European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
13:30   Uncertainty Quantification and Stochastic Models (Part 1)
Chair: Fred Wubs
13:30
25 mins
Numerical Simulation of the Herschel-Bulkey Model with Changing Flow Parameters
Sergio Gonzalez-Andrade
Abstract: In this talk, we discuss the numerical simulation of the Herschel-Bulkley flow where the viscosity and yield limit depend on a structure parameter. We are concerned with a generalized version of the Navier-Stokes-type system representing the Herschel-Bulkley viscoplastic model with flow parameters depending on a scalar variable. This variable, which can represent several phenomena: temperature effects, cristalization, etc, obeys a first-order transient differential equation. Thus, we are focused on the following system of coupled PDEs \begin{equation}\label{Housgen}\tag{$\mathcal{HG}$} \begin{array}{ccl} \partial_t \mathbf{u} +\mathbf{u}\cdot\nabla\mathbf{u}= \Div\boldsymbol{\sigma} + \mathbf{f},&\mbox{in $\Omega$}\vspace{0.2cm}\\\boldsymbol{\sigma} = -p\cdot \mathbf{I} + \widehat{\boldsymbol{\sigma}},&\mbox{in $\Omega$}\vspace{0.2cm}\\\widehat{\boldsymbol{\sigma}}= \mu(\lambda)|\mathcal{E}\mathbf{u}|^{r-2}\mathcal{E}\mathbf{u}+ g(\lambda)\dfrac{\mathcal{E}\mathbf{u}}{|\mathcal{E}\mathbf{u}|} ,&\mbox{if $\mathcal{E}\mathbf{u}=0$}\vspace{0.2cm}\\|\widehat{\boldsymbol{\sigma}}|\leq g(\lambda),&\mbox{if $\mathcal{E}\mathbf{u}\neq 0$}\vspace{0.2cm}\\\nabla\cdot \mathbf{u} =0,&\mbox{in $\Omega$}\vspace{0.2cm}\\ \partial_t\lambda -\kappa\Delta \lambda+\mathbf{u}\cdot \nabla \lambda =a(1-\lambda) - b|\mathcal{E}\mathbf{u}|^m,&\mbox{in $\Omega$}\vspace{0.2cm}\\ +B.C. \end{array} \end{equation} Here, $\lambda$ stands for the structure parameter, and $\mu$ and $g$ are specific functions representing the flow parameters, viscosity and yield stress, respectively, under the action of $\lambda$. We study a mixed formulation for the generalized Herschel-Bulkey resulting model \eqref{Housgen} and propose a finite element discretization for the resulting coupled system of PDEs. Next, since the variational formulation of this system involves slantly differentiable functions, we propose a semismooth Newton algorithm for the numerical resolution of such coupled system. Further, we implement this algorithm in a high performance computer. We carry on several numerical experiments, and we compare our results with benchmark experiments for the purpose of validating our method.
13:55
25 mins
Machine Learning for Closure Models in Multiphase Flow Applications
Jurriaan Buist, Benjamin Sanderse, Yous van Halder, Barry Koren, GertJan van Heijst
Abstract: Multiphase flows are described by the multiphase Navier-Stokes equations. Numerically solving these equations is computationally expensive, and performing many simulations for the purpose of design, optimization and uncertainty quantification is often prohibitively expensive. A simplified model, the so-called two-fluid model, can be derived from a spatial averaging process. The averaging process introduces a closure problem, which is represented by unknown friction terms in the two-fluid model. Correctly modeling these friction terms is a long-standing problem in two-fluid model development. In this work we take a new approach, and learn the closure terms in the two-fluid model from a set of unsteady high-fidelity simulations conducted with the open source code Gerris. These form the training data for a neural network. The neural network provides a functional relation between the two-fluid model's resolved quantities and the closure terms, which are added as source terms to the two-fluid model. With the addition of the locally defined interfacial slope as an input to the closure terms, our novel trained two-fluid model reproduces the dynamic behavior of high fidelity simulations better than the two-fluid model using a conventional set of closure terms.
14:20
25 mins
Information-based Variational Model Reduction of high-dimensional Reaction Networks
Markos Katsoulakis, Pedro Vilanova
Abstract: In this work we present new scalable, path space information theory-based variational methods for the efficient model reduction of high-dimensional reaction networks. The proposed methodology combines, (a) information theoretic tools for sensitivity analysis that allow us to identify the proper coarse variables of the reaction network, with (b) variational approximate inference methods for training a best-fit reduced model. This approach takes advantage of both physicochemical modelling and data-based approaches and allows to construct optimal parameterized reduced dynamics in the number of species, reactions and parameters, while controlling the information loss due to the reduction. We demonstrate the effectiveness of our model reduction method on several complex, high-dimensional biochemical reaction networks from recent literature.
14:45
25 mins
A Multilevel Monte Carlo Asymptotic-Preserving Particle Scheme for Kinetic Equations
Emil Loevbak, Giovanni Samaey, Stefan Vandewalle
Abstract: Particle simulations are required in many application domains. Often such simulations suffer from time-scale separation: the particles have fast characteristic dynamics, requiring small time steps, while the population-level (macroscopic) behaviour of large ensembles of such particles occurs on a much slower time scale, implying long time horizons for the simulation. This time-scale separation results in a high computational cost for classical simulation schemes. We consider hyperbolic transport equations, in which we introduce a scaling parameter ε (related to the mean free particle path) to characterize the time-scale separation. In the limit where this scaling parameter tends to zero, the hyperbolic transport equation converges to a parabolic diffusion equation. Simulations of the transport equation in the small ε region however, suffer from extreme time step reduction constraints to maintain stability. Asymptotic-preserving schemes, such as that proposed in [1], avoid this issue, but do so while adding a model error which is linear in the time step size. In recent work [3], we reduced this model error, while leveraging the computational advantage of the asymptotic-preserving scheme, using multilevel Monte Carlo [2]. The method first computes a coarse estimate with a large time step size, and correspondingly large bias. This estimate is then improved upon via a hierarchy of increasingly accurate Monte Carlo simulations using increasingly finer time step. In general, this approach reduces the computational cost, compared to directly simulating using a finer time step. However, the bias corrections and variances of differences of coupled simulations in this scheme have a different structure than typically observed in multilevel Monte Carlo applications, providing some novel insights into both the asymptotic-preserving schemes and efficient multilevel Monte Carlo simulations in this setting. In this talk, we will present some of these insights. Additionally, we will present an approach for selecting a suitable set of levels in the multilevel Monte Carlo scheme. References [1] G. Dimarco, L. Pareschi and G. Samaey. Asymptotic-Preserving Monte Carlo methods for transport equations in the diffusive limit. SIAM Journal on Scientific Computing, 40: A504–A528, 2018. [2] M.B. Giles. Multilevel Monte Carlo Path Simulation. Operations Research, 56(3): 607–617, 2008. [3] E. Løvbak, G. Samaey, S. Vandewalle. A Multilevel Monte Carlo Asymptotic-Preserving Particle Method for Kinetic Equations in the Diffusion Limit. Submitted, arXiv:1902.04347, 2019.