Joakim Beck

ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS

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.

Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients

In 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.

Sequential design with mutual information for computer experiments (MICE): Emulation of a tsunami model

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 a tsunami simulator with overall gains of up to 20% in that case.

Surrogate based Optimisation for Design of Pressure Swing Adsorption Systems

Abstract Pressure swing adsorption (PSA) is a cyclic adsorption process for gas separation and purification. PSA offers a broad range of design possibilities influencing the device behaviour. In the last decade much attention has been devoted towards simulation and optimisation of various PSA cycles. The PSA beds are modelled with hyperbolic/parabolic partial differential algebraic equations and the separation performance should be assessed at cyclic steady state (CSS). Detailed mathematical models together with the CSS constraint makes design difficult. We propose a surrogate based optimisation procedure based on kriging for the design of PSA systems. The numerical implementation is tested with a genetic algorithm, with a multi-start sequential quadratic programming method and with an efficient global optimisation algorithm. The case study is the design of a dual piston PSA system for the separation of a binary gas mixture of N 2 and CO 2 .

Fast Bayesian experimental design: Laplace-based importance sampling for the expected information gain

Abstract 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 for a specified 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 single-loop Monte Carlo method that uses the Laplace approximation of the return value of the inner loop. The first demonstration 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 sparse-grid isogeometric solver

Giancarlo Sangalli and Lorenzo Tamellini were partially supported by the European Research Council through the FP7 ERC consolidator grant n. 616563HIGEOM and by the GNCS 2017 project “Simulazione numerica di problemi di Interazione Fluido-Struttura (FSI) con metodi agli elementi finiti ed isogeometrici”. Lorenzo Tamellini also received support from the scholarship “Isogeometric method” granted by the Universita di Pavia and by the European Union’s Horizon 2020 research and innovation program through the grant no. 680448 CAxMan. Joakim Beck received support from the KAUST CRG3 Award Ref:2281 and the KAUST CRG4 Award Ref:2584.