Bivariate Hermite Interpolation Research Articles

Bivariate Hermite interpolation by a limiting case of the cross approximation algorithm

Computing low-rank approximations of a given function is a key step for implementing efficiently numerous algorithms in various fields, including the discretisation of non-local integral operators and Isogeometric Analysis. The adaptive cross approximation (ACA) algorithm is an efficient method requiring few computational resources introduced by Bebendorf. We introduce in the present paper the new paradigm of approximating the given function by a piecewise low-rank function with C1-regularity. The proposed approximation is based on the ACA algorithm and our main contribution is the extension of the interpolation property characterising this algorithm to Hermite interpolation. Therefore, we introduce a new method for low-rank Hermite interpolation using a limiting case of the ACA algorithm. The proposed method has full approximation order. We then propose a piecewise low-rank approximation with adaptive refinement using either the ACA algorithm or our new method to compute each piece. We finally compare the results obtained for the two methods.

Bivariate Hermite interpolation on the exponential curve

We study Hermite interpolation problems on the exponential curve $$y=e^x$$ in $$\mathbb {R}^2$$ . We construct some kind of regular Hermite interpolation schemes and investigate continuity properties of interpolation polynomials with respect to interpolation conditions.

On bivariate Hermite interpolation and the limit of certain bivariate Lagrange projectors

On bivariate Hermite interpolation and the limit of certain bivariate Lagrange projectors

A new formula for bivariate Hermite interpolation on variable step grids and its application to image interpolation.

In this paper, we present a novel formula of the bivariate Hermite interpolating (BHI) polynomial in the case of support points arranged on a grid with variable step. This expression is applicable when interpolation of a bivariate function is required, given its value and the values of its partial derivatives of arbitrarily high order, at the support points. The proposed formula is a generalization of an existing formula for the bivariate Hermite polynomial. It is also algebraically much simpler, thus can be computed more efficiently. In order to apply Hermite interpolation to image interpolation, we simplify the proposed (BHI) to handle support points on a regular unit-step grid. The values of image partial derivatives are arithmetically approximated using compact finite differences. The proposed method is being assessed in a number of image interpolation experiments that include a synthetic image, for which the values of the partial derivatives are computed analytically, as well as a collection of images from different medical modalities. The proposed BHI with up to second-order image partial derivatives, outperforms the convolution-based interpolation methods, as well as generalized interpolation methods with the same number of support points that was compared with, in the majority of image interpolation experiments. The computational load of the proposed BHI is calculated and its behaviour with respect to its controlling parameters is investigated.

Taylorian points of an algebraic curve and bivariate Hermite interpolation

We introduce and study the notion of Taylorian points of algebraic curves in C 2 , which enables us to define intrinsic Taylor interpolation polyno- mials on curves. These polynomials in turn lead to the construction of a well- behaved Hermitian scheme on curves, of which we give several examples. We show that such Hermitian schemes can be collected to obtain Hermitian bivariate polynomial interpolation schemes.

Characteristic 2 approach to bivariate interpolation problems

We investigate bivariate Hermite interpolation problems in characteristic 2. Given a nonnegative integer t, we describe all the sub-linear systems generated by monomials, in which there is no curve passing through a general point with multiplicity at least 2t. As an application, we show that a certain linear system of plane curves with ten base points is non-special.

Case study in bivariate Hermite interpolation

In this article we investigate the minimal dimension of a subspace of C 1 ( R 2 ) needed to interpolate an arbitrary function and some of its prescribed partial derivatives at two arbitrary points. The subspace in question may depend on the derivatives, but not on the location of the points. Several results of this type are known for Lagrange interpolation. As far as I know, this is the first such study for Hermite Interpolation.

On the poisedness of Bojanov–Xu interpolation

In this paper, we consider the bivariate Hermite interpolation introduced by Bojanov and Xu [SIAM J. Numer. Anal. 39(5) (2002) 1780–1793]. The nodes of the interpolation with Π 2 k - δ , where δ = 0 or 1 , are the intersection points of 2 k + 1 distinct rays from the origin with a multiset of k + 1 - δ concentric circles. Parameters are the values and successive radial derivatives, whenever the corresponding circle is multiple. The poisedness of this interpolation was proved only for the set of equidistant rays [Bojanov and Xu, 2002] and its counterparts with other conic sections [Hakopian and Ismail, East J. Approx. 9 (2003) 251–267]. We show that the poisedness of this ( k + 1 - δ ) ( 2 k + 1 ) dimensional Hermite interpolation problem is equivalent to the poisedness of certain 2 k + 1 dimensional Lagrange interpolation problems. Then the poisedness of Bojanov–Xu interpolation for a wide family of sets of rays satisfying some simple conditions is established. Our results hold also with above circles replaced by ellipses, hyperbolas, and pairs of parallel lines. Next a conjecture [Hakopian and Ismail, J. Approx. Theory 116 (2002) 76–99] concerning a poisedness relation between the Bojanov–Xu interpolation, with set of rays symmetric about x -axis, and certain univariate lacunary interpolations is established. At the end the poisedness for a wide class of lacunary interpolations is obtained.

On the Regularity of Multivariate Hermite Interpolation

In this paper a result due to Gevorgian, Sahakian, and the author concerning the regularity of bivariate Hermite interpolation is generalized in two directions: in the bivariate case and for arbitrary dimensions. Also a notion of independence (preregularity) of interpolation conditions is discussed and a relation on the independence on different dimensions is indicated. As corollaries combinatorial inequalities are obtained. At the end a pair of related number inequalities is presented.

On Bivariate Hermite Interpolation with Minimal Degree Polynomials

A Newton-type approach is used to deal with bivariate polynomial Hermite interpolation problems when the data are distributed in the intersections of two families of straight lines, as a generalization of regular grids. The interpolation operator is degree-reducing and the interpolation space is a minimal degree space. Integral remainder formulas are given for the Lagrange case, then extended to the Hermite case, and finally used to obtain error estimates.

Bivariate Hermite Interpolation and Numerical Curves

In this paper, Hermite interpolation by bivariate algebraic polynomials of total degree ⩽nis considered. The interpolation parameters are the values of a function and its partial derivatives up to some ordernν−1 at the nodeszν=(xν, yν),ν=1, …, s, wherenνis the multiplicity ofzν. The sequence N={n1, …, ns; n} of multiplicities associated with the degree of interpolating polynomials is investigated. Some results of the paper were announced in [GHS93].

Bivariate hermite interpolation and applications to Algebraic Geometry

We try to solve the bivariate interpolation problem (1.3) for polynomials (1.1), whereS is a lower set of lattice points, and for theq-th interpolation knot,A q is the set of orders of derivatives that appear in (1.3). The number of coefficients |S| is equal to the number of equations Σ|A q |. If this is possible for all knots in general position, the problem is almost always solvable (=a.a.s.). We seek to determine whether (1.3) is a.a.s. An algorithm is given which often gives a positive answer to this. It can be applied to the solution of a problem of Hirschowitz in Algebraic Geometry. We prove that for Hermite conditions (1.3) (when allA q are lower triangles of orderp) andP is of total degreen, (1.3) is a.a.s. for allp=1, 2, 3 and alln, except for the two casesp=1,n=2 andp=1,n=4.