Related papers: Backward Error Measures for Roots of Polynomials
We propose generalized resubstitution error estimators for regression, a broad family of estimators, each corresponding to a choice of empirical probability measures and loss function. The usual sum of squares criterion is a special case…
We associate to an $N$-sample of a given rotationally invariant probability measure $\mu_0$ with compact support in the complex plane, a polynomial $P_N$ with roots given by the sample. Then, for $t \in (0,1)$, we consider the empirical…
In indirect measurements, the measurand is determined by solving an inverse problem which requires a model of the measurement process. Such models are often approximations and introduce systematic errors leading to a bias of the posterior…
We provide a robust and general algorithm for computing distribution functions associated to induced orthogonal polynomial measures. We leverage several tools for orthogonal polynomials to provide a spectrally-accurate method for a broad…
We propose algorithms for addressing the bias of the posterior mean when used as an estimator of parameters. These algorithms build upon the recently proposed Bayesian infinitesimal jackknife approximation (Giordano and Broderick (2023))…
The Inverse Problem for the estimation of a point-wise approximation error occurring at the discretization and solving of the system of partial differential equations is addressed. The set of the differences between the numerical solutions…
We establish a generalization of Anush Tserunyan and Jenna Zomback's 2024 Backward Ergodic Theorem. We remove the countable-to-one assumption and thus provide a backward ergodic theorem for arbitrary measure-preserving transformations.…
We introduce and study a variant of the Wasserstein distance on the space of probability measures, specially designed to deal with measures whose support has a dendritic, or treelike structure with a particular direction of orientation. Our…
In the reduced basis method, the evaluation of the a posteriori estimator can become very sensitive to round-off errors. In this note, the origin of the loss of accuracy is revealed, and a solution to this problem is proposed and…
The slope of the best fit line from minimizing the sum of the squared oblique errors is the root of a polynomial of degree four. This geometric view of measurement errors is used to give insight into the performance of various slope…
We study the extent of independence needed to approximate the product of bounded random variables in expectation, a natural question that has applications in pseudorandomness and min-wise independent hashing. For random variables whose…
We show in this paper that the roots $x_1$ and $x_2$ of a scalar quadratic polynomial $ax^2+bx+c=0$ with real or complex coefficients $a$, $b$ $c$ can be computed in a element-wise mixed stable manner, measured in a relative sense. We also…
The a posteriori error estimates are studied for a class of nonlinear stead-state Poisson-Nernst-Planck equations, which are a coupled system consisting of the Nernst-Planck equation and the Poisson equation. Both the global upper bounds…
Motivated by problems in contact mechanics, we propose a duality approach for computing approximations and associated a posteriori error bounds to solutions of variational inequalities of the first kind. The proposed approach improves upon…
We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the K-theory of Grassmannians share many common properties. Our Grothendieck measures are analogs…
In this paper we show how to find the exact error (not just an estimate of the error) of a conforming mixed approximation by using the functional type a posteriori error estimates in the spirit of Repin. The error is measured in a mixed…
We study convergence rates of variational posterior distributions for nonparametric and high-dimensional inference. We formulate general conditions on prior, likelihood, and variational class that characterize the convergence rates. Under…
Motivated by numerical methods for solving parametric partial differential equations, this paper studies the approximation of multivariate analytic functions by algebraic polynomials. We introduce various anisotropic model classes based on…
We present reduced basis approximations and rigorous a posteriori error bounds for the instationary Stokes equations. We shall discuss both a method based on the standard formulation as well as a method based on a penalty approach, which…
We present a new method to calculate analytically the roots of the general complex polynomial of degree three. Thismethod is based on the approach of appropriated changes of variable involving an arbitrary parameter. The advantageof this…