Tobias Grafke

Nayef Shkeir, Tobias Grafke

Abstract

Sampling Gaussian measures restricted to nonlinear manifolds in high dimensions requires a proposal kernel that respects both geometric constraints and dimensional scaling requirements. We introduce tangent preconditioned Crank-Nicolson (pCN), which combines a manifold projection with a cancellation of the ambient Gaussian contribution, leaving only a geometric residual. Considering a shifted sphere with $\sqrt d$ scaling of radius and centre as a tractable benchmark embedded manifold, we prove that uncorrected tangent pCN results in a proposal noise scale of order $d^{-1/6}$ (compared to $d^{-1/2}$ for a tangent random-walk). We establish asymptotic optimal tuning for the geodesic expected squared jump distance which is maximised at an acceptance probability of 0.574 for the family of drift-corrected tangent proposals containing tangent pCN and tangent MALA. We also prove a joint diffusion limit for the parallel and orthogonal coordinates to the sphere's centre vector of tangent pCN. The orthogonal diffusion speed matches the limiting geodesic expected squared jump distance and is maximised at acceptance probability 0.574, whereas the parallel speed is maximised at 0.712, demonstrating that optimal tuning depends on the observable. We then identify an intrinsic precision correction characterised by potential Hessian and Ricci curvature that yields a proposal kernel with dimension-independent fixed step size limit. We discuss a generalisation to arbitrary hypersurfaces and present nondegenerate scaling of tangent pCN applied to conditioned diffusions under resolution refinement.

arXiv

Nayef Shkeir, Tobias Grafke, J. Comp. Phys. 569 (2027), 115447

Abstract

Matrix evolution equations occur in many applications, such as dynamical Lyapunov/Sylvester systems or Riccati equations in optimization and stochastic control, machine learning or data assimilation. In many such problems, the dominant stability restriction is imposed by a stiff linear term, making standard explicit integrators impractical. Exponential time differencing (ETD) is known to produce highly stable numerical schemes by treating the linear term in an exact fashion. In particular, for stiff problems, ETD methods are the methods of choice. We extend ETD to matrix-valued evolution equations of the form $\dot Q=LQ+QR+N(Q,t)$ by deriving explicit matrix-ETD (METD) schemes. When L and R commute, we construct an explicit p-th order METDp family and prove order-p global convergence under standard assumptions; for the non-commuting case, we develop a Baker-Campbell-Hausdorff (BCH)-based extension. This allows us to produce highly efficient and stable integration schemes. We demonstrate efficiency and applicability on stiff PDE-derived and large-scale matrix dynamics, including an Allen-Cahn system, turbulent jet fluctuation statistics, and continuous graph neural networks. We further show that the scheme is more accurate, stable, and efficient than competing schemes in large-scale high-rank stiff systems.

doi:10.1016/j.jcp.2026.115447

arXiv

Jiayao Shao, Tobias Grafke, and Robert S MacKay, Phys. Rev. E 114 (2026), 024210

Abstract

Stochastic dynamical systems allow modeling of transitions induced by random disturbances, in particular from an attracting equilibrium and crossing the stable manifold of a saddle. While the small-noise limit is well-described by the large deviation principle, existing computational methods often struggle with stochastic forcings any more complicated than nondegenerate Gaussian white noise, and with unbounded time intervals. The primary innovations of this work are extending the framework to cater for colored and degenerate forcing and unbounded time horizons—scenarios that are physically realistic but numerically challenging. We cater for degenerate noise by using the Hamiltonian optimal control method. We cater for a class of colored noises by using linear filters on white noise. We cater for infinite time horizon by introducing the method of division (MOD), a computational approach for approximating rare transition events, including their most likely paths and the exponential scaling laws of their transition rates. The effectiveness of MOD and the above approaches to colored noise, is demonstrated by illustration on two examples: an inverted double-well potential and a simplified roll-heave model for ship capsize.

doi:10.1103/w8dd-9cqg

arXiv

T. Schorlepp and T. Grafke, SIAM JUQ 14 (2026) 3.

Abstract

We show how to efficiently compute asymptotically sharp estimates of extreme event probabilities in stochastic differential equations (SDEs) with small multiplicative Brownian noise. The underlying approximation is known as sharp large deviation theory or precise Laplace asymptotics in mathematics, the second-order reliability method (SORM) in reliability engineering, and the instanton or optimal fluctuation method with 1-loop corrections in physics. It is based on approximating the tail probability in question with the most probable realization of the stochastic process, and local perturbations around this realization. We first recall and contextualize the relevant classical theoretical result on precise Laplace asymptotics of diffusion processes [G. Ben Arous, Stochastics, 25 (1988), pp. 125–153], and then show how to compute the involved infinite-dimensional quantities—operator traces and Carleman–Fredholm determinants—numerically in a way that is scalable with respect to the time discretization and remains feasible in high spatial dimensions. Using tools from automatic differentiation, we achieve a straightforward black-box numerical computation of the SORM estimates in JAX. The method is illustrated in examples of SDEs and stochastic partial differential equations, including a two-dimensional random advection-diffusion model of a passive scalar. We thereby demonstrate that it is possible to obtain efficient and accurate SORM estimates for very high-dimensional problems, as long as the infinite-dimensional structure of the problem is correctly taken into account. Our JAX implementation of the method is made publicly available.

doi:10.1137/25M1743867

arXiv

T. Grafke, S. Scholtes, A. Wagner, M. Westdickenberg, Math. Mod. Meth. Appl. Sci. 36 (7) (2026) 1381–1414

Abstract

We explore recent progress and open questions concerning local minima and saddle points of the Cahn-Hilliard energy in $d\ge 2$ and the critical parameter regime of large system size and mean value close to $-1$. We employ the String Method of E, Ren, and Vanden-Eijnden — a numerical algorithm for computing transition pathways in complex systems — in $d=2$ to gain additional insight into the properties of the minima and saddle point. Motivated by the numerical observations, we adapt a method of Caffarelli and Spruck to study convexity of level sets in $d\ge2$.

doi:10.1142/S0218202526500235

arXiv

A. Svirsky, T. Grafke, and A. Frishman, Phys. Rev. Lett. 136 (2026), 124001

Abstract

Pipe flow is a canonical example where turbulence first appears intermittently in space and time, taking the form of localized structures termed puffs. Turbulence spreads via puff self-replication, which must out-compete puff decays to sustain it. Here we study the self-replication process, a transition from one to two puffs, using direct numerical simulations. We identify an edge state on the phase space boundary between the two states, demonstrate that it mediates the transition, and show that self-replication follows a previously proposed mechanism, with the edge state as its tipping point.

doi:10.1103/wvdb-69my

arXiv

Jelle Soons, Tobias Grafke, René M. van Westen, and Henk A. Dijkstra, J. Phys: Complexity 6 (2026), 035015

Abstract

Recently the global average temperature has temporarily exceeded the 1.5°C goal of the Paris Agreement, and so an overshoot of various climate tipping elements becomes increasingly likely. In this study we analyze the physical processes of an overshoot of the Atlantic Meridional Overturning Circulation (AMOC), one of the major tipping elements, using a conceptual box model. Here either the atmospheric temperature above the North Atlantic, or the freshwater forcing into the North Atlantic overshoot their respective critical boundaries. In both cases a higher forcing rate can prevent a collapse of the AMOC, since a higher rate of forcing causes initially a fresher North Atlantic, which in turn results in a higher northward transport by the subtropical gyre supplementing the salinity loss in time. For small exceedance amplitudes the AMOC is still resilient as the forcing rates can be low and so other state variables outside of the North Atlantic can adjust. Contrarily, for larger overshoots the trajectories are dynamically similar and we find a lower limit in volume and exceedance time for respectively freshwater and temperature forcing in order to prevent a collapse. Moreover, for a large overshoot an increased air-sea temperature coupling has a destabilizing effect, while the reverse holds for an overshoot close to the tipping point. The understanding of the physics of the AMOC overshoot behavior is important for interpreting results of Earth System Models and for evaluating the effects of mitigation and intervention strategies.

doi:10.1088/2632-072X/ae06e7

arXiv

Paolo Bernuzzi, Tobias Grafke, SIAM Multiscale Model. Simul. 23(3) (2025) 1274

Abstract

Noise-induced transitions between multistable states happen in a multitude of systems, such as species extinction in biology, protein folding, or tipping points in climate science. Large deviation theory is the rigorous language to describe such transitions for non-equilibrium systems in the small noise limit. At its core, it requires the computation of the most likely transition pathway, solution to a PDE constrained optimization problem. Standard methods struggle to compute the minimiser in the particular coexistence of (1) multistability, i.e. coexistence of multiple long-lived states, and (2) degenerate noise, i.e. stochastic forcing acting only on a small subset of the system's degrees of freedom. In this paper, we demonstrate how to adapt existing methods to compute the large deviation minimiser in this setting by combining ideas from optimal control, large deviation theory, and numerical optimisation. We show the efficiency of the introduced method in various applications in biology, medicine, and fluid dynamics, including the transition to turbulence in subcritical pipe flow.

doi:10.1137/24M169713X

arXiv

Summary: In stochastic systems, multiple long-lived states might coexists, in which the system spends a large amount of time before spontaneously and often catastrophically switching to a different long-lived state. For example, in molecular dynamics, the reactant and the product state both correspond to local minima in an energy landscape, but thermal activation pushed the system across the separating energy barrier, corresponding to the chemical reaction. Similarly, Earth's climate might spend a long time in a certain long-lived state, but climate change might push it across a tipping point, such that the state of the climate abruptly changes. Often, these changes are hard to predict and rare on the intrinsic time scale of the system. In this project, algorithms are developed to simulate only those trajectories of the stochastic system that exhibit the transition. This transition path sampling or bridge sampling allows to generate for example conditional statistics, analyze physical causes responsible for the transition, or even develop early warning signs for an anticipated transition in progress.

Relevant publications

1. T. Grafke, "Sampling conditioned diffusions via Pathspace Projected Monte Carlo", ArXiv (2025) (link)

2. T. Grafke, A. Laio, "Metadynamics for transition paths in irreversible dynamics", Multiscale Modeling & Simulation, Vol. 22, Iss. 1 (2024) (link)