Report on Current Developments in the Research Area

General Direction of the Field

The recent developments in the research area are marked by a significant shift towards the integration of stochastic elements and advanced numerical methods in the analysis and solution of partial differential equations (PDEs), particularly in the context of finance, epidemiology, and inverse problems. The field is moving towards more sophisticated and efficient computational techniques that can handle the complexities introduced by randomness, nonlinearity, and high-dimensionality.

One of the key trends is the application of time-fractional diffusion equations with stochastic parameters. This approach allows for a more nuanced modeling of diffusion processes, where the randomness in the diffusivity can lead to more realistic and accurate predictions. The focus is on developing efficient numerical methods to estimate expected values, which are crucial for decision-making in various fields such as finance and epidemiology. High-order quasi-Monte Carlo methods and Galerkin finite elements are being combined to achieve this, with a strong emphasis on error analysis and computational efficiency.

Another notable trend is the development of second-order implicit-explicit (IMEX) Runge-Kutta schemes for solving parabolic PDEs, particularly in finance. These schemes are designed to handle the mixed derivatives and non-regular initial conditions that are common in financial models. The IMEX approach allows for the efficient treatment of both advection and diffusion terms, overcoming the limitations of purely explicit or implicit methods. This has led to more accurate and stable numerical solutions, particularly for option pricing and other financial applications.

In the realm of epidemiology, there is a growing interest in the numerical approximation of reproduction numbers for age-structured models. The use of pseudospectral methods to discretize infinite-dimensional operators and approximate these critical parameters is proving to be a powerful tool. The convergence of these approximations is being rigorously studied, with a focus on the regularity of model coefficients and the resulting convergence order. This work is not only advancing the theoretical understanding of these models but also providing practical tools for public health decision-making.

Inverse problems, particularly those of Cauchy-type, are also seeing innovative approaches. A probabilistic framework is being developed to analyze and solve these problems, leveraging elliptic measures and Monte Carlo simulations. This approach is designed to handle the inherent instability of inverse problems and provide accurate approximations even in high-dimensional settings or when the solution or domain exhibits singularities or complex geometries. The probabilistic error analysis and explicit error bounds are key contributions that enhance the reliability and applicability of these methods.

Noteworthy Papers

• Time-fractional diffusion equations with randomness: Efficient estimation of expected values using high-order quasi-Monte Carlo methods and Galerkin finite elements.
• IMEX-RK finite volume methods for nonlinear 1d parabolic PDEs: Second-order accuracy for nonlinear problems with non-regular initial conditions, applicable to option pricing.
• Convergence of pseudospectral approximation of reproduction numbers: Rigorous proof of convergence for age-structured models, with applications to epidemiology.
• Probabilistic approach to Cauchy-type inverse problems: Novel framework for analyzing and solving inverse problems, with explicit error bounds and GPU-based simulations.