Sparse spectral surrogate models for deterministic and stochastic computer simulations

Abstract

Computer simulations are an invaluable tool for modeling and investigating real-world phenomena and processes. However, as any model, simulations are affected by uncertainty caused by imperfect knowledge or natural variability of their parameters, initial conditions, or input values. This leads to uncertainty in the model response, which needs to be quantified to make subsequent conclusions and decisions trustworthy. To alleviate the considerable cost of uncertainty analyses for expensive computational models, the latter are often replaced by surrogates, i.e., by approximations with an explicit functional form that can be created based on a rather small number of model evaluations, and can be evaluated at low cost. Some computer models are affected by uncertainty only through their input parameters: for fixed values of the inputs, they always produce the same response. These models are called deterministic simulators. In contrast, models that feature inherent stochasticity are called stochastic simulators. The latter generate a different result each time they are run even if their input parameters are held at fixed values. In other words, they behave like random fields whose index set is the space of input parameters. In this thesis, we investigate spectral surrogate models, which are a class of global non-intrusive methods that expand the computational model onto an orthonormal basis of a suitable function space. We focus on sparse expansions, i.e., representations that only include a small finite subset of the basis elements. Sparse representations are typically computed by regression with sparsity-encouraging constraints, often using ideas originating from the field of compressed sensing. In particular, for deterministic simulators we explore the popular sparse polynomial chaos expansions (PCE) method, which utilizes a polynomial basis that is orthonormal with respect to the distribution of the input variables. We conduct an extensive literature survey as well as a benchmark of several promising methods on multiple models of varying dimensionality and complexity. The benchmark results are aggregated and visualized in a novel way to extract reliable recommendations about which methods should be used in practice. We also investigate the recently proposed Poincaré chaos expansions, which rely on a generally non-polynomial basis consisting of eigenfunctions of a specific differential operator connected to the Poincaré inequality. By construction, this basis is well suited for derivative-based global sensitivity analysis, which we explore both analytically and numerically. Furthermore, we propose a new surrogate model for stochastic simulators. Taking the random function view of a stochastic simulator, we approximate its trajectories by sparse PCE and perform Karhunen-Loève expansion on them. The latter is a well-known spectral representation for a random field which separately characterizes its spatial and stochastic variation. The joint distribution of the random coefficients is inferred using the marginal-copula framework. The resulting surrogate model is able to approximate marginal distributions, mean, and covariance function of the stochastic simulator, and can generate new trajectories.