text: 395,68,1295304450136051713,1190175298106675200,Jonas Latz,"Error analysis for probabilities of rare events with approximate models - a new paper in the arXiv (<LINK>) by Fabian Wagner, Iason Papaioannou, Elisabeth Ullmann, and myself. A #thread. #research #Mathematics #numerics (1/7) An important task in, e.g. #structuralengineering and #environmentalengineering, is the estimation of the probability of a system failure, e.g., the probability of groundwater pollution in case a radioactive waste repository is damaged. We call this probability P_f. (2/7) P_f will usually be in the range [1E-9, 1E-6]. Moreover, as in the example mentioned above, P_f often depends on mathematical models, like a #PDE or #ODE. When estimating P_f, e.g. using a sampling or optimisation method, this model needs to be approximated as well. (3/7) In this paper, we are not interested in the accuracy of a sampling method, but in how accurate we need to approximate the underlying mathematical model (ODE or PDE) to get a reasonable approximation of P_f. We denote the probability of failure with approximate model by P_h. (4/7) Past results indicate that the error |P_f - P_h| scales like the approximation error in the mathematical model (Elfverson et al. 2016; <LINK>). This implies that (up to an unknown constant) the model needs to be approximated super accurately. (5/7) In our paper, we show that actually |P_f - P_h|/P_hFORM behaves like the model approximation error. Here, P_hFORM is the approximation of P_h using the First Order Reliability Method (FORM), an optimisation based approach towards the estimation of rare events. (6/7) This shows that the model only needs to be approximated moderately precisely. Moreover, it gives an immediate algorithmic procedure to estimate the error in P_f. The paper contains a rigorous analysis alongside with numerical experiments. (7/7)",https://arxiv.org/abs/2008.06368,"The estimation of the probability of rare events is an important task in reliability and risk assessment. We consider failure events that are expressed in terms of a limit-state function, which depends on the solution of a partial differential equation (PDE). In many applications, the PDE cannot be solved analytically. We can only evaluate an approximation of the exact PDE solution. Therefore, the probability of rare events is estimated with respect to an approximation of the limit-state function. This leads to an approximation error in the estimate of the probability of rare events. Indeed, we prove an error bound for the approximation error of the probability of failure, which behaves like the discretization accuracy of the PDE multiplied by an approximation of the probability of failure, the first order reliability method (FORM) estimate. This bound requires convexity of the failure domain. For non-convex failure domains, we prove an error bound for the relative error of the FORM estimate. Hence, we derive a relationship between the required accuracy of the probability of rare events estimate and the PDE discretization level. This relationship can be used to guide practicable reliability analyses and, for instance, multilevel methods. ",Error analysis for probabilities of rare events with approximate models,7,"['Error analysis for probabilities of rare events with approximate models - a new paper in the arXiv (<LINK>) by Fabian Wagner, Iason Papaioannou, Elisabeth Ullmann, and myself. A #thread. #research #Mathematics #numerics (1/7)', 'An important task in, e.g. #structuralengineering and #environmentalengineering, is the estimation of the probability of a system failure, e.g., the probability of groundwater pollution in case a radioactive waste repository is damaged. We call this probability P_f. (2/7)', 'P_f will usually be in the range [1E-9, 1E-6]. Moreover, as in the example mentioned above, P_f often depends on mathematical models, like a #PDE or #ODE. When estimating P_f, e.g. using a sampling or optimisation method, this model needs to be approximated as well. (3/7)', 'In this paper, we are not interested in the accuracy of a sampling method, but in how accurate we need to approximate the underlying mathematical model (ODE or PDE) to get a reasonable approximation of P_f. We denote the probability of failure with approximate model by P_h. (4/7)', 'Past results indicate that the error |P_f - P_h| scales like the approximation error in the mathematical model (Elfverson et al. 2016; https://t.co/z8koIHlEOS). This implies that (up to an unknown constant) the model needs to be approximated super accurately. (5/7)', 'In our paper, we show that actually |P_f - P_h|/P_hFORM behaves like the model approximation error. Here, P_hFORM is the approximation of P_h using the First Order Reliability Method (FORM), an optimisation based approach towards the estimation of rare events. (6/7)', 'This shows that the model only needs to be approximated moderately precisely. Moreover, it gives an immediate algorithmic procedure to estimate the error in P_f. The paper contains a rigorous analysis alongside with numerical experiments. (7/7)']",20,08,1813