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We develop an a posteriori error analysis for a novel quantity of interest (QoI) evolutionary partial differential equations (PDEs). Specifically, the QoI is the first time at which a functional of the solution to the PDE achieves a threshold value signifying a particular event, and differs from classical QoIs which are modeled as bounded linear functionals. We use Taylor's theorem and adjoint based analysis to derive computable and accurate error estimates for linear parabolic and hyperbolic PDEs. Specifically, the heat equation and linearized shallow water equations (SWE) are used for the parabolic and hyperbolic cases, respectively. Numerical examples illustrate the accuracy of the error estimates.

A quasi-second order scheme is developed to obtain approximate solutions of the shallow water equationswith bathymetry. The scheme is based on a staggered finite volume scheme for the space discretization:the scalar unknowns are located in the discretisation cells while the vector unknowns are located on theedges (in 2D) or faces (in 3D) of the mesh. A MUSCL-like interpolation for the discrete convectionoperators in the water height and momentum equations is performed in order to improve the precisionof the scheme. The time discretization is performed either by a first order segregated forward Eulerscheme in time or by the second order Heun scheme. Both schemes are shown to preserve the waterheight positivity under a CFL condition and an important state equilibrium known as the lake at rest.Using some recent Lax-Wendroff type results for staggered grids, these schemes are shown to be Lax-consistent with the weak formulation of the continuous equations; besides, the forward Euler schemeis shown to be consistent with a weak entropy inequality. Numerical results confirm the efficiency andaccuracy of the schemes.

The goal of this paper is to construct ergodic estimators for the parameters in the double exponential Ornstein-Uhlenbeck process, observed at discrete time instants with time step size h. The existence and uniqueness, the strong consistency, and the asymptotic normality of the estimators are obtained for arbitrarily fixed time step size h. A simulation method of the double exponential Ornstein-Uhlenbeck process is proposed and some numerical simulations are performed to demonstrate the effectiveness of the proposed estimators.

We propose a new data-driven approach for learning the fundamental solutions (i.e. Green's functions) of various linear partial differential equations (PDEs) given sample pairs of input-output functions. Building off the theory of functional linear regression (FLR), we estimate the best-fit Green's function and bias term of the fundamental solution in a reproducing kernel Hilbert space (RKHS) which allows us to regularize their smoothness and impose various structural constraints. We use a general representer theorem for operator RKHSs to approximate the original infinite-dimensional regression problem by a finite-dimensional one, reducing the search space to a parametric class of Green's functions. In order to study the prediction error of our Green's function estimator, we extend prior results on FLR with scalar outputs to the case with functional outputs. Furthermore, our rates of convergence hold even in the misspecified setting when the data is generated by a nonlinear PDE under certain constraints. Finally, we demonstrate applications of our method to several linear PDEs including the Poisson, Helmholtz, Schr\"{o}dinger, Fokker-Planck, and heat equation and highlight its ability to extrapolate to more finely sampled meshes without any additional training.

Random forests remain among the most popular off-the-shelf supervised learning algorithms. Despite their well-documented empirical success, however, until recently, few theoretical results were available to describe their performance and behavior. In this work we push beyond recent work on consistency and asymptotic normality by establishing rates of convergence for random forests and other supervised learning ensembles. We develop the notion of generalized U-statistics and show that within this framework, random forest predictions can potentially remain asymptotically normal for larger subsample sizes than previously established. We also provide Berry-Esseen bounds in order to quantify the rate at which this convergence occurs, making explicit the roles of the subsample size and the number of trees in determining the distribution of random forest predictions.

Efficient and robust iterative solvers for strong anisotropic elliptic equations are very challenging. In this paper a block preconditioning method is introduced to solve the linear algebraic systems of a class of micro-macro asymptotic-preserving (MMAP) scheme. MMAP method was developed by Degond {\it et al.} in 2012 where the discrete matrix has a $2\times2$ block structure. By the approximate Schur complement a series of block preconditioners are constructed. We first analyze a natural approximate Schur complement that is the coefficient matrix of the original non-AP discretization. However it tends to be singular for very small anisotropic parameters. We then improve it by using more suitable approximation for boundary rows of the exact Schur complement. With these block preconditioners, preconditioned GMRES iterative method is developed to solve the discrete equations. Several numerical tests show that block preconditioning methods can be a robust strategy with respect to grid refinement and the anisotropic strengths.

This work deals with a number of questions relative to the discrete and continuous adjoint fields associated with the compressible Euler equations and classical aerodynamic functions. The consistency of the discrete adjoint equations with the corresponding continuous adjoint partial differential equation is one of them. It is has been established or at least discussed only for a handful of numerical schemes and a contribution of this article is to give the adjoint consistency conditions for the 2D Jameson-Schmidt-Turkel scheme in cell-centred finite-volume formulation. The consistency issue is also studied here from a new heuristic point of view by discretizing the continuous adjoint equation for the discrete flow and adjoint fields. Both points of view prove to provide useful information. Besides, it has been often noted that discrete or continuous inviscid lift and drag adjoint exhibit numerical divergence close to the wall and stagnation streamline for a wide range of subsonic and transonic flow conditions. This is analyzed here using the physical source term perturbation method introduced in reference [Giles and Pierce, AIAA Paper 97-1850, 1997]. With this point of view, the fourth physical source term of appears to be the only one responsible for this behavior. It is also demonstrated that the numerical divergence of the adjoint variables corresponds to the response of the flow to the convected increment of stagnation pressure and diminution of entropy created at the source and the resulting change in lift and drag.

Regula Falsi, or the method of false position, is a numerical method for finding an approximate solution to f(x) = 0 on a finite interval [a, b], where f is a real-valued continuous function on [a, b] and satisfies f(a)f(b) < 0. Previous studies proved the convergence of this method under certain assumptions about the function f, such as both the first and second derivatives of f do not change the sign on the interval [a, b]. In this paper, we remove those assumptions and prove the convergence of the method for all continuous functions.

The aim of this study is the weak convergence rate of a temporal and spatial discretization scheme for stochastic Cahn-Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler scheme is used in time. The presence of the unbounded operator in front of the nonlinear term and the lack of the associated Kolmogorov equations make the error analysis much more challenging and demanding. To overcome these difficulties, we further exploit a novel approach proposed in [7] and combine it with Malliavin calculus to obtain an improved weak rate of convergence, in comparison with the corresponding strong convergence rates. The techniques used here are quite general and hence have the potential to be applied to other non-Markovian equations. As a byproduct the rate of the strong error can also be easily obtained.

A computationally efficient high-order solver is developed to compute the wall distances, which are typically used for turbulence modelling, peripheral flow simulations, Computer Aided Design (CAD) etc. The wall distances are computed by solving the differential equations namely: Eikonal, Hamilton-Jacobi (H-J) and Poisson. The computational benefit of using high-order schemes (explicit/compact schemes) for wall-distance solvers, both in terms of accuracy and computational time, has been demonstrated. A new H-J formulation based on the localized artificial diffusivity (LAD) approach has been proposed, which has produced results with an accuracy comparable to that of the Eikonal formulation. When compared to the baseline H-J solver using upwind schemes, the solution accuracy has improved by an order of magnitude and the calculations are $\approx$ 5 times faster using the modified H-J formulation. A modified curvature correction has also been implemented into the H-J solver to account for the near-wall errors due to concave/convex wall curvatures. The performance of the solver using different schemes has been tested both on the steady canonical test cases and the unsteady test cases like `piston-cylinder arrangement', `bouncing cube' and `burning of a star grain propellant' where the wall-distance evolves with time.