Publications

This work is concerned with the numerical computation of Gross–Pitaevskii ground states using the tensor train (TT) format. We regard the problem as the minimization of the Gross–Pitaevskii energy functional under a unit mass constraint, and propose a first-order Riemannian optimization scheme that preserves both the mass constraint and the low-rank representation throughout the optimization, by operating on the manifold of unit-norm functions of fixed TT-rank. On this manifold, we derive the energy-adaptive Riemannian gradient and show how to compute it efficiently in the TT format. Spatial discretization is performed using a spectral method with numerical integration, which allows the required computations to be carried out efficiently while preserving the low-rank format. The method is shown to substantially reduce computational time compared to full-rank computations, while maintaining accuracy for both single- and multicomponent Gross–Pitaevskii equations.

A JAX implementation of the Self-Scaled Broyden family of quasi-Newton methods, building on the Optimistix optimisation library. It includes BFGS, DFP, Broyden and their self-scaled variants (SSBFGS, SSDFP, SSBroyden), together with a Zoom line search satisfying the strong Wolfe conditions. The note documents the implementation to ease the adoption of these optimisers within the JAX community.

Natural Gradient Descent (NGD) has emerged as a promising optimization algorithm for training neural network-based solvers for PDEs, such as Physics-Informed Neural Networks (PINNs). However, its practical use is often limited by the high computational cost of solving linear systems involving the Gramian matrix. While matrix-free NGD methods based on the conjugate gradient (CG) method avoid explicit matrix inversion, the ill-conditioning of the Gramian significantly slows the convergence of CG. In this work, we extend matrix-free NGD to broader classes of problems than previously considered and propose the use of Randomized Numerical Linear Algebra (RandNLA) techniques for efficient preconditioning of the inner CG solver. The resulting algorithms demonstrate substantial performance improvements over existing NGD-based methods on a range of PDE problems discretized using neural networks, and offer competitive results compared to other state-of-the-art optimizers.

In structural dynamics, mass lumping techniques are commonly employed for improving the efficiency of explicit time integration schemes and increasing their critical time step constrained by the largest discrete frequency of the system. For immersogeometric methods, it was first observed in 2020 that for sufficiently smooth spline discretizations, the largest frequency was not affected by small trimmed elements if the mass matrix was lumped, a finding later supported by several independent numerical studies. This article provides a rigorous theoretical analysis aimed at unraveling this property. By combining linear algebra with functional analysis, we derive sharp analytical estimates capturing the behavior of the largest discrete frequency for lumped mass approximations and various trimming configurations. Additionally, we also provide estimates for the smallest discrete frequency, which has lately drawn closer scrutiny. Our estimates are then confirmed numerically for 1D and 2D problems.

This work is concerned with the numerical solution of large-scale symmetric positive definite matrix equations of the form A₁XB₁ᵀ + A₂XB₂ᵀ + ⋯ + AℓXBℓᵀ = F, as they arise from discretized partial differential equations and control problems. One often finds that X admits good low-rank approximations, in particular when the right-hand side F has low rank. For ℓ > 2, several existing methods combine a classical iterative method with low-rank truncation. We consider a more direct approach that approximates X on manifolds of fixed-rank matrices through Riemannian CG, and propose several novel preconditioning strategies, including a change of metric in the ambient space, preconditioning the Riemannian gradient, and a variant of ADI on the tangent space. Combined with a strategy for adapting the rank of the approximation, the resulting method is competitive for a number of examples representative of typical applications.