Vector-valued Approximation Research Articles

Error estimates of divergence-free generalized moving least squares (Div-Free GMLS) derivatives approximations in Sobolev spaces

The divergence-free generalized moving least squares (Div-Free GMLS) approximation has recently been utilized to solve some incompressible fluid flows problems. In our recent work (Mohammadi and Dehghan (2021) [28]), we have presented its formulation more precisely, and also the error estimates of derivatives have been carried out in L∞(Ω), where Ω⊂Rd is a bounded set satisfying an interior cone condition. However, the error estimates of this vector-valued approximation in Sobolev spaces are not done. So, in this paper we make the error estimates of Div-Free GMLS derivatives approximations in Lq(Ω), where 1≤q≤∞, using a stable local divergence-free polynomial reproduction property. Note that, the method is a direct approximants of exact derivatives of a divergence-free vector field, which possesses the optimal rates of convergence. This vector-valued technique can also be developed to find the numerical solution of the incompressible fluid flows problems easier than the other available mesh-dependent methods. Finally, we have shown how the proposed approximation can recover the velocity field variable of the well-known Darcy's problem in a two-dimensional space.

Existence results and optimization over the set of efficient solutions in vector-valued approximation theory

Existence results and optimization over the set of efficient solutions in vector-valued approximation theory

Approximating the derivative of manifold-valued functions

We consider the approximation of manifold-valued functions by embedding the manifold into a higher dimensional space, applying a vector-valued approximation operator and projecting the resulting vector back to the manifold. It is well known that the approximation error for manifold-valued functions is close to the approximation error for vector-valued functions. This is not true anymore if we consider the derivatives of such functions. In our paper we give pre-asymptotic error bounds for the approximation of the derivative of manifold-valued function. In particular, we provide explicit constants that depend on the reach of the embedded manifold.

A reconstructed discontinuous approximation on unfitted meshes to [formula omitted] and [formula omitted] interface problems

We propose a high-order discontinuous Galerkin finite element method for solving the H(div)- and H(curl)-elliptic interface problems on unfitted meshes. The vector-valued approximation space is constructed by the patch reconstruction with at most d degrees of freedom per element. The C2-smooth interface is allowed to intersect elements in a quite general setup. The patch reconstruction provides the stability near the interface naturally without any additional stabilization strategy. The method is based on the symmetric interior penalty method and the optimal and suboptimal convergence rates under the energy norm and L2 norm are derived. Numerical examples in both two and three dimensions are presented to demonstrate the accuracy of the method.

A divergence-free generalized moving least squares approximation with its application

An approximation based on moving least squares for vector-valued functions satisfying the divergence-free property was employed in [67] by Trask, Maxey, and Hu. They changed the traditional polynomial bases in moving least squares approximation to the vector-valued polynomial basis functions satisfying the divergence-free property for constructing a new vector-valued approximation. In this paper, we adopt [67], but we consider another approach, i.e., a direct method based on the generalized moving least squares (GMLS) approximation for vector-valued functions, which satisfy the divergence-free property. This GMLS approximation produces diffuse or uncertain derivatives for each component of the proposed vector-valued approximation satisfying the divergence-free property. Using the new approach, the linear functionals such as the derivatives act only on each row of the vector-valued polynomial basis functions, which reduces the computational cost comparing with the divergence-free moving least squares approximation. The pointwise error estimates of the presented approximation are obtained for the bounded sub-domains in Rd(d≥2) with an interior cone condition. Some numerical results are provided to confirm the obtained theoretical results. As an application, the Cahn-Hilliard-Hele-Shaw (CHHS) equation is solved numerically via a stabilized semi-implicit scheme in time that holds the mass conservative and energy dissipative properties and also the GMLS approximation in space. Besides, the developed vector-valued approximation has been used to compute the Leray-Helmholtz projection of the advective velocity.

Best Local Weighted Approximation. An Approach with Abstract Seminorms

We extend results given by Zó and Cuenya in 2007 about a general approach to problems of best vector-valued approximation on small regions from a finite dimensional subspace of polynomials of some degree. This approach is called best local approximation. We consider a weighted local approximation of a vector-valued function on the origin and a weighted best local approximation of a real-valued function on several points, similar to classical problems in best local approximation with balanced neighborhood.

On convergence of subspaces generated by dilations of polynomials. An application to best local approximation

We study the convergence of a net of subspaces generated by dilations of polynomials in a finite dimensional subspace. As a consequence, we extend the results given by Z´o and Cuenya [Advanced Courses of Mathematical Analysis II (Granada, 2004), 193–213, World Scientific, 2007] on a general approach to the problems of best vector-valued approximation on small regions from a finite dimensional subspace of polynomials.

On Vector-Valued Approximation of State Constrained Optimal Control Problems for Nonlinear Hyperbolic Conservation Laws

We study one class of nonlinear fluid dynamic models with controls in the initial condition and the source term. The model is described by a nonlinear inhomogeneous hyperbolic conservation law with state and control constraints. We consider the case when the greatest lower bound of the cost functional can be unattainable on the set ? of admissible pairs or the set ? is possibly empty. Using the methods of vector-valued optimization theory, we show that this optimal control problem admits the existence of the so-called weakened approximate solution which can be interpreted as generalized solution to some vector optimization problem of special form.

Optimal discriminating designs for several competing regression models

The problem of constructing optimal discriminating designs for a class of regression models is considered. We investigate a version of the $T_p$-optimality criterion as introduced by Atkinson and Fedorov [Biometrika 62 (1975a) 289-303]. The numerical construction of optimal designs is very hard and challenging, if the number of pairwise comparisons is larger than 2. It is demonstrated that optimal designs with respect to this type of criteria can be obtained by solving (nonlinear) vector-valued approximation problems. We use a characterization of the best approximations to develop an efficient algorithm for the determination of the optimal discriminating designs. The new procedure is compared with the currently available methods in several numerical examples, and we demonstrate that the new method can find optimal discriminating designs in situations where the currently available procedures fail.

Uniqueness in Vector-Valued Approximation

We consider when a finite dimensional subspace is a unicity space in the problem of best vector-valued approximation. Different choices of mixed norms are discussed.

Row convergence theorems for generalised inverse vector-valued Padé approximants

The approximants mentioned in the title are related to vector-valued continued fractions and the vector ϵ-algorithm devised by Wynn in 1963. Here we establish a unitary invariance property of these approximants and describe how the classical (1-dimensional) Padé approximants can be obtained as a special case. The main results of the paper consist of De Montessus—De Ballore type convergence theorems for row sequences (having fixed denominator degree) of vector-valued approximants to meromorphic vector functions.

Vector-Valued Approximation and its Application to Fitting Exponential Decay Curves

Vector-Valued Approximation and its Application to Fitting Exponential Decay Curves

Vector-valued approximation and its application to fitting exponential decay curves

This paper deals with characterization of best approximations to vector-valued functions. The approximations are themselves vector-valued functions with components depending nonlinearly on the approximation parameters. The constraint is imposed that certain of the parameters should be identical for all components. An application to exponential approximation is discussed in some detail.