James Foster

Andraž Jelinčič, James Foster, Patrick Kidger · oxford

Despite the success of adaptive time-stepping in ODE simulation, it has so far seen few applications for Stochastic Differential Equations (SDEs). To simulate SDEs adaptively, methods such as the Virtual Brownian Tree (VBT) have been developed, which can generate Brownian motion (BM) non-chronologically. However, in most applications, knowing only the values of Brownian motion is not enough to achieve a high order of convergence; for that, we must compute time-integrals of BM such as $\int_s^t W_r \, dr$. With the aim of using high order SDE solvers adaptively, we extend the VBT to generate these integrals of BM in addition to the Brownian increments. A JAX-based implementation of our construction is included in the popular Diffrax library (https://github.com/patrick-kidger/diffrax). Since the entire Brownian path produced by VBT is uniquely determined by a single PRNG seed, previously generated samples need not be stored, which results in a constant memory footprint and enables experiment repeatability and strong error estimation. Based on binary search, the VBT's time complexity is logarithmic in the tolerance parameter $\varepsilon$. Unlike the original VBT algorithm, which was only precise at some dyadic times, we prove that our construction exactly matches the joint distribution of the Brownian motion and its time integrals at any query times, provided they are at least $\varepsilon$ apart. We present two applications of adaptive high order solvers enabled by our new VBT. Using adaptive solvers to simulate a high-volatility CIR model, we achieve more than twice the convergence order of constant stepping. We apply an adaptive third order underdamped or kinetic Langevin solver to an MCMC problem, where our approach outperforms the No U-Turn Sampler, while using only a tenth of its function evaluations.

David Smith, Frederik Geth, Elliott Vercoe et al.

For the modeling, design and planning of future energy transmission networks, it is vital for stakeholders to access faithful and useful power flow data, while provably maintaining the privacy of business confidentiality of service providers. This critical challenge has recently been somewhat addressed in [1]. This paper significantly extends this existing work. First, we reduce the potential leakage information by proposing a fundamentally different post-processing method, using public information of grid losses rather than power dispatch, which achieve a higher level of privacy protection. Second, we protect more sensitive parameters, i.e., branch shunt susceptance in addition to series impedance (complete pi-model). This protects power flow data for the transmission high-voltage networks, using differentially private transformations that maintain the optimal power flow consistent with, and faithful to, expected model behaviour. Third, we tested our approach at a larger scale than previous work, using the PGLib-OPF test cases [10]. This resulted in the successful obfuscation of up to a 4700-bus system, which can be successfully solved with faithfulness of parameters and good utility to data analysts. Our approach addresses a more feasible and realistic scenario, and provides higher than state-of-the-art privacy guarantees, while maintaining solvability, fidelity and feasibility of the system.