In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its partial derivatives , a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute $\epsilon$ -approximate inverse Cholesky factors of the kernel matrices with complexity $O(N\log ^d(N/\epsilon ))$ in space and $O(N\log ^{2d}(N/\epsilon ))$ in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm’s near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.

The particle motion associated to the α-relaxation in supercooled liquids is still challenging scientists due to its difficulty to be probed experimentally. By combining synchrotron techniques, we report the existence of microscopic structure-dynamics relationships in Pt_42.5Cu₂₇Ni_9.5P₂₁ and Pd_42.5Cu₂₇Ni_9.5P₂₁ liquids which allows us to disentangle structural and kinetic contributions to the α-process. While the two alloys show similar kinetic fragilities, their structural fragilities differ and correlate with the temperature dependence of the stretching parameter describing the decay of the density fluctuations. This implies that the evolution of dynamical heterogeneities in supercooled alloys is determined by the rigidity of the melt structure. We find also that the atomic motion not only reflects the topological order but also the chemical short-range order, which can lead to a surprising slowdown of the α-process at the mesoscopic length scale. These results will contribute to the comprehension of the glass transition, which is still missing.

The FeSe compound is the simplest high-temperature superconductor (HTSc) possible, and relatively cheap, not containing any rare-earth material. Although the transition temperature, T_c, is just below 10 K, the upper critical fields are comparable with other HTSc. Preparing FeSe using solid-state sintering yields samples exhibiting strong ferromagnetic hysteresis loops (MHLs), and the superconducting contribution is only visible after subtracting MHLs from above T_c. Due to the complicated phase diagram, the samples are a mixture of several phases, the superconducting β-FeSe, and the non-superconducting δ-FeSe and γ-FeSe. Furthermore, antiferromagnetic Fe₇Se₈ and ferromagnetic α-Fe may be contained, depending directly on the Se loss during the sintering process. Here, we show MHLs measured up to ±7 T and determine the magnetic characteristics, together with the amount of superconductivity determined from M(T) measurements. We also performed a thorough analysis of the microstructures in order to establish a relation between microstructure and the resulting sample properties.

In recent studies, many groups have investigated the interaction of dislocations and grain boundaries by bi-crystals and micro-specimen experiments. Partially, these experiments were combined with supplementary simulations by discrete dislocation dynamics, but quantitative data for the grain boundary resistance against slip transfer is still missing. In this feasibility study with first results, we use stage-I-fatigue cracks as highly localised sources for dislocations with well-known Burgers vectors to study the interaction between dislocations in the plastic zone in front of the crack tip and selected grain boundaries. The stress concentration at the grain boundary is calculated with the dislocation-free zone model of fracture using the dislocation density distribution in the plastic zone from slip trace height profile measurements by atomic force microscopy. The grain boundary resistance values calculated from common geometric models are compared to the local stress distribution at the grain boundaries. Hence, it is possible to quantify the grain boundary resistance and to combine geometric and stress approach for grain boundary resistance against slip transfer to a self-contained concept. As a result, the prediction of the grain boundary resistance effect based on a critical stress concept is possible with knowledge of the geometric parameters of the grain boundary only, namely the orientations of both participating grains and the orientation of the grain boundary plane.