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We formulate a data-driven method for constructing finite volume discretizations of a dynamical system's underlying Continuity / Fokker-Planck equation. A method is employed that allows for flexibility in partitioning state space, generalizes to function spaces, applies to arbitrarily long sequences of time-series data, is robust to noise, and quantifies uncertainty with respect to finite sample effects. After applying the method, one is left with Markov states (cell centers) and a random matrix approximation to the generator. When used in tandem, they emulate the statistics of the underlying system. We apply the method to the Lorenz equations (a three-dimensional ordinary differential equation) and a modified Held-Suarez atmospheric simulation (a Flux-Differencing Discontinuous Galerkin discretization of the compressible Euler equations with gravity and rotation on a thin spherical shell). We show that a coarse discretization captures many essential statistical properties of the system, such as steady state moments, time autocorrelations, and residency times for subsets of state space.

In this paper, we propose a mesh-free numerical method for solving elliptic PDEs on unknown manifolds, identified with randomly sampled point cloud data. The PDE solver is formulated as a spectral method where the test function space is the span of the leading eigenfunctions of the Laplacian operator, which are approximated from the point cloud data. While the framework is flexible for any test functional space, we will consider the eigensolutions of a weighted Laplacian obtained from a symmetric Radial Basis Function (RBF) method induced by a weak approximation of a weighted Laplacian on an appropriate Hilbert space. Especially, we consider a test function space that encodes the geometry of the data yet does not require us to identify and use the sampling density of the point cloud. To attain a more accurate approximation of the expansion coefficients, we adopt a second-order tangent space estimation method to improve the RBF interpolation accuracy in estimating the tangential derivatives. This spectral framework allows us to efficiently solve the PDE many times subjected to different parameters, which reduces the computational cost in the related inverse problem applications. In a well-posed elliptic PDE setting with randomly sampled point cloud data, we provide a theoretical analysis to demonstrate the convergent of the proposed solver as the sample size increases. We also report some numerical studies that show the convergence of the spectral solver on simple manifolds and unknown, rough surfaces. Our numerical results suggest that the proposed method is more accurate than a graph Laplacian-based solver on smooth manifolds. On rough manifolds, these two approaches are comparable. Due to the flexibility of the framework, we empirically found improved accuracies in both smoothed and unsmoothed Stanford bunny domains by blending the graph Laplacian eigensolutions and RBF interpolator.

In this paper, we prove that, under precise spectral assumptions, some finite difference approximations of scalar leftgoing transport equations on the positive half-line with numerical boundary conditions are $\ell^1$-stable but $\ell^q$-unstable for any $q>1$. The proof relies on the accurate description of the Green's function for a particular family of finite rank perturbations of Toeplitz operators whose essential spectrum belongs to the closed unit disk and with a simple eigenvalue of modulus $1$ embedded into the essential spectrum.

This paper extends an a posteriori error estimator for the elastic, Frank-Oseen model of liquid crystals, derived in [9], to include electric and flexoelectric effects. The problem involves a nonlinear coupled system of equations with a local unit-length constraint imposed via a penalty method. The proposed estimator is proven to be a reliable estimate of global approximation error. The performance of the coupled error estimator as a guide for adaptive refinement is shown in the numerical results, where the adapted grids successfully yield substantial reductions in computational work and comparable or better conformance to important physical laws.

The propagation of charged particles through a scattering medium in the presence of a magnetic field can be described by a Fokker-Planck equation with Lorentz force. This model is studied both, from a theoretical and a numerical point of view. A particular trace estimate is derived for the relevant function spaces to clarify the meaning of boundary values. Existence of a weak solution is then proven by the Rothe method. In the second step of our investigations, a fully practicable discretization scheme is proposed based on implicit time-stepping through the energy levels and a spherical-harmonics finite-element discretization with respect to the remaining variables. A full error analysis of the resulting scheme is given, and numerical results are presented to illustrate the theoretical results and the performance of the proposed method.

In this work, we study non-asymptotic bounds on correlation between two time realizations of stable linear systems with isotropic Gaussian noise. Consequently, via sampling from a sub-trajectory and using \emph{Talagrands'} inequality, we show that empirical averages of reward concentrate around steady state (dynamical system mixes to when closed loop system is stable under linear feedback policy ) reward , with high-probability. As opposed to common belief of larger the spectral radius stronger the correlation between samples, \emph{large discrepancy between algebraic and geometric multiplicity of system eigenvalues leads to large invariant subspaces related to system-transition matrix}; once the system enters the large invariant subspace it will travel away from origin for a while before coming close to a unit ball centered at origin where an isotropic Gaussian noise can with high probability allow it to escape the current invariant subspace it resides in, leading to \emph{bottlenecks} between different invariant subspaces that span $\mathbb{R}^{n}$, to be precise : system initiated in a large invariant subspace will be stuck there for a long-time: log-linear in dimension of the invariant subspace and inversely to log of inverse of magnitude of the eigenvalue. In the problem of Ordinary Least Squares estimate of system transition matrix via a single trajectory, this phenomenon is even more evident if spectrum of transition matrix associated to large invariant subspace is explosive and small invariant subspaces correspond to stable eigenvalues. Our analysis provide first interpretable and geometric explanation into intricacies of learning and concentration for random dynamical systems on continuous, high dimensional state space; exposing us to surprises in high dimensions

We consider the measurement model $Y = AX,$ where $X$ and, hence, $Y$ are random variables and $A$ is an a priori known tall matrix. At each time instance, a sample of one of $Y$'s coordinates is available, and the goal is to estimate $\mu := \mathbb{E}[X]$ via these samples. However, the challenge is that a small but unknown subset of $Y$'s coordinates are controlled by adversaries with infinite power: they can return any real number each time they are queried for a sample. For such an adversarial setting, we propose the first asynchronous online algorithm that converges to $\mu$ almost surely. We prove this result using a novel differential inclusion based two-timescale analysis. Two key highlights of our proof include: (a) the use of a novel Lyapunov function for showing that $\mu$ is the unique global attractor for our algorithm's limiting dynamics, and (b) the use of martingale and stopping time theory to show that our algorithm's iterates are almost surely bounded.

In this paper we investigate the stability properties of the so-called gBBKS and GeCo methods, which belong to the class of nonstandard schemes and preserve the positivity as well as all linear invariants of the underlying system of ordinary differential equations for any step size. A stability investigation for these methods, which are outside the class of general linear methods, is challenging since the iterates are always generated by a nonlinear map even for linear problems. Recently, a stability theorem was derived presenting criteria for understanding such schemes. For the analysis, the schemes are applied to general linear equations and proven to be generated by $\mathcal C^1$-maps with locally Lipschitz continuous first derivatives. As a result, the above mentioned stability theorem can be applied to investigate the Lyapunov stability of non-hyperbolic fixed points of the numerical method by analyzing the spectrum of the corresponding Jacobian of the generating map. In addition, if a fixed point is proven to be stable, the theorem guarantees the local convergence of the iterates towards it. In the case of first and second order gBBKS schemes the stability domain coincides with that of the underlying Runge--Kutta method. Furthermore, while the first order GeCo scheme converts steady states to stable fixed points for all step sizes and all linear test problems of finite size, the second order GeCo scheme has a bounded stability region for the considered test problems. Finally, all theoretical predictions from the stability analysis are validated numerically.

In the present paper we consider the initial data, external force, viscosity coefficients, and heat conductivity coefficient as random data for the compressible Navier--Stokes--Fourier system. The Monte Carlo method, which is frequently used for the approximation of statistical moments, is combined with a suitable deterministic discretisation method in physical space and time. Under the assumption that numerical densities and temperatures are bounded in probability, we prove the convergence of random finite volume solutions to a statistical strong solution by applying genuine stochastic compactness arguments. Further, we show the convergence and error estimates for the Monte Carlo estimators of the expectation and deviation. We present several numerical results to illustrate the theoretical results.

Using the equivalent inclusion method (a method strongly related to the Hashin-Shtrikman variational principle) as a surrogate model, we propose a variance reduction strategy for the numerical homogenization of random composites made of inclusions (or rather inhomogeneities) embedded in a homogeneous matrix. The efficiency of this strategy is demonstrated within the framework of two-dimensional, linear conductivity. Significant computational gains vs full-field simulations are obtained even for high contrast values. We also show that our strategy allows to investigate the influence of parameters of the microstructure on the macroscopic response. Our strategy readily extends to three-dimensional problems and to linear elasticity. Attention is paid to the computational cost of the surrogate model. In particular, an inexpensive approximation of the so-called influence tensors (that are used to compute the surrogate model) is proposed.