The paper is devoted to the construction of an optimal interpolation formula in $K_2(P_2)$ Hilbert space. Here the interpolation formula consists of a linear combination $\sum_{β=0}^NC_β(z)φ(x_β)$ of given values of a function $φ$ from the space $K_2(P_2)$. The difference between functions and the interpolation formula is considered as a linear functional called the error functional. The error of the interpolation formula is estimated by the norm of the error functional. We obtain the optimal interpolation formula by minimizing the norm of the error functional by coefficients $C_β(z)$ of the interpolation formula. The obtained optimal interpolation formula is exact for trigonometric functions $\sinωx$ and $\cosωx$. At the end of the paper, we give some numerical results which confirm our theoretical results.
The paper is concerned with classic kernel interpolation methods, in addition to approximation methods that are augmented by gradient measurements. To apply kernel interpolation using radial basis functions (RBFs) in a stable way, we propose a type of quasi-optimal interpolation points, searching from a large set of \textit{candidate} points, using a procedure similar to designing Fekete points or power function maximizing points that use pivot from a Cholesky decomposition. The proposed quasi-optimal points result in a smaller condition number, and thus mitigates the instability of the interp…
The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the Hausdorff distance and have not been studied yet in the literature. Parametric polynomial curves of low degree are used and a geometric continuity is prescribed at the boundary points of the circular arc. A general theory about the existence and the uniqueness of the optimal approximant is presented and a rigorous analysis is done for some special cases for which the degree of the polynomial curve and the order of the geometric smoothness differ by two. This in…
Optimal estimation for the Fujino-Morley interpolation error constants
Graduate School of Science and Technology, Niigata University（新潟大学科学技术研究科）；Department of Applied Mathematics, National Cheng Kung University（国立成功大学应用数学系）；Department of Mathematics, National Cheng Kung University（国立成功大学数学系）
The aim of this paper is to derive a refined first-order expansion formula in Rn, the goal being to get an optimal reduced remainder, compared to the one obtained by usual Taylor's formula. For a given function, the formula we derived is obtained by introducing a linear combination of the first derivatives, computed at $n+1$ equally spaced points. We show how this formula can be applied to two important applications: the interpolation error and the finite elements error estimates. In both cases, we illustrate under which conditions a significant improvement of the errors can be obtained, namel…
We introduce and discuss shape-based models for finding the best interpolation data in the compression of images with noise. The aim is to reconstruct missing regions by means of minimizing a data fitting term in the $L^2$-norm between the images and their reconstructed counterparts using time-dependent PDE inpainting. We analyze the proposed models in the framework of the $Γ$-convergence from two different points of view. First, we consider a continuous stationary PDE model, obtained by focusing on the first iteration of the discretized time-dependent PDE, and get pointwise information on the…
We prove stability bounds for Stokes-like virtual element spaces in two and three dimensions. Such bounds are also instrumental in deriving optimal interpolation estimates. Furthermore, we develop some numerical tests in order to investigate the behaviour of the stability constants also from the practical side.
A genetic algorithm procedure is demonstrated that refines the selection of interpolation points of the discrete empirical interpolation method (DEIM) when used for constructing reduced order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs) with proper orthogonal decomposition. The method achieves nearly optimal interpolation points with only a few generations of the search, making it potentially useful for {\em online} refinement of the sparse sampling used to construct a projection of the nonlinear terms. With the genetic algorithm, points are opt…
A Multi-Fidelity Parametric Framework for Reduced-Order Modeling using Optimal Transport-based Interpolation: Applications to Diffused-Interface Two-Phase Flows
$\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems by multivariate rational interpolation
Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）；Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；Virginia Tech（弗吉尼亚理工大学）
