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An incremental approach for computation of convex hull for data points in two-dimensions is presented. The algorithm is not output-sensitive and costs a time that is linear in the size of data points at input. Graham's scan is applied only on a subset of the data points, represented at the extremal of the dataset. Points are classified for extremal, in proportion with the modular distance, about an imaginary point interior to the region bounded by convex hull of the dataset assumed for origin or center in polar coordinate. A subset of the data is arrived by terminating at until an event of no change in maximal points is observed per bin, for iteratively and exponentially decreasing intervals.

We introduce a prior for the parameters of univariate continuous distributions, based on the Wasserstein information matrix, which is invariant under reparameterisations. We briefly discuss the links between the proposed prior with information geometry. We present several examples where we can either obtain this prior in closed-form, or propose a numerically tractable approximation for cases where the prior is not available in closed-form. Since this prior is improper in some cases, we present sufficient conditions for the propriety of the posterior distribution for general classes of models.

In this paper we propose a deep learning based numerical scheme for strongly coupled FBSDE, stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the means square error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDE, fails for a simple linear-quadratic control problem, and motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.

Proving linear inequalities and identities of Shannon's information measures, possibly with linear constraints on the information measures, is an important problem in information theory. For this purpose, ITIP and other variant algorithms have been developed and implemented, which are all based on solving a linear program (LP). In particular, an identity $f = 0$ is verified by solving two LPs, one for $f \ge 0$ and one for $f \le 0$. In this paper, we develop a set of algorithms that can be implemented by symbolic computation. Based on these algorithms, procedures for verifying linear information inequalities and identities are devised. Compared with LP-based algorithms, our procedures can produce analytical proofs that are both human-verifiable and free of numerical errors. Our procedures are also more efficient computationally. For constrained inequalities, by taking advantage of the algebraic structure of the problem, the size of the LP that needs to be solved can be significantly reduced. For identities, instead of solving two LPs, the identity can be verified directly with very little computation.

In the present paper, we study the analyticity of the leftmost eigenvalue of the linear elliptic partial differential operator with random coefficient and analyze the convergence rate of the quasi-Monte Carlo method for approximation of the expectation of this quantity. The random coefficient is assumed to be represented by an affine expansion $a_0(\boldsymbol{x})+\sum_{j\in \mathbb{N}}y_ja_j(\boldsymbol{x})$, where elements of the parameter vector $\boldsymbol{y}=(y_j)_{j\in \mathbb{N}}\in U^\infty$ are independent and identically uniformly distributed on $U:=[-\frac{1}{2},\frac{1}{2}]$. Under the assumption $ \|\sum_{j\in \mathbb{N}}\rho_j|a_j|\|_{L_\infty(D)} <\infty$ with some positive sequence $(\rho_j)_{j\in \mathbb{N}}\in \ell_p(\mathbb{N})$ for $p\in (0,1]$ we show that for any $\boldsymbol{y}\in U^\infty$, the elliptic partial differential operator has a countably infinite number of eigenvalues $(\lambda_j(\boldsymbol{y}))_{j\in \mathbb{N}}$ which can be ordered non-decreasingly. Moreover, the spectral gap $\lambda_2(\boldsymbol{y})-\lambda_1(\boldsymbol{y})$ is uniformly positive in $U^\infty$. From this, we prove the holomorphic extension property of $\lambda_1(\boldsymbol{y})$ to a complex domain in $\mathbb{C}^\infty$ and estimate mixed derivatives of $\lambda_1(\boldsymbol{y})$ with respect to the parameters $\boldsymbol{y}$ by using Cauchy's formula for analytic functions. Based on these bounds we prove the dimension-independent convergence rate of the quasi-Monte Carlo method to approximate the expectation of $\lambda_1(\boldsymbol{y})$. In this case, the computational cost of fast component-by-component algorithm for generating quasi-Monte Carlo $N$-points scales linearly in terms of integration dimension.

In this article, we deal with the efficient computation of the Wright function in the cases of interest for the expression of solutions of some fractional differential equations. The proposed algorithm is based on the inversion of the Laplace transform of a particular expression of the Wright function for which we discuss in detail the error analysis. We also present a code package that implements the algorithm proposed here in different programming languages. The analysis and implementation are accompanied by an extensive set of numerical experiments that validate both the theoretical estimates of the error and the applicability of the proposed method for representing the solutions of fractional differential equations.

Instrumental variable models allow us to identify a causal function between covariates X and a response Y, even in the presence of unobserved confounding. Most of the existing estimators assume that the error term in the response Y and the hidden confounders are uncorrelated with the instruments Z. This is often motivated by a graphical separation, an argument that also justifies independence. Posing an independence condition, however, leads to strictly stronger identifiability results. We connect to existing literature in econometrics and provide a practical method for exploiting independence that can be combined with any gradient-based learning procedure. We see that even in identifiable settings, taking into account higher moments may yield better finite sample results. Furthermore, we exploit the independence for distribution generalization. We prove that the proposed estimator is invariant to distributional shifts on the instruments and worst-case optimal whenever these shifts are sufficiently strong. These results hold even in the under-identified case where the instruments are not sufficiently rich to identify the causal function.

We show that the probability of the exceptional set decays exponentially for a broad class of randomized algorithms approximating solutions of ODEs, admitting a certain error decomposition. This class includes randomized explicit and implicit Euler schemes, and the randomized two-stage Runge-Kutta scheme (under inexact information). We design a confidence interval for the exact solution of an IVP and perform numerical experiments to illustrate the theoretical results.

We discuss two approaches for the formulation and implementation of space-time discontinuous Galerkin spectral element methods (DG-SEM). In one, time is treated as an additional coordinate direction and a Galerkin procedure is applied to the entire problem. In the other, the method of lines is used with DG-SEM in space and the fully implicit Runge-Kutta method Lobatto IIIC in time. The two approaches are mathematically equivalent in the sense that they lead to the same discrete solution. However, in practice they differ in several important respects, including the terminology used to describe them, the structure of the resulting software, and the interaction with nonlinear solvers. Challenges and merits of the two approaches are discussed with the goal of providing the practitioner with sufficient consideration to choose which path to follow. Additionally, implementations of the two methods are provided as a starting point for further development. Numerical experiments validate the theoretical accuracy of these codes and demonstrate their utility, even for 4D problems.

A general class of KdV-type wave equations regularized with a convolution-type nonlocality in space is considered. The class differs from the class of the nonlinear nonlocal unidirectional wave equations previously studied by the addition of a linear convolution term involving third-order derivative. To solve the Cauchy problem we propose a semi-discrete numerical method based on a uniform spatial discretization, that is an extension of a previously published work of the present authors. We prove uniform convergence of the numerical method as the mesh size goes to zero. We also prove that the localization error resulting from localization to a finite domain is significantly less than a given threshold if the finite domain is large enough. To illustrate the theoretical results, some numerical experiments are carried out for the Rosenau-KdV equation, the Rosenau-BBM-KdV equation and a convolution-type integro-differential equation. The experiments conducted for three particular choices of the kernel function confirm the error estimates that we provide.