A bi-fidelity method for the multiscale Boltzmann equation with random parameters

Abstract

In this paper, we study the multiscale Boltzmann equation with multi-dimensional random parameters by a bi-fidelity stochastic collocation (SC) method developed in [51,65,66]. By choosing the compressible Euler system as the low-fidelity model, we adapt the bi-fidelity SC method to combine computational efficiency of the lowfidelity model with high accuracy of the high-fidelity (Boltzmann) model. With only a small number of high-fidelity asymptotic-preserving solver runs for the Boltzmann equation, the bi-fidelity approximation can capture well the macroscopic quantities of the solution to the Boltzmann equation in the random space. A priori estimate on the accuracy between the high-and bi-fidelity solutions together with a convergence analysis is established. Finally, we present extensive numerical experiments to verify the efficiency and accuracy of our proposed method.

Computing Black Scholes with Uncertain Volatility—A Machine Learning Approach

“…The application of Bi-Fidelity techniques to various problems is an active area of research [26,27]. In the setting of uncertainty quantification for PDE models, it is frequently described in the context of uncertainty quantification via Stochastic Collocation methods; see, e.g., [28,29] for the general procedure or [30][31][32][33] for applications. The combination with the stochastic Galerkin method works similarly; however, it is not very common in literature.…”

“…However, important characteristics of the solution must be shared between the models. Now one can proceed with the typical Bi-Fidelity algorithm as described in [28][29][30]. Below, the application of this algorithm is explained, where the volatility is assumed to depend on L = 2 random variables Θ 1 , Θ 2 for a better readability.…”

“…The following three steps describe the offline learning phase of the algorithm in which the stored approximation data are generated [28][29][30]. These steps have to be executed only once.…”

“…Assume now that a certain volatility model z is given and the corresponding Bi-Fidelity solution of the Black Scholes equation with uncertain volatility shall be computed. This is performed in the online phase as follows [28][29][30]:…”

Multi-fidelity methods for uncertainty propagation in kinetic equations

“…However, the choice of the samples for the high fidelity solver remains arbitrary. The second methodology, is based on a simpler bi-fidelity setting where a single low-fidelity model is used to effectively inform the selection of representative points in the parameter space and then employ this information to construct accurate approximations to high-fidelity solutions [32,55,63,[105][106][107]. As a result, it does not necessarily require that the low-fidelity and high-fidelity to reside in the same physical space.…”