Computational Mathematics, Statistics, and Machine Learning

We focus on developing numerical methods to solve complex partial differential equations (PDEs) in fields like materials science, fluid mechanics, and thermodynamics. This includes developing probabilistic frameworks to account for uncertainty in material properties and geometry at microscopic scales, particularly in simulations of materials with random or heterogeneous structures. Using Monte Carlo methods, we simulate failure mechanisms by modelling the statistical distribution of material responses under stress, allowing for predictions that incorporate the variability inherent in real-world materials. Additionally, adaptive multilevel Monte Carlo methods (efficient solving PDEs by dynamically refining computational meshes based on error estimates) are used to achieve high accuracy with optimal computational resources.

Beyond stochastic methods, we develop deterministic techniques to improve the computational efficiency of solving large, complex systems of equations. This includes high-order finite element methods for fluid and wave dynamics, where stability and accuracy are critical, as well as the application of reduced-order modeling techniques such as proper orthogonal decomposition. Using reduced-order models, we can decrease the computational load for large-scale simulations, making it feasible to model complex fluid and heat transfer systems more efficiently. Collectively, these efforts contribute to advancing computational tools for simulating real-world phenomena across a variety of engineering and scientific applications.

References:

1) M. Abbaszadeh, M. Dehghan, A. Khodadadian, N. Noii, C. Heitzinger and T. Wick, A reduced-order variational multiscale interpolating element free Galerkin technique based on proper orthogonal decomposition for solving Navier-Stokes equations coupled with a heat transfer equation: Nonstationary incompressible Boussinesq equations. Journal of Computational Physics, 426, 109875, 2021, DOI: 10.1016/j.jcp.2020.109875

2) M. Parvizi, A. Khodadadian and M.R. Eslahchi, Analysis of Ciarlet-Raviart mixed finite element methods for solving damped Boussinesq equation, Journal of Computational and Applied Mathematics, 379, 112818, 2020, DOI: 10.1016/j.cam.2020.112818

3) A. Khodadadian, M. Parvizi and C. Heitzinger, An adaptive multilevel Monte Carlo algorithm for the stochastic drift-diffusion-Poisson system, Computer Methods in Applied Mechanics and Engineering, 368, 113163, 2020, DOI: 10.1016/j.cma.2020.113163

4) N. Noii, A. Khodadadian and F. Aldakheel, Probabilistic failure mechanisms via Monte Carlo simulations of complex microstructures, Computer Methods in Applied Mechanics and Engineering, 399, 115358, 2022, DOI: 10.1016/j.cma.2022.115358

Traditional regime-switching models are widely used for volatility modelling in empirical economics and finance research for their ability to identify and account for the impact of latent regimes or states on the behaviour of the interested variables. Recently, there are several different ways of modelling endogenous regime changes. By different constructions, the resulting state transition probabilities are time-varying and dependent on the lagged values of the observed time series.

By considering regime switching chains, we proposed the general idea of using Reducible Diffusions (RDs) with nonlinear time-varying transformations for modeling financial and economic variables. Our application suggests that from an empirical point of view time-varying transformations are highly relevant and statistically significant. We expect that the proposed models can describe more truthfully the dynamic time-varying behavior of economic and financial variables and potentially improve out-of-sample forecasts significantly [Bu, R., J. Cheng, and K. Hadri. (2016). Reducible Diffusions with Time-Varying Transformations with Application to Short-Term Interest Rates. Economic Modelling 52: 266–277.].

We examine model specification in regime-switching continuous-time diffusions for modeling S&P 500 Volatility Index (VIX) [Bu, R., Cheng, J., & Hadri, K. (2017). Specification analysis in regime-switching Continuous-time diffusion models for market volatility. Studies in Nonlinear Dynamics and Econometrics, 21(1), 65–80.]. Our investigation is carried out under two nonlinear diffusion frameworks, the NLDCEV and the CIRCEV frameworks, and our focus is on the nonlinearity in regime-dependent drift and diffusion terms, the switching components, and the endogeneity in regime changes. While we find strong evidence of regime-switching effects, models with a switching diffusion term capture the VIX dynamics considerably better than models with only a switching drift, confirming the presence and importance of volatility regimes. Strong evidence of nonlinear endogeneity in regime changes is also detected.

We generalize the latent-factor-driven endogenous regime-switching Gaussian model of Chang, et al., Journal of Econometrics, 2017, 196, 127–143 by allowing the state-dependent conditional distributions to be non-Gaussian. Our setup is more general and promises substantially broader relevance and applicability to empirical studies. We provide evidence to justify our generalization by a simulation study and a real data application. Our simulation results confirm that when the state-dependent dynamics are misspecified, the bias of model parameter estimates, the power of the likelihood ratio test against endogenous regime changes, and the quality of the extracted latent factor all deteriorate quite considerably. [Bu, R., J. Cheng, and Jawadi. Fredj. (2020). A latent-factor-driven endogenous regime-switching non-Gaussian model: Evidence from simulation and application. International Journal of Finance & Economics, DOI: 10.1002/ijfe.2192].

Current research is focussed in understanding bivariate time series models driven by either threshold/smooth transition chains or (exo and endo) regime switching models with latent vector autoregressive factors, which allows for (un)synchronized switching and endogenous feedback. This will allow us to consider dynamics of several financial factors and the interactions among them. Several projects are based on this idea.

Another key topic we are interested in is risk management. It is vital for clearinghouses to employ appropriate market instruments (Margin, capital requirement and price limits) in order to strike a delicate balance between increasing futures price stability, not impairing price discovery, and facilitating futures growth. A framework is proposed that is rooted in extreme value theory to study the performance of margin, capital requirement and price limits and their interactions in the presence of clearing firms’ risk preferences. By applying the concept of self-enforcing contracts, we incorporate clearing firms’ risk attitudes into the framework. The efficacy of these market instruments under three risk measures (i.e. VaRs, ESs and SRMs) is studied, while the latter two risk measures present two approaches to gauge potential losses in the form of capital requirement [Cheng, J., Hong, Y., & Tao J (2015) How do risk attitudes of clearing firms matter for managing default exposure in futures markets? The European Journal of Finance, 22(10)]. The results cast new light on the economic rationale of price limits, which is examined now during the Coronavirus outbreak.

Collaborators

• Dr Ruijun Bu, Liverpool University, UK
• Prof Fredj Jawadi, University of Lille, France