Computational Mathematics and Applications Seminar

We present an exponential-integrator-based multi-index Monte Carlo (MIMC) method for the weak approximation of mild solutions to semilinear stochastic partial differential equations (SPDEs). Theoretical results on multi-index coupled solutions of the SPDE are provided, demonstrating their stability and the satisfaction of multiplicative error estimates. Leveraging this theory, we develop a tractable MIMC algorithm. Numerical experiments illustrate that MIMC outperforms alternative approaches, such as multilevel Monte Carlo, particularly in low-regularity settings.

There is a mystery at the heart of operator learning: how can one recover a non-self-adjoint operator from data without probing the adjoint? Current practical approaches suggest that one can accurately recover an operator while only using data generated by the forward action of the operator without access to the adjoint. However, naively, it seems essential to sample the action of the adjoint for learning time-dependent PDEs. 

In this talk, we will first explore connections with low-rank matrix recovery problems in numerical linear algebra. Then, we will show that one can approximate a family of non-self-adjoint infinite-dimensional compact operators via projection onto a Fourier basis without querying the adjoint.

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We introduce a novel approach to global optimization  via continuous-time dynamic programming and Hamilton-Jacobi-Bellman (HJB) PDEs. For non-convex, non-smooth objective functions,  we reformulate global optimization as an infinite horizon, optimal asymptotic stabilization control problem. The solution to the associated HJB PDE provides a value function which corresponds to a (quasi)convexification of the original objective.  Using the gradient of the value function, we obtain a  feedback law driving any initial guess towards the global optimizer without requiring derivatives of the original objective. We then demonstrate that this HJB control law can be integrated into other global optimization frameworks to improve its performance and robustness. 

 Many optimal control, estimation and design problems can be formulated as so-called dynamic optimization problems, which are optimization problems with differential equations and other constraints. State-of-the-art methods based on collocation, which enforce the differential equations at only a finite set of points, can struggle to solve certain dynamic optimization problems, such as those with high-index differential algebraic equations, consistent overdetermined constraints or problems with singular arcs. We show how numerical methods based on integrating the differential equation residuals can be used to solve dynamic optimization problems where collocation methods fail. Furthermore, we show that integrated residual methods can be computationally more efficient than direct collocation.

This seminar takes place at RAL (Rutherford Appleton Lab). 

The Landau equation stands as one of the fundamental equations in kinetic theory and plays a key role in plasma physics. However, computing it presents significant challenges due to the complexity of the Landau operator,  the dimensionality, and the need to preserve the physical properties of the solution. In this presentation, I will introduce deep learning assisted particle methods aimed at addressing some of these challenges. These methods combine the benefits of traditional structure-preserving techniques with the approximation power of neural networks, aiming to handle high dimensional problems with minimal training. 

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