Victoria Volodina

Many mathematical models suffer from model discrepancy, posing a significant challenge to their use in clinical decision-making. In this article, we consider methods for addressing this issue. In the first approach, a mathematical model is treated as a black box system, and model discrepancy is defined as an independent and additive term that accounts for the difference between the physical phenomena and the model representation. A Gaussian Process (GP) is commonly used to capture the model discrepancy. An alternative approach is to construct a hybrid grey box model by filling in the incomplete parts of the mathematical model with a neural network. The neural network is used to learn the missing processes by comparing the observations with the model output. To enhance interpretability, the outputs of this non-parametric model can then be regressed into a symbolic form to obtain the learned model. We compare and discuss the effectiveness of these approaches in handling model discrepancy using clinical data from the ICU and the Siggaard–Andersen oxygen status algorithm. This article is part of the theme issue ‘Uncertainty quantification for healthcare and biological systems (Part 2)’.

Uncertainty quantification (UQ) is an essential aspect of computational modelling and statistical prediction. Multiple applications, including geophysics, climate science and aerospace engineering, incorporate UQ in the development and translation of new technologies. In contrast, the application of UQ to biological and healthcare models is understudied and suffers from several critical knowledge gaps. In an era of personalized medicine, patient-specific modelling, and digital twins , a lack of UQ understanding and appropriate implementation of UQ methodology limits the success of modelling and simulation in a clinical setting. The main contribution of our review article is to emphasize the importance and current deficiencies of UQ in the development of computational frameworks for healthcare and biological systems. As the introduction to the special issue on this topic, we provide an overview of UQ methodologies, their applications in non-biological and biological systems and the current gaps and opportunities for UQ development, as later highlighted by authors publishing in the special issue. This article is part of the theme issue ‘Uncertainty quantification for healthcare and biological systems (Part 1)’.

Gaussian Process (GP) emulators are widely used to approximate complex computer model behaviour across the input space. Motivated by the problem of coupling computer models, recently progress has been made in the theory of the analysis of networks of connected GP emulators. In this paper, we combine these recent methodological advances with classical state-space models to construct a Bayesian decision support system. This approach gives a coherent probability model that produces predictions with the measure of uncertainty in terms of two first moments and enables the propagation of uncertainty from individual decision components. This methodology is used to produce a decision support tool for a UK county council considering low carbon technologies to transform its infrastructure to reach a net-zero carbon target. In particular, we demonstrate how to couple information from an energy model, a heating demand model, and gas and electricity price time-series to quantitatively assess the impact on operational costs of various policy choices and changes in the energy market.

Majorisation, also called rearrangement inequalities, yields a type of stochastic ordering in which two or more distributions can be then compared. This method provides a representation of the peakedness of probability distributions and is also independent of the location of probabilities. These properties make majorisation a good candidate as a theory for uncertainty. We demonstrate that this approach is also dimension free by obtaining univariate decreasing rearrangements from multivariate distributions, thus we can consider the ordering of two, or more, distributions with different support. We present operations including inverse mixing and maximise/minimise to combine and analyse uncertainties associated with different distribution functions. We illustrate these methods for empirical examples with applications to scenario analysis and simulations.

District heating is a network of pipes through which heat is delivered from a centralised source. It is expected to play an important role in the decarbonisation of the energy sector in the coming years. In district heating, heat is traditionally generated through fossil fuels, often with combined heat and power (CHP) units. However, increasingly, waste heat is being used as a low carbon alternative, either directly or, for low temperature sources, via a heat pump. The design of district heating often has competing objectives: the need for inexpensive energy and meeting low carbon targets. In addition, the planning of district heating schemes is subject to multiple sources of uncertainty such as variability in heat demand and energy prices. This paper proposes a decision support tool to analyse and compare system designs for district heating under uncertainty using stochastic ordering (dominance). Contrary to traditional uncertainty metrics that provide statistical summaries and impose total ordering, stochastic ordering is a partial ordering and operates with full probability distributions. In our analysis, we apply the orderings in the mean and dispersion to the waste heat recovery problem in Brunswick, Germany.

Urban Waste Heat Recovery, heat recovery from low-temperature urban sources such as data centres and metro systems, has a great deal of potential in terms of meeting domestic and commercial heat demands whilst significantly reducing carbon emissions. Urban sources of heat are advantageous in that they tend to be close to areas of high heat demand and are therefore highly suitable as inputs to existing and newly constructed district heating networks. This paper has two main focuses. Firstly, the issue of efficiency in waste heat recovery is addressed with a focus on Technical, Economic, Social, and Environmental (TESE) efficiencies, which we consider should be given equal consideration. Secondly, we address the question of contractual efficiency and argue that contracts should be underpinned by mathematical modelling. We then focus on the contractual relationship between the owner of the waste heat and the district heating operator and consider the question of waste heat pricing. We suggest a profit sharing approach in which the price per unit of waste heat is allowed to vary according to important aspects such as demand and the electricity price. A demonstration of this approach is presented using a simple model of a waste heat recovery system that extracts heat from a data centre in Brunswick, Germany.

Many computer models possess high-dimensional input spaces and substantial computational time to produce a single model evaluation. Although such models are often ‘deterministic’, these models suffer from a wide range of uncertainties. We argue that uncertainty quantification is crucial for computer model validation and reproducibility. We present a statistical framework, termed history matching, for performing global parameter search by comparing model output to the observed data. We employ Gaussian process (GP) emulators to produce fast predictions about model behaviour at the arbitrary input parameter settings allowing output uncertainty distributions to be calculated. History matching identifies sets of input parameters that give rise to acceptable matches between observed data and model output given our representation of uncertainties. Modellers could proceed by simulating computer models’ outputs of interest at these identified parameter settings and producing a range of predictions. The variability in model results is crucial for inter-model comparison as well as model development. We illustrate the performance of emulation and history matching on a simple one-dimensional toy model and in application to a climate model. This article is part of the theme issue ‘Reliability and reproducibility in computational science: implementing verification, validation and uncertainty quantification in silico ’.