In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids.

In calculating expected information gain in optimal Bayesian experimental design, the computation of the inner loop in the classical double-loop Monte Carlo requires a large number of samples and suffers from underflow if the number of samples is small. These drawbacks can be avoided by using an importance sampling approach. We present a computationally efficient method for optimal Bayesian experimental design that introduces importance sampling based on the Laplace method to the inner loop. We derive the optimal values for the method parameters in which the average computational cost is minimized according to the desired error tolerance. We use three numerical examples to demonstrate the computational efficiency of our method compared with the classical double-loop Monte Carlo, and a more recent single-loop Monte Carlo method that uses the Laplace method as an approximation of the return value of the inner loop. The first example is a scalar problem that is linear in the uncertain parameter. The second example is a nonlinear scalar problem. The third example deals with the optimal sensor placement for an electrical impedance tomography experiment to recover the fiber orientation in laminate composites.

a b s t r a c tIn this work we consider quasi-optimal versions of the Stochastic Galerkin method for solving linear elliptic PDEs with stochastic coefficients. In particular, we consider the case of a finite number N of random inputs and an analytic dependence of the solution of the PDE with respect to the parameters in a polydisc of the complex plane C N . We show that a quasi-optimal approximation is given by a Galerkin projection on a weighted (anisotropic) total degree space and prove a (sub)exponential convergence rate. As a specific application we consider a thermal conduction problem with non-overlapping inclusions of random conductivity. Numerical results show the sharpness of our estimates.

Abstract. Computer simulators can be computationally intensive to run over a large number of input values, as required for optimization and various uncertainty quantification tasks. The standard paradigm for the design and analysis of computer experiments is to employ Gaussian random fields to model computer simulators. Gaussian process models are trained on input-output data obtained from simulation runs at various input values. Following this approach, we propose a sequential design algorithm MICE (mutual information for computer experiments) that adaptively selects the input values at which to run the computer simulator in order to maximize the expected information gain (mutual information) over the input space. The superior computational efficiency of the MICE algorithm compared to other algorithms is demonstrated by test functions and by a tsunami simulator with overall gains of up to 20% in that case.Key words. active learning, best linear unbiased prediction, Gaussian process, shallow water equations AMS subject classifications. 60G15, 62M20, 62K99, 65Y20, 91B74DOI. 10.1137/1409896131. Introduction. Computer experiments are widely employed to study physical processes [31,36] and involve running a computer simulator which mimics the physical process at various input values. When the computer simulator is computationally expensive to run, say, minutes, hours, or even days, often on a high performance cluster, only a limited number of simulation runs can be afforded, making the planning of such experiments even more important. Surrogate models, also known as emulators, are often used as means for designing and analyzing computer experiments [31]. Emulators are statistical models that have been used to approximate the input-output behavior of computer simulators for making probabilistic predictions. In this setting, we want to find a design of computer experiments that with minimal computational effort leads to a surrogate model with a good overall fit. We restrict our attention to deterministic computer simulators with a scalar output. In design of experiments it is customary to use space-filling designs [36] such as uniform designs, multilayer designs, maximin (Mm)-and minimax (mM)-distance designs, and Latin hypercube designs (LHD). Space-filling designs treat all regions of the design space as equally important, but are "one shot" designs that may waste computations over some unnecessary regions of the input space. A variety of adaptive designs have been proposed which can take advantage of information collected during the experimental design process [21,31], typically in the form of input-output

Vacuum/pressure swing adsorption is an attractive and often energy efficient separation process for some applications. However, there is often a trade-off between the different objectives: purity, recovery and power consumption. Identifying those trade-offs is possible through use of multi-objective optimisation methods but this is computationally challenging due to the size of the search space and the need for high fidelity simulations due to the inherently dynamic nature of the process. This paper presents the use of surrogate modelling to address the computational requirements of high fidelity simulations needed to evaluate alternative designs. We present SbNSGA-II ALM, surrogate based NSGA-II, a robust and fast multi-objective optimisation method based on kriging surrogate models and NSGA-II with Active Learning MacKay (ALM) design criteria. The method is evaluated by application to an industrially relevant case study: a two column six step system for CO 2 /N 2 separation. A 5 times reduction in computational effort is observed.

Abstract. Over the past 20 years, analyzing the abundance of the isotope chlorine-36 (36Cl) has emerged as a popular tool for geologic dating. In particular, it has been observed that 36Cl measurements along a fault plane can be used to study the timings of past ground displacements during earthquakes, which in turn can be used to improve existing seismic hazard assessment. This approach requires accurate simulations of 36Cl accumulation for a set of fault-scarp rock samples, which are progressively exhumed during earthquakes, in order to infer displacement histories from 36Cl measurements. While the physical models underlying such simulations have continuously been improved, the inverse problem of recovering displacement histories from 36Cl measurements is still mostly solved on an ad hoc basis. The current work resolves this situation by providing a MATLAB implementation of a fast, automatic, and flexible Bayesian Markov-chain Monte Carlo algorithm for the inverse problem, and provides a validation of the 36Cl approach to inference of earthquakes from the demise of the Last Glacial Maximum until present. To demonstrate its performance, we apply our algorithm to a synthetic case to verify identifiability, and to the Fiamignano and Frattura faults in the Italian Apennines in order to infer their earthquake displacement histories and to provide seismic hazard assessments. The results suggest high variability in slip rates for both faults, and large displacements on the Fiamignano fault at times when the Colosseum and other ancient buildings in Rome were damaged.

In this work we explore the extension of the quasi-optimal sparse grids method proposed in our previous work "On the optimal polynomial approximation of stochastic PDEs by Galerkin and Collocation methods" to a Darcy problem where the permeability is modeled as a lognormal random field. We propose an explicit a-priori/a-posteriori procedure for the construction of such quasi-optimal grid and show its effectivenenss on a numerical example. In this approach, the two main ingredients are an estimate of the decay of the Hermite coefficients of the solution and an efficient nested quadrature rule with respect to the Gaussian weight.

This paper proposes an extension of the Multi-Index Stochastic Collocation (MISC) method for forward uncertainty quantification (UQ) problems in computational domains of shape other than a square or cube, by exploiting isogeometric analysis (IGA) techniques. Introducing IGA solvers to the MISC algorithm is very natural since they are tensor-based PDE solvers, which are precisely what is required by the MISC machinery. Moreover, the combination-technique formulation of MISC allows the straight-forward reuse of existing implementations of IGA solvers. We present numerical results to showcase the effectiveness of the proposed approach. Highlights• Isogeometric solvers used in a MISC framework for forward UQ problems.• The combination-technique formulation of the method allows straight-forward reuse of legacy IGA solvers.