Numerical Algorithms

Let p, n ∈ ℕ with 2p ≥ n + 2, and let I a be a polyharmonic spline of order p on the grid ℤ × aℤ n which satisfies the interpolating conditions $I_{a}\left( j,am\right) =d_{j}\left( am\right) $ for j ∈ ℤ, m ∈ ℤ n where the functions d j : ℝ n → ℝ and the parameter a > 0 are given. Let $B_{s}\left( \mathbb{R}^{n}\right) $ be the set of all integrable functions f : ℝ n → ℂ such that the integral $$ \left\| f\right\| _{s}:=\int_{\mathbb{R}^{n}}\left| \widehat{f}\left( \xi\right) \right| \left( 1+\left| \xi\right| ^{s}\right) d\xi $$ is finite. The main result states that for given $\mathbb{\sigma}\geq0$ there exists a constant c>0 such that whenever $d_{j}\in B_{2p}\left( \mathbb{R}^{n}\right) \cap C\left( \mathbb{R}^{n}\right) ,$ j ∈ ℤ, satisfy $\left\| d_{j}\right\| _{2p}\leq D\cdot\left( 1+\left| j\right| ^{\mathbb{\sigma}}\right) $ for all j ∈ ℤ there exists a polyspline S : ℝ n+1 → ℂ of order p on strips such that $$ \left| S\left( t,y\right) -I_{a}\left( t,y\right) \right| \leq a^{2p-1}c\cdot D\cdot\left( 1+\left| t\right| ^{\mathbb{\sigma}}\right) $$ for all y ∈ ℝ n , t ∈ ℝ and all 0 < a ≤ 1.

This paper introduces a new class of weak second-order explicit stabilized stochastic Runge-Kutta methods for stiff Itô stochastic differential equations. The convergence and mean-square stability properties of our new methods are analyzed. The numerical results of two examples are presented to confirm our theoretical results.

In this present paper, we propose a modified form of generalized system of variational inequalities and introduce an iterative scheme for finding a common element of the set of fixed points of nonexpansive mapping and the solution set of the proposed problem in the framework of real Hilbert spaces. We prove a strong convergence theorem of the proposed iterative scheme. Applying our main result, we prove strong convergence theorems of the standard constrained convex optimization problem and the split feasibility problem. In support of our main result, a numerical example is also presented.

An optimization-based tool for flow predictions in natural rivers is introduced assuming that some physical characteristics of a river within a spatial-time domain $$[x_{\min }, x_{\max }] \times [t_{\min }, t_{\textrm{today}}]$$ are known. In particular, it is assumed that the bed elevation and width of the river are known at a finite number of stations in $$[x_{\min }, x_{\max }]$$ and that the flow-rate at $$x=x_{\min }$$ is known for a finite number of time instants in $$[t_{\min },t_{\textrm{today}}]$$ . Using these data, given $$t_{\textrm{future}} > t_{\textrm{today}}$$ and a forecast of the flow-rate at $$x=x_{\min }$$ and $$t=t_{\textrm{future}}$$ , a regression-based algorithm informed by partial differential equations produces predictions for all state variables (water elevation, depth, transversal wetted area, and flow-rate) for all $$x \in [x_{\min }, x_{\max }]$$ and $$t=t_{\textrm{future}}$$ . The algorithm proceeds by solving a constrained optimization problem that takes into account the available data and the fulfillment of Saint-Venant equations for one-dimensional channels. The effectiveness of this approach is corroborated with flow predictions of a natural river.

The zero set of one general multivariate exponential polynomial with interval coefficients is enclosed by unions and intersections of closed half-spaces. Tighter enclosures are derived in the bivariate case. Common zeros of polynomial systems can be located by an appropriate intersection of these enclosure sets in an appropriate space. The resulting domains are directly brought into polynomial equation solvers.

In this paper, we propose a new branch-and-bound algorithm for the general quadratic problems with box constraints. We, first, transform the problem into a separable form by D. C. decomposition and Cholesky factorization of a positive definite matrix. Then a lower bounding technique is derived and a branch-and-bound algorithm is presented based on the lower bounding and rectangular bisection. Finally, preliminary computational results are reported.

A system identification based on physical laws often involves a parameter estimation. Before performing an estimation problem, it is necessary to investigate its identifiability. This investigation leads often to painful calculations. Generally, the numerical computation of the parameters does not use these calculus. In this contribution we propose least-squares methods to link identifiability approaches with numerical parameter estimation.

We consider two Krylov subspace methods for solving linear systems, which are the minimal residual method and the orthogonal residual method. These two methods are studied without referring to any particular implementations. By using the Petrov–Galerkin condition, we describe the residual norms of these two methods in terms of Krylov vectors, and the relationship between there two norms. We define the Ritz singular values, and prove that the convergence of these two methods is governed by the convergence of the Ritz singular values.