Journal of Mathematics in Industry 

About: Journal of Mathematics in Industry is an academic journal published by Springer Nature. The journal publishes majorly in the area(s): Computer science & Finite element method. It has an ISSN identifier of 2190-5983. It is also open access. Over the lifetime, 154 publications have been published receiving 1699 citations.

Certified reduced basis approximation for parametrized partial differential equations and applications

Abstract: Reduction strategies, such as model order reduction (MOR) or reduced basis (RB) methods, in scientific computing may become crucial in applications of increasing complexity. In this paper we review the reduced basis methods (built upon a high-fidelity ‘truth’ finite element approximation) for a rapid and reliable approximation of parametrized partial differential equations, and comment on their potential impact on applications of industrial interest. The essential ingredients of RB methodology are: a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform a competitive Offline-Online splitting in the computational procedure, and a rigorous a posteriori error estimation used for both the basis selection and the certification of the solution. The combination of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (for example, optimization, control or parameter identification). After a brief excursus on the methodology, we focus on linear elliptic and parabolic problems, discussing some extensions to more general classes of problems and several perspectives of the ongoing research. We present some results from applications dealing with heat and mass transfer, conduction-convection phenomena, and thermal treatments.

Adjoint methods for car aerodynamics

Abstract: The adjoint method has long been considered as the tool of choice for gradient-based optimisation in computational fluid dynamics (CFD). It is the independence of the computational cost from the number of design variables that makes it particularly attractive for problems with large design spaces. Originally developed by Lions and Pironneau in the 70’s, the adjoint method has evolved towards a standard tool within the development processes of the aeronautical industries. Its uptake in the automotive industry, however, lags behind. The first systematic applications of adjoint methods in automotive CFD have interestingly not taken place in the classical shape design arena, but in a relatively young discipline of sensitivity-based optimisation: fluid dynamic topology optimisation. While being an established concept in structure mechanics for decades already, its transfer to fluid dynamics took place just ten years ago. We demonstrate that specifically for ducted flow applications, like airducts for cabin ventilation or engine intake ports, it constitutes a very powerful tool and has matured over the last years to a level that allows its systematic usage for various automotive applications. To drive adjoint-based shape optimisation to the same degree of maturity and robustness for car applications is the subject of ongoing research collaborations between academia and the car industry. Achievements and challenges encountered during these efforts are presented.

Beyond just “flattening the curve”: Optimal control of epidemics with purely non-pharmaceutical interventions

Abstract: When effective medical treatment and vaccination are not available, non-pharmaceutical interventions such as social distancing, home quarantine and far-reaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptible-exposed-infectious-recovered) model and continuous-time optimal control theory, we compute the optimal non-pharmaceutical intervention strategy for the case that a vaccine is never found and complete containment (eradication of the epidemic) is impossible. In this case, the optimal control must meet competing requirements: First, the minimization of disease-related deaths, and, second, the establishment of a sufficient degree of natural immunity at the end of the measures, in order to exclude a second wave. Moreover, the socio-economic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a single-intervention scenario that goes beyond heuristically motivated interventions and simple “flattening of the curve”. Careful analysis of the computed control strategy reveals, however, that the obtained solution is in fact a tightrope walk close to the stability boundary of the system, where socio-economic costs and the risk of a new outbreak must be constantly balanced against one another. The model system is calibrated to reproduce the initial exponential growth phase of the COVID-19 pandemic in Germany.

Quantifying the role of social distancing, personal protection and case detection in mitigating COVID-19 outbreak in Ontario, Canada.

Abstract: Public health interventions have been implemented to mitigate the spread of coronavirus disease 2019 (COVID-19) in Ontario, Canada; however, the quantification of their effectiveness remains to be done and is important to determine if some of the social distancing measures can be relaxed without resulting in a second wave. We aim to equip local public health decision- and policy-makers with mathematical model-based quantification of implemented public health measures and estimation of the trend of COVID-19 in Ontario to inform future actions in terms of outbreak control and de-escalation of social distancing. Our estimates confirm that (1) social distancing measures have helped mitigate transmission by reducing daily infection contact rate, but the disease transmission probability per contact remains as high as 0.145 and case detection rate was so low that the effective reproduction number remained higher than the threshold for disease control until the closure of non-essential business in the Province; (2) improvement in case detection rate and closure of non-essential business had resulted in further reduction of the effective control number to under the threshold. We predict the number of confirmed cases according to different control efficacies including a combination of reducing further contact rates and transmission probability per contact. We show that improved case detection rate plays a decisive role to reduce the effective reproduction number, and there is still much room in terms of improving personal protection measures to compensate for the strict social distancing measures.

A neural network-based framework for financial model calibration

Abstract: A data-driven approach called CaNN (Calibration Neural Network) is proposed to calibrate financial asset price models using an Artificial Neural Network (ANN). Determining optimal values of the model parameters is formulated as training hidden neurons within a machine learning framework, based on available financial option prices. The framework consists of two parts: a forward pass in which we train the weights of the ANN off-line, valuing options under many different asset model parameter settings; and a backward pass, in which we evaluate the trained ANN-solver on-line, aiming to find the weights of the neurons in the input layer. The rapid on-line learning of implied volatility by ANNs, in combination with the use of an adapted parallel global optimization method, tackles the computation bottleneck and provides a fast and reliable technique for calibrating model parameters while avoiding, as much as possible, getting stuck in local minima. Numerical experiments confirm that this machine-learning framework can be employed to calibrate parameters of high-dimensional stochastic volatility models efficiently and accurately.