C. Michler

R. E. Heath, I. M. Gamba, P. J. Morrison et al.

A discontinuous Galerkin method for approximating the Vlasov-Poisson system of equations describing the time evolution of a collisionless plasma is proposed. The method is mass conservative and, in the case that piecewise constant functions are used as a basis, the method preserves the positivity of the electron distribution function and weakly enforces continuity of the electric field through mesh interfaces and boundary conditions. The performance of the method is investigated by computing several examples and error estimates associated system's approximation are stated. In particular, computed results are benchmarked against established theoretical results for linear advection and the phenomenon of linear Landau damping for both the Maxwell and Lorentz distributions. Moreover, two nonlinear problems are considered: nonlinear Landau damping and a version of the two-stream instability are computed. For the latter, fine scale details of the resulting long-time BGK-like state are presented. Conservation laws are examined and various comparisons to theory are made. The results obtained demonstrate that the discontinuous Galerkin method is a viable option for integrating the Vlasov-Poisson system.

Antonio Briola, Marwin Schmidt, Fabio Caccioli et al.

High-dimensional data often exhibit dependencies among variables that violate the isotropic-noise assumption under which principal component analysis (PCA) is optimal. For cases where the noise is not independent and identically distributed across features (i.e., the covariance is not spherical) we introduce Graph Regularized PCA (GR-PCA). It is a graph-based regularization of PCA that incorporates the dependency structure of the data features by learning a sparse precision graph and biasing loadings toward the low-frequency Fourier modes of the corresponding graph Laplacian. Consequently, high-frequency signals are suppressed, while graph-coherent low-frequency ones are preserved, yielding interpretable principal components aligned with conditional relationships. We evaluate GR-PCA on synthetic data spanning diverse graph topologies, signal-to-noise ratios, and sparsity levels. Compared to mainstream alternatives, it concentrates variance on the intended support, produces loadings with lower graph-Laplacian energy, and remains competitive in out-of-sample reconstruction. When high-frequency signals are present, the graph Laplacian penalty prevents overfitting, reducing the reconstruction accuracy but improving structural fidelity. The advantage over PCA is most pronounced when high-frequency signals are graph-correlated, whereas PCA remains competitive when such signals are nearly rotationally invariant. The procedure is simple to implement, modular with respect to the precision estimator, and scalable, providing a practical route to structure-aware dimensionality reduction that improves structural fidelity without sacrificing predictive performance.

Yuanrong Wang, Vignesh Raja Swaminathan, Nikita P. Granger et al.

In this paper, we discuss the Dutch power market, which is comprised of a day-ahead market and an intraday balancing market that operates like an auction. Due to fluctuations in power supply and demand, there is often an imbalance that leads to different prices in the two markets, providing an opportunity for arbitrage. To address this issue, we restructure the problem and propose a collaborative dual-agent reinforcement learning approach for this bi-level simulation and optimization of European power arbitrage trading. We also introduce two new implementations designed to incorporate domain-specific knowledge by imitating the trading behaviours of power traders. By utilizing reward engineering to imitate domain expertise, we are able to reform the reward system for the RL agent, which improves convergence during training and enhances overall performance. Additionally, the tranching of orders increases bidding success rates and significantly boosts profit and loss (P&L). Our study demonstrates that by leveraging domain expertise in a general learning problem, the performance can be improved substantially, and the final integrated approach leads to a three-fold improvement in cumulative P&L compared to the original agent. Furthermore, our methodology outperforms the highest benchmark policy by around 50% while maintaining efficient computational performance.