Probability bounds analysis for nonlinear dynamic process models

Abstract

Dynamic process models frequently involve uncertain parameters and inputs. Propagating these uncertainties rigorously through a mathematical model to determine their effect on system states and outputs is a challenging problem. In this work, we describe a new approach, based on the use of Taylor model methods, for the rigorous propagation of uncertainties through nonlinear systems of ordinary differential equations (ODEs). We concentrate on uncertainties whose distribution is not known precisely, but can be bounded by a probability box (p-box), and show how to use p-boxes in the context of Taylor models. This allows us to obtain p-box representations of the uncertainties in the state variable outputs of a nonlinear ODE model. Examples having two to three uncertain parameters or initial states and focused on reaction process dynamics are used to demonstrate the potential of this approach. Using this method, rigorous probability bounds can be determined at a computational cost that is significantly less than that required by Monte Carlo analysis.

Imprecise probabilities in engineering analyses

“…Probability bounds analysis has been used in uncertainty computations in many contexts including series system failure analysis and system reliability [140], quantication of margins of uncertainty [103], nite-element structural [172,112,173,104], dierential equations of chemical reactions [174], engineering design [123,175], validation [176,113], pharmacokinetics [177], human health and ecological risk assessments at Superfund sites [178,179,180], and even global circulation models [181]. The Wikipedia page for probability bounds analysis lists over two dozen applications of the method to various engineering problems.…”

Static Analysis of Programs with Imprecise Probabilistic Inputs

“…Our abstract domain is precise enough to correctly bound the output variables and the figure shows that we can also, with high probability, discard some values in the resulting interval. For example, we could show that P (x 1 ≤ 1.13) ≤ 0.0552, which is an even more precise enclosure than the one obtained by RiskCALC [13]. Our analysis took 104s on a 1.6Ghz laptop.…”

“…We now use an example from [13] to test the precision and performance of our abstract domain on arithmetic operations. The problem is to compute bounds on the solution of the differential equationṡ…”

“…As in [13], we used VSPODE [28] to obtain a Taylor model polynomial that expresses the solution at t f = 20 as an order 5 polynomial of θ 1 and θ 2 . We then used the probabilistic affine forms to evaluate the Horner form of this polynomial.…”

Sensitivity Analysis of Uncertain Dynamic Systems Using Set-Valued Integration

“…Values for all the other parameters are the same as in [12]. The problem involves bounding the derivatives of the final conversion X(t N ) at t N = 900 s with respect to the uncertain parameters p = (T (0), T Table 4.…”