Academic literature on the topic 'Data approximation'

We consider linear Schrödinger equation perturbed by delta distribution with singular potential and the initial data. Due to the singularities appearing in the equation, we introduce two kinds of approximations: the parameter's approximation for potential and the initial data given by mollifiers of different growth and the approximation for the Green function for Schrödinger equation with regularized derivatives. These approximations reduce the perturbed Schrödinger equation to the family of singular integral equations. We prove the existence-uniqueness theorems in Colombeau space [Formula: see text], 1 ≤ p,q ≤ ∞, employing novel stability estimates (w.r.) to singular perturbations for ε → 0, which imply the statements in the framework of Colombeau generalized functions. In particular, we prove the existence-uniqueness result in [Formula: see text] and [Formula: see text] algebra of Colombeau.

A dynamic geometric data stream is a sequence of m ADD/REMOVE operations of points from a discrete geometric space {1,…, Δ} d ?. ADD (p) inserts a point p from {1,…, Δ} d into the current point set P , REMOVE(p) deletes p from P . We develop low-storage data structures to (i) maintain ε-nets and ε-approximations of range spaces of P with small VC-dimension and (ii) maintain a (1 + ε)-approximation of the weight of the Euclidean minimum spanning tree of P . Our data structure for ε-nets uses [Formula: see text] bits of memory and returns with probability 1 – δ a set of [Formula: see text] points that is an e-net for an arbitrary fixed finite range space with VC-dimension [Formula: see text]. Our data structure for ε-approximations uses [Formula: see text] bits of memory and returns with probability 1 – δ a set of [Formula: see text] points that is an ε-approximation for an arbitrary fixed finite range space with VC-dimension [Formula: see text]. The data structure for the approximation of the weight of a Euclidean minimum spanning tree uses O ( log (1/δ)( log Δ/ε) O ( d )) space and is correct with probability at least 1 – δ. Our results are based on a new data structure that maintains a set of elements chosen (almost) uniformly at random from P .

We develop a separable-kernel decomposition method for approximating the double-square-root (DSR) continuation operator in one-way migrations in this paper. This new approach is a further development of separable approximations of the single-square-root (SSR) operator. The separable approximation of the DSR operator generally involves solving a complicated nonlinear system of integral equations. Instead of solving this nonlinear system directly, our new method consists of repeatedly applying the separable-kernel technique developed for the two-variable SSR operator to the multivariable DSR operator. Numerical experiments demonstrate the efficiency of the proposed method. We illustrate the fast convergence of the obtained separable approximation. We also demonstrate the capability of this novel approximation for imaging an area with geologic complexities through synthetic data.

The paper obtains the exact distribution of Bookstein's shape variables under his plausible model for landmark data. We consider its properties including invariances, marginal distributions and the relationship with Kendall's uniform measure. Particular cases for triangles and quadrilaterals are highlighted. A normal approximation to the distribution is obtained, extending Bookstein's result for three landmarks. The adequacy of these approximations is also studied.