted so that the model predictions are in better agreement with experimental data. This adjustment is referred to as model calibration. In the remainder of this dissertation, the terms “model parameter estimation” and “model calibration” are used synonymously.
Having calibrated the model, the fifth step is model validation which answers the question - Is the mathematical equation an accurate representation of reality?. The process of model validation aims to quantify the deviation of the model from reality (referred to as model form error or model discrepancy term or model inadequacy function) and hence, assess the predictive capability of the model.
Hence, there are three major activities related to the quantification of model un- certainty - model verification, calibration, and validation. Note that the steps of verification, validation, and calibration are not necessarily in a fixed sequence; differ- ent sequences might be suitable for different problems and there might be iterations between some of the steps. For example, it may be desirable to perform calibration before and after validation. The three topics of verification, validation, and calibra- tion are discussed in detail in the following sections.
verification [103–105] and estimation of solution approximation error [91, 106–109]
have been investigated by several researchers.
In general, the solution approximation error is composed of both deterministic and stochastic terms [108]. For example, the discretization error arising in finite element analysis is deterministic, while the surrogate model error that arises as a result of replacing the finite element analysis with a surrogate model is stochastic.
In the context of uncertainty propagation, deterministic errors can be addressed by correcting the bias, and the corrected solutions are used to train the surrogate model;
the stochastic errors of the surrogate model can be addressed through sampling based on their estimated distributions. As a result, the overall solution approximation error is also stochastic.
The true solution of the mathematical equation can be computed as a function of the model inputs and parameters as y(x;θ) = yc(x;θ) +Gse(x;θ). Since Gse(x;θ) is stochastic, y is stochastic even for given values of x and θ. The remainder of this subsection discusses the estimation of discretization error and surrogate model uncertainty.
4.2.1 Discretization Error
The most common type of solution approximation error is due to discretization in finite element analysis, and methods of convergence analysis [110], a-posteriori error estimation [111], Richardson extrapolation [109, 112, 113], etc. have been studied for the estimation of discretization error. Rebba et al. [91] state that the method of a- posterior error estimation [111, 114, 115] quantifies only a surrogate measure of error to facilitate adaptive mesh refinement, but does not compute the actual discretization error. On the other hand, the method of Richardson extrapolation has been found to come closest to quantifying the actual discretization error [91, 109, 112].
Lethdenote the mesh size used in finite element analysis and Ψ the corresponding prediction. Let y denote the “true” solution of the mathematical equation which is obtained as h tends to zero. According to the basic Richardson extrapolation [112], the relation between h and yc can be expressed as:
y=yc+Ahp (4.1)
In Eq. 4.1, p is the order of convergence, A is the polynomial coefficient. In order to estimate the true solution y, three different mesh sizes (h1 < h2 < h3) are considered and the corresponding finite element solutions (yc(h1) = Ψ1, yc(h2) = Ψ2, yc(h3) = Ψ3) are calculated. Eq. 4.1 has three unknownsp, A, and y, which can be estimated based on the three mesh solutions. Mesh doubling/halving is commonly done to simplify the equations. Ifr = hh32 = hh21, then the discretization error (ǫh) and the true solution can be calculated as:
y= Ψ1−ǫh
Ψ2−Ψ1 =ǫh(rp−1) p log(r) = log(ΨΨ32−−ΨΨ11)
(4.2)
The solutions Ψ1 , Ψ2 , Ψ3 are dependent on both x and θ and hence the error estimate ǫh and the true solution y are also functions of both x and θ. Since the discretization error is a deterministic quantity, it needs to be corrected for, in the context of uncertainty propagation.
The use of Richardson extrapolation requires uniform meshing and uniform con- vergence, thereby limiting the applicability of this method in practical finite element analysis. Recently, Rangavajhala et al. [109] has developed a method to overcome these limitations by extending the Richardson extrapolation methodology from a polynomial relation (Eq. 4.1) to a more flexible Gaussian process extrapolation; this
GP is used to extrapolate to h = 0 in order to estimate the discretization error. In that case, due to the uncertainty associated with GP interpolation, the discretiza- tion error is also stochastic; therefore, the training points (in particular, the output values) for the surrogate model are stochastic, and it is necessary to account for this uncertainty while constructing the surrogate model. Rasmussen [48, 51, 80] discusses constructing GP models when the training point values are stochastic.
4.2.2 Surrogate Model Uncertainty
Another type of solution approximation error arises when the underlying model is replaced with a surrogate model for fast uncertainty propagation and/or model calibration. Surrogate model error is stochastic, even for a given realization of inputs and parameters. As discusses earlier in Section 2.8, different types of surrogate mod- eling techniques (regression models [27], polynomial chaos expansions [116], radial basis functions [117], support vector machines [46], relevance vector machines [47], Gaussian processes [118]) are available in the literature, and the quantification of the surrogate model error is different for different types of surrogate models. Methods of the quantification of this error (for different surrogate models) are well-established in the literature.
As stated earlier in Section 2.8, this dissertation uses the Gaussian process model as a surrogate to replace expensive computer simulations. There are three important reasons why a Gaussian process model has been used in this research work:
1. The GP model is capable of capturing highly nonlinear relationships that exist between input and output variables without the need for an explicit functional form. Hence, a closed form expression (as in polynomial type regression meth- ods) need not be assumed.
2. For a non-parametric interpolation technique, this method requires fewer sample
points (usually 30 or less) as against methods such as kernel estimation and non- parametric multiplicative regression.
3. A GP model provides a direct estimate of the variance in the output prediction.
Gaussian process interpolation method was explained earlier in Section 2.8 and the Gaussian process prediction (mean and variance) was given by Eq. 2.19. It was emphasized that the choice of training points is important for the construction of the GP model. In Section 2.8, the training points were created based on input-output data of the expensive computer model. Now, the difference is that the training values need to be generated by considering the model parameters (θ) in addition to the inputs (x), because the value of the model parameter is also necessary to execute the model and compute the model output.
Once the surrogate model is constructed, the expected value and variance of the Gaussian process prediction can then be used to draw multiple samples for uncertainty analysis, thereby including the effect of surrogate model uncertainty in uncertainty propagation.