The development of reduced order models for complex applications, offering the promise for rapid and accurate evaluation of the output of complex models under parameterized variation, remains a very active research area. Applications are found in problems which require many evaluations, sampled over a potentially large parameter space, such as in optimization, control, uncertainty quantification and applications where near real-time response is needed.

However, many challenges remain to secure the flexibility, robustness, and efficiency needed for general large-scale applications, in particular for nonlinear and/or time-dependent problems.

After giving a brief general introduction to reduced order models, we discuss developments in two different directions. In the first part, we discuss recent developments of reduced methods that conserve chosen invariants for nonlinear time-dependent problems. We pay particular attention to the development of reduced models for Hamiltonian problems and propose a greedy approach to build the basis. As we shall demonstrate, attention to the construction of the basis must be paid not only to ensure accuracy but also to ensure stability of the reduced model. Time permitting, we shall also briefly discuss how to extend the approach to include more general dissipative problems through the notion of port-Hamiltonians, resulting in reduced models that remain stable even in the limit of vanishing viscosity and also touch on extensions to Euler and Navier-Stokes equations.

The second part of the talk discusses the combination of reduced order modeling for nonlinear problems with the use of neural networks to overcome known problems of on-line efficiency for general nonlinear problems. We discuss the general idea in which training of the neural network becomes part of the offline part and demonstrate its potential through a number of examples, including for the incompressible Navier-Stokes equations with geometric variations.

This work has been done with in collaboration with B.F. Afkram (EPFL, CH), N. Ripamonti EPFL, CH) and S. Ubbiali (USI, CH).

The rapidly increasing number of cores in high-performance computing systems causes a multitude of challenges for developers of numerical methods. New parallel algorithms are required to unlock future growth in computing power for applications and energy efficiency and algorithm-based fault tolerance are becoming increasingly important. So far, most approaches to parallelise the numerical solution of partial differential equations focussed on spatial solvers, leaving time as a bottleneck. Recently, however, time stepping methods that offer some degree of concurrency, so-called parallel-in-time integration methods, have started to receive more attention.

I will introduce two different numerical algorithms, Parareal (by Lions et al., 2001) and PFASST (by Emmett and Minion, 2012), that allow to exploit concurrency along the time dimension in parallel computer simulations solving partial differential equations. Performance results for both methods on different architectures and for different equations will be presented. The PFASST algorithm is based on merging ideas from Parareal, spectral deferred corrections (SDC, an iterative approach to derive high-order time stepping methods by Dutt et al. 2000) and nonlinear multi-grid. Performance results for PFASST on close to half a million cores will illustrate the potential of the approach. Algorithmic modifications like IPFASST will be introduced that can further reduce solution times. Also, recent results showing how parallel-in-time integration can provide algorithm-based tolerance against hardware faults will be shown.

In micro-fluidics, at small scales where inertial effects become negligible, surface to volume ratios are large and the interfacial processes are extremely important for the overall dynamics. Integral
equation based methods are attractive for the simulations of e.g. droplet-based microfluidics, with tiny water drops dispersed in oil, stabilized by surfactants. In boundary integral formulations for
Stokes flow, jumps in pressure and velocity gradients are naturally taken care of, viscosity ratios enter only in coefficients of the equations, and only the drop surfaces must be discretized and not the volume inside nor in between.

We present numerical methods for drops with insoluble surfactants, both in two and three dimensions. We discretize the integral equations using Nyström methods, and special care is taken in the evaluation of singular and also nearly singular integrals that is needed in the case of close drop interactions. A spectral method is used to solve the advection-diffusion equation on each drop surface that describes the evolution of surfactant concentration. The drop velocity and surfactant concentration couple together through an equation of state for the surface tension coefficient. An adaptive time-stepping strategy is developed for the coupled problem, with the constraint to minimize the number of Stokes solves, since this is the computationally most expensive part.

For high quality discretization of the drops throughout the simulations, a hybrid method is used in two dimensions, offering an arc-length parameterization of the interface. In three dimensions, a
reparameterization procedure is developed to optimize the spherical harmonics representation of the drop, while conserving the drop volume and amount of surfactant.

We present results from some validation tests and illustrate the ability of the numerical methods in different challenging problems.

Decision trees usefully represent the sparse, high dimensional and noisy nature of chemical data from experiments. Having learned a function from this data, we may want to thereafter optimise the function, e.g. for picking the best catalyst for a chemical process. This work studies a mixed-integer non-linear optimisation problem involving: (i) gradient boosted trees modelling catalyst behaviour, (ii) penalty functions mitigating risk, and (iii) penalties enforcing chemical composition constraints. We develop several heuristic methods to find feasible solutions, and an exact, branch and bound algorithm that leverages structural properties of the gradient boost trees and penalty functions. We computationally test our methods on an industrial instance from BASF.
This work was completed in collaboration with Mr Miten Mistry and Dr Dimitris Letsios at Imperial College London and Dr Robert Lee and Dr Gerhard Krennrich from BASF.
 

We present discrete methods for computing low-rank approximations of time-dependent tensors that are the solution of a differential equation. The approximation format can be Tucker, tensor trains, MPS or hierarchical tensors. We will consider two types of discrete integrators: projection methods based on quasi-optimal metric projection, and splitting methods based on inexact solutions of substeps. For both approaches we show numerically and theoretically that their behaviour is superior compared to standard methods applied to the so-called gauged equations. In particular, the error bounds are robust in the presence of small singular values of the tensor’s matricisations. Based on joint work with Emil Kieri, Christian Lubich, and Hanna Walach.

Solving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the non standard interface conditions are naturally defined at the boundary between elements. In this work we introduce the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. We present the detailed analysis of the hybrid discontinuous Galerkin method for the Stokes problem with non standard boundary conditions. This analysis is supported by numerical evidence. In addition, the advantage of the new preconditioners over more classical choices is also supported by numerical experiments.

This work was done in collaboration with G. Barrenechea, M. Bosy (Univ. Strathclyde) and F. Nataf, P-H Tournier (Univ of Paris VI)