In this work, cascading transmission line failures are studied through a dynamical model of the power system operating under fixed conditions. The power grid is modeled as a stochastic dynamical system where first-principles electromechanical dynamics are excited by small Gaussian disturbances in demand and generation around a specified operating point. In this context, a single line failure is interpreted in a large deviation context as a first escape event across a surface in phase space defined by line security constraints. The resulting system of stochastic differential equations admits a transverse decomposition of the drift, which leads to considerable simplification in evaluating the quasipotential (rate function) and, consequently, computation of exit rates. Tractable expressions for the rate of transmission line failure in a restricted network are derived from large deviation theory arguments and validated against numerical simulations. Extensions to realistic settings are considered, and individual line failure models are aggregated into a Markov model of cascading failure inspired by chemical kinetics. Cascades are generated by traversing a graph composed of weighted edges representing transitions to degraded network topologies. Numerical results indicate that the Markov model can produce cascades with qualitative power-law properties similar to those observed in empirical cascades.

Compression, Inversion, and Approximate PCA of Dense Kernel Matrices at Near-Linear Computational Complexity

We design a variational asymptotic preserving scheme for the Vlasov--Poisson--Fokker--Planck system with the high field scaling, which describes the Brownian motion of a large system of particles in a surrounding bath. Our scheme builds on an implicit-explicit framework, wherein the stiff terms coming from the collision and field effects are solved implicitly while the convection terms are solved explicitly. To treat the implicit part, we propose a variational approach by viewing it as a Wasserstein gradient flow of the relative entropy, and solve it via a proximal quasi-Newton method. In so doing we get positivity and asymptotic preservation for free. The method is also massively parallelizable and thus suitable for high dimensional problems. We further show that the convergence of our implicit solver is uniform across different scales. A suite of numerical examples are presented at the end to validate the performance of the proposed scheme.

This paper is concerned with an inverse random source problem for the three-dimensional time-harmonic Maxwell equations. The source is assumed to be a centered complex-valued Gaussian vector field ...

In the first part of the paper, we propose and rigorously analyze a mixed\nfinite element method for the approximation of the periodic strong solution to\nthe fully nonlinear second-order Hamilton--Jacobi--Bellman equation with\ncoefficients satisfying the Cordes condition. These problems arise as the\ncorrector problems in the homogenization of Hamilton--Jacobi--Bellman\nequations. The second part of the paper focuses on the numerical homogenization\nof such equations, more precisely on the numerical approximation of the\neffective Hamiltonian. Numerical experiments demonstrate the approximation\nscheme for the effective Hamiltonian and the numerical solution of the\nhomogenized problem.\n

In this paper we propose an efficient method to compress a high dimensional function into a tensor ring format, based on alternating least squares (ALS). Since the function has size exponential in $d$, where $d$ is the number of dimensions, we propose an efficient sampling scheme to obtain $O(d)$ important samples in order to learn the tensor ring. Furthermore, we devise an initialization method for ALS that allows fast convergence in practice. Numerical examples show that to approximate a function with similar accuracy, the tensor ring format provided by the proposed method has fewer parameters than the tensor-train format and also better respects the structure of the original function.

This article addresses the problem of estimating the Koopman generator of a Markov process. The direct computation of the infinitesimal generator is not easy because of the discretization of the state space, in particular because of the trade-off inherent in the choice of the best lag time to study the process. Short lag times implies a strong discretization of the state space and a consequent loss of Markovianity. Large lag times bypass events on fast timescales. We propose a method to approximate the generator with the computation of the Newton polynomial extrapolation. This technique is a multistep approach which uses as its input Koopman transfer operators evaluated for a series of lag times. Thus, the estimated infinitesimal generator combines information from different time resolutions and does not bias only fast- or slow-decaying dynamics. We show that the multiscale Newton method can improve the estimation of the generator in comparison to the computation using finite difference or matrix logarithm methods.

Transition path theory (TPT) for diffusion processes is a framework for analysing the transitions of multiscale ergodic diffusion processes between disjoint metastable subsets of state space. Most methods for applying TPT involve the construction of a Markov state model on a discretisation of state space that approximates the underlying diffusion process. However, the assumption of Markovianity is difficult to verify in practice, and there are to date no known error bounds or convergence results for these methods. We propose a Monte Carlo method for approximating the forward committor, probability current, and streamlines from TPT for diffusion processes. Our method uses only sample trajectory data and partitions of state space based on Voronoi tessellations. It does not require the construction of a Markovian approximating process. We rigorously prove error bounds for the approximate TPT objects and use these bounds to show convergence to their exact counterparts in the limit of arbitrarily fine discretisation. We illustrate some features of our method by application to a process that solves the Smoluchowski equation on a triple-well potential.

We develop an asymptotic analysis of target fluxes in the three-dimensional (3D) narrow capture problem. The latter concerns a diffusive search process in which the targets are much smaller than the size of the search domain. The small target assumption allows us to use matched asymptotic expansions and Green's functions to solve the diffusion equation in Laplace space. In particular, we derive an asymptotic expansion of the Laplace transformed flux into each target in powers of the nondimensionalized target size $\epsilon$. One major advantage of working directly with fluxes is that one can generate statistical quantities, such as splitting probabilities and conditional first passage time moments, without having to solve a separate boundary value problem in each case. However, in order to derive asymptotic expansions of these quantities, it is necessary to eliminate Green's function singularities that arise in the limit $s\rightarrow 0$, where $s$ is the Laplace variable. We achieve this by considering a triple expansion in $\epsilon$, $s$, and $\Lambda\sim \epsilon /s$. This allows us to perform partial summations over infinite power series in $\Lambda$, which leads to multiplicative factors of the form $\Lambda^n/(1+\Lambda)^n $. Since $\Lambda^n/(1+\Lambda)^n \rightarrow 1$ as $s\rightarrow 0$, the singularities in $s$ are eliminated. We then show how corresponding asymptotic expansions of the splitting probabilities and conditional mean first passage times (MFPTs) can be derived in the small-$s$ limit. The resulting expressions agree with previous asymptotic expansions derived by solving a separate boundary value problem for each statistical quantity, although each expansion was only carried out to second order in the expansions. Here we also determine the third order contributions, which are $O(\epsilon^2)$ and $O(\epsilon)$ in the cases of the splitting probabilities and conditional (MFPTs), respectively. Finally, we illustrate the theory by considering a pair of targets in a spherical search domain, for which the Green's functions can be calculated explicitly.

The Fokker--Planck approximation for an elementary linear, two-dimensional kinetic model endowed with a mass-preserving integral collision process is numerically studied, along with its diffusive limit. In order to set up a well-balanced discretization relying on an $S$-matrix, exact steady states of the continuous equation are derived. The ability of the scheme to keep these stationary solutions invariant produces the discretization of the local differential operator which mimics the collision process. The aforementioned scheme can be reformulated as an implicit-explicit one, which is proved to be both well-balanced and asymptotic-preserving in the diffusion limit. Several numerical benchmarks, computed on coarse grids, are displayed so as to illustrate the results.