Uncertainty quantification and optimal decisions

Abstract

A mathematical model can be analysed to construct policies for action that are close to optimal for the model. If the model is accurate, such policies will be close to optimal when implemented in the real world. In this paper, the different aspects of an ideal workflow are reviewed: modelling, forecasting, evaluating forecasts, data assimilation and constructing control policies for decision-making. The example of the oil industry is used to motivate the discussion, and other examples, such as weather forecasting and precision agriculture, are used to argue that the same mathematical ideas apply in different contexts. Particular emphasis is placed on (i) uncertainty quantification in forecasting and (ii) how decisions are optimized and made robust to uncertainty in models and judgements. This necessitates full use of the relevant data and by balancing costs and benefits into the long term may suggest policies quite different from those relevant to the short term.

Likely equilibria of stochastic hyperelastic spherical shells and tubes

“…where λ 1 , λ 2 , λ 3 represent the radial, tangential, and azimuthal stretch ratios, respectively, and df /dR denotes differentiation of f with respect to R. The radial equation of equilibrium is [7] dP 11 dR…”

“…Unavoidably, uncertainties are attached to material properties. For natural and industrial elastic materials, uncertainties in the mechanical responses generally arise from the inherent micro-structural inhomogeneity, sample-to-sample intrinsic variability, and lack of data, which are sparse, inferred from indirect measurements, and contaminated by noise [16][17][18][19]. Stochastic approaches are thus growing in importance as a tool in many disciplines, such as materials science, engineering and biomechanics, where understanding the variability in the mechanical behaviour of materials is critical.…”

Likely Cavitation in Stochastic Elasticity

“…Within this framework, hyperelastic materials are the class of material models described by a strain-energy function characterised by a set of deterministic model parameters. In addition, for solid elastic materials, uncertainties in the observational data generally arise from the inherent variation in material properties and testing protocols [7,10,20,38]. In view of these uncertainties, recently, stochastic representations of isotropic incompressible hyperelastic materials characterised by a stochastic strain-energy function, for which the model parameters are random variables following standard probability laws, were proposed in [52], while compressible versions of these models were constructed in [53].…”

Stochastic isotropic hyperelastic materials: constitutive calibration and model selection

“…Next, following [23][24][25][26][27], for the random nonlinear shear modulus µ(a 0 ), defined by (3.7), we set the mathematical expectations: 12) where, by the constraint (3.11), the mean value µ(a 0 ) is fixed and greater than zero, and the logarithmic constraint (3.12) implies that both µ(a 0 ) and µ(a 0 ) −1 are second-order random variables (i.e. they have finite mean and finite variance).…”

“…Recently, there has been a growing interest in probability and statistical techniques for engineering and biomedical applications, where the calibration of models using available data and the quantification of uncertainties in model parameters are of utmost importance [4][5][6]. There are, however, many challenges introduced by the consideration and quantification of uncertainties in mathematical models, and their use in making predictions, some of which are discussed in [7][8][9][10][11][12].…”