Inverse Problems and Numerical Methods in Stochastic Differential Equations

The field of inverse problems and numerical methods in stochastic differential equations is experiencing significant developments, driven by advancements in solving ill-posed inverse problems and designing effective numerical schemes. Researchers are focusing on innovative approaches to address the challenges posed by stochastic wave equations, heat equations, and other complex systems. A key area of research is the development of novel numerical methods, such as iterative regularization techniques and modified Euler approximations, which can efficiently solve inverse problems and provide stable solutions. Another important direction is the use of data-driven approaches, including graph Laplacian-based methods and moment analysis, to improve the accuracy and robustness of inverse problem solutions. Notably, some papers have made significant contributions to the field. For example, one paper proposes a time-domain PML method for the stochastic acoustic wave equation, while another presents a data-assisted iterative regularization method with early stopping. A third paper develops a novel computational framework for solving inverse source problems in stochastic wave equations. These innovative approaches and methods are advancing the field and offering promising applications in various domains, including seismic wave propagation analysis and financial market volatility modeling.

Sources

PML method for the time-domain stochastic acoustic wave equation and an inverse source problem

On the convergence of iterative regularization method assisted by the graph Laplacian with early stopping

Explicit modified Euler approximations of the A\"{i}t-Sahalia type model with Poisson jumps

The inverse source problem of stochastic wave equation

Fast reconstruction approaches for photoacoustic tomography with smoothing Sobolev/Mat\'ern priors

Moments, Time-Inversion and Source Identification for the Heat Equation

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