Uncertainty Quantification andBayesian Inversion

Abstract

Uncertainty estimation arises at least implicitly in any kind of modeling of the real – or phenomenological – world, and it is desirable to actually quantify the uncertainty in probabilistic terms. Here the emphasis is on uncertain systems, typically modeled by partial differential equations, where the randomness is assumed spatial. Traditional computational approaches usually use some form of perturbation or Monte Carlo simulation. Here the emphasis is on recent methods based on stochastic functional or spectral approximations. Quantifying the uncertainty in mathematical models due to uncertainties in their parameters, combined with the traditional ability to predict the system behavior given external excitations or loadings, is called solving the stochastic forward problem . Given this capability, one has the possibility to use this to estimate or identify unknown and hence uncertain parameters in the mathematical model by observing the system response. This is called a solution to the inverse problem , and a short introduction into inverse problems in a Bayesian setting using the functional or spectral approximations is provided.

Considerations on the identifiability of fracture and bond properties of reinforced concrete

“…In most cases, however, computing the full posterior density is not essential and one only needs its estimates, for example, the expected value. Subsequently, according to Reference 51, the problem of evaluating the posterior density from Equation (2) can be restated as a problem of computing the conditional expectation (CE) 18,20–22,52 defined as where $$$ \mathfrak{B}:= \sigma \left({Y}_Q\right) $$$ is the Borel sub‐$$$ \sigma $$$‐algebra, $$$ \mathfrak{B}\subset \mathfrak{F} $$$, generated by the measurement operator $$$ {Y}_Q $$$. For Equation (3) to hold, random variables $$$ \boldsymbol{q} $$$ need to have finite variance, implying that they belong to the Hilbert space .…”

“…49,50 In most cases, however, computing the full posterior density is not essential and one only needs its estimates, for example, the expected value. Subsequently, according to Reference 51, the problem of evaluating the posterior density from Equation (2) can be restated as a problem of computing the conditional expectation (CE) 18,[20][21][22]52 defined as…”