## Research objective

Using modern mathematical tools and high-performance computing, we can now incorporate elements of uncertainty in simulations to account for lack of knowledge about input parameters, variability in operating conditions, or inappropriate modelling assumptions. The aim of uncertainty quantification (UQ) is to provide more accurate predictions about systems’ behaviour and enable replacement of physical experiments with non-destructive virtual testing.

The objective of our work is to develop such rigorous systematic mathematical methods in both data analytics, modelling and simulation. Errors in computational engineering arise from multiple sources: numerical schemes, boundary conditions, measurements, etc. We focus on non-intrusive approaches which enable use of existing codes and straightforward parallel evaluation for quantities of interest.

## Modelling in an uncertain world

## Case study

## Uncertainty quantification at multi-scale interface

We looked at parametric uncertainties related to modelling particle interactions through Lenard–Jones potentials and their effect on the substance viscosity which in turns affects the velocity profiles of the fluid dynamics system. With the use of surrogate model techniques computational costs related to propagation of uncertainties through a chain of models can be substantially reduced. In some cases the speedup factor we achieved was as high as 100.

## Case study

## Waves through heterogenous media

We consider the very challenging problem of uncertainty quantification for acoustic wave propagation in a heterogeneous medium with prescribed covariance. A multi-level Monte Carlo (MLMC) method is used along with a novel technique for efficient generation of random fields with a given rank. The strategy of MLMC based on telescoping sums allows for reduced computational cost of sampling with respect to the classical approach or general polynomial chaos which suffers from the course of dimensionality.

Multi-scale problems are one of the greatest challenges in computational engineering. The objective is to understand how large systems, such as a turbine or pump, are affected by small-scale phonemena such as molecular interaactions. It is very common that the input parameters used in modelling vary or are associated with measurement errors. Therefore, we need to calculate these inaccuracies to obtain realistic predictions.

## Publications

[1] M.J. Zimoń, R. Sawko, D.R. Emerson, C. Thompson,

“Uncertainty Quantification at the Molecular–Continuum Model Interface,”

*Fluids* **2**, 12, 2017.

[2] H.N. Najm,

“Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics,”

*Annual Review of Fluid Mechanics* **41**, 35-52, 2009.

[3] M. Salloum, K. Sargsyan, R. Jones, H.N. Najm, B. Debusschere,

“Quantifying sampling noise and parametric uncertainty in atomistic-to-continuum simulations using surrogate models,”

*Multiscale Model. Simul.* **13**, 953-976, 2015.