Interpolation Conditions Research Articles

Contour interpolation: A case study in Modularity of Mind

In his monograph Modularity of Mind (1983), philosopher Jerry Fodor argued that mental architecture can be partly decomposed into computational organs termed modules, which were characterized as having nine co-occurring features such as automaticity, domain specificity, and informational encapsulation. Do modules exist? Debates thus far have been framed very generally with few, if any, detailed case studies. The topic is important because it has direct implications on current debates in cognitive science and because it potentially provides a viable framework from which to further understand and make hypotheses about the mind’s structure and function. Here, the case is made for the modularity of contour interpolation, which is a perceptual process that represents non-visible edges on the basis of how surrounding visible edges are spatiotemporally configured. There is substantial evidence that interpolation is domain specific, mandatory, fast, and developmentally well-sequenced; that it produces representationally impoverished outputs; that it relies upon a relatively fixed neural architecture that can be selectively impaired; that it is encapsulated from belief and expectation; and that its inner workings cannot be fathomed through conscious introspection. Upon differentiating contour interpolation from a higher-order contour representational ability (“contour abstraction”) and upon accommodating seemingly inconsistent experimental results, it is argued that interpolation is modular to the extent that the initiating conditions for interpolation are strong. As interpolated contours become more salient, the modularity features emerge. The empirical data, taken as a whole, show that at least certain parts of the mind are modularly organized.

Variational time discretization of Riemannian splines

We investigate a generalization of cubic splines to Riemannian manifolds. Spline curves are defined as minimizers of the spline energy - a combination of the Riemannian path energy and the time integral of the squared covariant derivative of the path velocity - under suitable interpolation conditions. A variational time discretization for the spline energy leads to a constrained optimization problem over discrete paths on the manifold. Existence of continuous and discrete spline curves is established using the direct method in the calculus of variations. Furthermore, the convergence of discrete spline paths to a continuous spline curve follows from the $\Gamma$-convergence of the discrete to the continuous spline energy. Finally, selected example settings are discussed, including splines on embedded finite-dimensional manifolds, on a high-dimensional manifold of discrete shells with applications in surface processing, and on the infinite-dimensional shape manifold of viscous rods.

An Efficient Algorithm Based on Extrapolation for the Solution of Nonlinear Parabolic Equations

AbstractTwo numerical procedures are developed to approximate the solution of one-dimensional parabolic equations using extrapolated collocation method. By defining two different end conditions and forcing cubic spline to satisfy the interpolation conditions along with one of the end conditions, we obtain fourth- (CBS4) and sixth-order (CBS6) approximations to the solution in spatial direction. Also in time direction, a weighted finite-difference discretization is used to approximate the solution at each time level. The convergence analysis of the methods is discussed in detail and some error bounds are obtained theoretically. Finally, some different examples of Burgers’ equations with applications in fluid mechanics as well as convection–diffusion problems with applications in transport problems are solved to show the applicability and good performance of the procedures.

Greedy Sampling of Graph Signals

Sampling is a fundamental topic in graph signal processing, having found applications in estimation, clustering, and video compression. In contrast to traditional signal processing, the irregularity of the signal domain makes selecting a sampling set non-trivial and hard to analyze. Indeed, though conditions for graph signal interpolation from noiseless samples exist, they do not lead to a unique sampling set. The presence of noise makes choosing among these sampling sets a hard combinatorial problem. Although greedy sampling schemes are commonly used in practice, they have no performance guarantee. This work takes a twofold approach to address this issue. First, universal performance bounds are derived for the Bayesian estimation of graph signals from noisy samples. In contrast to currently available bounds, they are not restricted to specific sampling schemes and hold for any sampling sets. Second, this paper provides near-optimal guarantees for greedy sampling by introducing the concept of approximate submodularity and updating the classical greedy bound. It then provides explicit bounds on the approximate supermodularity of the interpolation mean-square error showing that it can be optimized with worst-case guarantees using greedy search even though it is not supermodular. Simulations illustrate the derived bound for different graph models and show an application of graph signal sampling to reduce the complexity of kernel principal component analysis.

Contour interpolation: A case study in Modularity of Mind

In his monograph Modularity of Mind (1983), philosopher Jerry Fodor argued that mental architecture can be partly decomposed into computational organs termed modules, which were characterized as having nine co-occurring features such as automaticity, domain specificity, and informational encapsulation. Do modules exist? Debates thus far have been framed very generally with few, if any, detailed case studies. The topic is important because it has direct implications on current debates in cognitive science and because it potentially provides a viable framework from which to further understand and make hypotheses about the mind's structure and function. Here, the case is made for the modularity of contour interpolation, which is a perceptual process that represents non-visible edges on the basis of how surrounding visible edges are spatiotemporally configured. There is substantial evidence that interpolation is domain specific, mandatory, fast, and developmentally well-sequenced; that it produces representationally impoverished outputs; that it relies upon a relatively fixed neural architecture that can be selectively impaired; that it is encapsulated from belief and expectation; and that its inner workings cannot be fathomed through conscious introspection. Upon differentiating contour interpolation from a higher-order contour representational ability (contour abstraction) and upon accommodating seemingly inconsistent experimental results, it is argued that interpolation is modular to the extent that the initiating conditions for interpolation are strong. As interpolated contours become more salient, the modularity features emerge. The empirical data, taken as a whole, show that at least certain parts of the mind are modularly organized.

Dynamic stationary crack analysis of isotropic solids and anisotropic composites by enhanced local enriched consecutive-interpolation elements

The recently developed local enriched consecutive-interpolation 4-node quadrilateral element (XCQ4) is extended to study transient dynamic stress intensity factors (DSIFs) for isotropic solids and anisotropic composite materials containing stationary cracks. The XCQ4 involves both nodal values and averaged nodal gradients as interpolation conditions to smooth the distribution of unknown variables. The possible physical properties of enriched nodes in consecutive-interpolation procedure (CIP) would be thought as non-locality feature, which could improve the accuracy of results and eliminate the non-smooth stresses among inter-elements. In XCQ4, the crack is determined by level set function and enriched by Heaviside and crack-tip enrichment functions with special anisotropic enriched crack-tip functions. Time-dependent discrete equations for dynamic cracks are solved by Newmark time integration scheme at each time step without considering the effects of velocity-based global damping matrix. The proposed method is verified through a series of numerical examples of transient fracture in both isotropic and anisotropic materials. Numerical DSIFs are compared with reference solutions available in literature. The behavior of dynamic response is explored in specimens with complex configuration under step and sine loads.

Spline parameterization method for 2D and 3D geometries based on T-mesh optimization

We present a method to obtain high quality spline parameterization of 2D and 3D geometries for their use in isogeometric analysis. As input data, the proposed method demands a boundary representation of the domain, and it constructs automatically a spline transformation between the physical and parametric domains. Parameterization of the interior of the object is obtained by deforming isomorphically an adapted parametric T-mesh onto the physical domain by applying a T-mesh untangling and smoothing procedure, which is the key of the method. Mesh optimization is based on the mean ratio shape quality measure. The spline representation of the geometry is calculated by imposing interpolation conditions using the data provided by one-to-one correspondence between the meshes of the parametric and physical domains. We give a detailed description of the proposed technique and show some examples. Also, we present some examples of the application of isogeometric analysis in geometries parameterized with our method.

Realization of functions on the symmetrized bidisc

We prove a realization formula and a model formula for analytic functions with modulus bounded by 1 on the symmetrized bidiscG=def{(z+w,zw):|z|<1,|w|<1}. As an application we prove a Pick-type theorem giving a criterion for the existence of such a function satisfying a finite set of interpolation conditions.

Polynomial wavelets on the unit circle

In this paper, we have considered polynomial wavelets on unit circle. The scaling functions are considered to be the fundamental polynomials of the Lagrange interpolants on the equally spaced nodes different from the n roots of unity, which satisfy certain interpolatory conditions.

Positive constrained approximation via RBF-based partition of unity method

In this paper, we discuss the problem of constructing Radial Basis Function (RBF)-based Partition of Unity (PU) interpolants that are positive if data values are positive. More specifically, we compute positive local approximants by adding up several constraints to the interpolation conditions. This approach, considering a global approximation problem and Compactly Supported RBFs (CSRBFs), has been previously proposed in Wu et al. (2010). Here, the use of the PU technique enables us to intervene only locally and as a consequence to reach a better accuracy. This is also due to the fact that we select the optimal number of positive constraints by means of an a priori error estimate and we do not restrict to the use of CSRBFs. Numerical experiments and applications to population dynamics are provided to illustrate the effectiveness of the method in applied sciences.

System identification of microwave filters from multiplexers by rational interpolation

Microwave multiplexers are multi-port structures composed of several two-port filters connected to a common junction. This paper addresses the de-embedding problem, in which the goal is to determine the filtering components given the measured scattering parameters of the overall multiplexer at several frequencies. Due to structural properties, the transmission zeros of the filters play a crucial role in this problem, and, consequently, in our approach. We propose a system identification algorithm for deriving a rational model of the filters’ scattering matrix. The approach is based on rational interpolation with derivative constraints, with the interpolation conditions being located precisely at the filters’ transmission zeros.

Boundary Nevanlinna--Pick Interpolation with Prescribed Peak Points. Application to Impedance Matching

We study a generalized Nevanlinna--Pick interpolation problem on the half-plane for rational functions of prescribed degree, where peak points are imposed and interpolation conditions may lie on the real axis. This generalizes previous work by Georgiou, Byrnes, Lindquist, and Megretski. The problem is motivated by the issue of broadband matching in electronics and microwave system design. We prove existence and uniqueness of a solution by differential-topological techniques. The approach is put to work numerically on a real example, using a continuation method.

Mixed meshless local Petrov–Galerkin collocation method for modeling of material discontinuity

A mixed meshless local Petrov---Galerkin (MLPG) collocation method is proposed for solving the two-dimensional boundary value problem of heterogeneous structures. The heterogeneous structures are defined by partitioning the total material domain into subdomains with different linear-elastic isotropic properties which define homogeneous materials. The discretization and approximation of unknown field variables is done for each homogeneous material independently, therein the interface of the homogeneous materials is discretized with overlapping nodes. For the approximation, the moving least square method with the imposed interpolation condition is utilized. The solution for the entire heterogeneous structure is obtained by enforcing displacement continuity and traction reciprocity conditions at the nodes representing the interface boundary. The accuracy and numerical efficiency of the proposed mixed MLPG collocation method is demonstrated by numerical examples. The obtained results are compared with a standard fully displacement (primal) meshless approach as well as with available analytical and numerical solutions. Excellent agreement of the solutions is exhibited and a more robust, superior and stable modeling of material discontinuity is achieved using the mixed method.

Quasi-static crack propagation simulation by an enhanced nodal gradient finite element with different enrichments

The recently developed consecutive-interpolation local enriched partition-of-unity method based on 4-node quadrilateral element (XCQ4) is used to study quasi-static crack propagation in 2-dimensional solids. For some of these problems numerical results have also been calculated with the standard extended finite element (XQ4), provided that the two approaches have the same number of degrees of freedom. In addition, two different versions of enrichment functions capturing the crack tip fields are taken into account, integrating into either the XCQ4 or XQ4. In each case, results have been computed with both settings and compared between each other. It is found that the numerical solution using the XCQ4 element has better accuracy than that found with the XQ4, and these solutions agree well with reference solutions available in literature. The underlying difference between the consecutive-interpolation basis functions and those for the traditional XQ4 is that the former approximation functions constructed by incorporating both nodal values and averaged nodal gradients obtained from linear shape function as interpolation conditions, enhancing and smoothing the stress fields and stress intensity factors. Additionally, the conditioning issue of the developed method is also numerically examined.

Implicit Volterra series interpolation for model reduction of bilinear systems

We propose a new interpolatory framework for model reduction of large-scale bilinear systems. The input–output representation of a bilinear system in frequency domain involves a series of multivariate transfer functions, each representing a subsystem of the bilinear system. If a weighted sum of these multivariate transfer functions associated with a reduced bilinear system interpolates a weighted sum of the original multivariate transfer functions, we say that the reduced system satisfies a Volterra series interpolation (Flagg and Gugercin, 2015). These interpolatory conditions can also ensure the necessary conditions for H2-optimal model reduction (Flagg and Gugercin, 2015; Benner and Breiten, 2012). We observe that, by carefully selecting the weights of the series, the Volterra series interpolatory conditions are transformed to the problem of interpolating a linear system with an affine parameter dependence. Such linear parametric systems can then be reduced by some method for parametric model order reduction.Linear systems where the affine parameter dependence is given as low-rank variation in the state matrix can be mapped into a non-parameterized multi-input multi-output linear system. This allows us to utilize the standard (non-parametric) linear IRKA (Gugercin et al., 2008) for the problem of parameterized/bilinear interpolation. Numerical results show that the approximations are of comparable accuracy to those obtained from the bilinear iterative rational Krylov algorithm (Benner and Breiten, 2012). The proposed approach, however, has the advantage that it reduces the computational costs as it involves computations associated with solving linear systems only.

Geometric conditions for multiple sampling and interpolation in the Fock space

We study multiple sampling, interpolation and uniqueness for the classical Fock spaces in the case of unbounded multiplicities. We show that there are no sequences which are simultaneously sampling and interpolating when the multiplicities tend to infinity. This answers partially a question posed by Brekke and Seip.

Multipoint interpolation of Volterra series and [formula omitted]-model reduction for a family of bilinear descriptor systems

In this paper, we investigate interpolatory model order reduction for large-scale bilinear descriptor systems. Recently, it was shown in Wyatt et al. (2013) for linear descriptor systems that directly extending the standard rational interpolation conditions used in H2 optimal model reduction to descriptor systems, in general, yields an unbounded error in the H2-norm. This is due to the possible mismatch of the polynomial part of the original and reduced-order systems. This conclusion holds for nonlinear systems as well. In this paper, we deal with bilinear descriptor systems and aim to pay attention to the polynomial part of the bilinear descriptor system along with interpolation. To this end, we have shown in Goyal et al. (2015) how to determine the polynomial part of each subsystem of the bilinear descriptor system explicitly, by assuming special structures of the system matrices. Considering the same structured bilinear descriptor systems, in this paper we first show how to achieve multipoint interpolation of the underlying Volterra series of bilinear descriptor systems while retaining the polynomial part of each subsystem of the bilinear system. Then, we extend the interpolation based first-order necessary conditions for H2 optimality to bilinear descriptor systems and propose an iterative scheme to obtain an H2 optimal reduced-order system. By means of various numerical examples, we demonstrate the efficiency of the proposed model order reduction technique and compare it with reduced bilinear systems obtained by using linear IRKA, the Loewner method for bilinear systems and POD-based approximations.

L2–Optimal Fopdt Models of High–Order Transfer Functions

AbstractNecessary conditions for the L2 optimality of a first order plus dead time (FOPDT) model of a high‐order plant are derived using classic analytic function theory. They are expressed as a set of three nonlinear equations that partly resemble the interpolation conditions valid for rational approximation. From these conditions a simple procedure to find the optimal FOPDT model is obtained. Examples taken from the relevant literature are worked out to show the performance of the method in comparison with alternative techniques.

The discretization for bivariate ideal interpolation

Carl de Boor conjectured that every ideal interpolant over complex field is the pointwise limit of Lagrange interpolants. Boris Shekhtman proved that the conjecture is true in two variables, and he also provided a counterexample for more than three variables. However, Shekhtman only mentioned the existence of some mathematical objects (without giving a method to compute them) in his proof of the bivariate case. For general interpolation condition functionals on the interpolation sites, we improve Shekhtman’s method to find a sequence of interpolation sites (also called discrete sites), such that the corresponding Lagrange interpolants converge to the given bivariate ideal interpolant. We discuss a special case where the multiplicity space is of breadth one. The results in this paper give a completely algorithmic way to realize Shekhtman’s method.

Multiple Sampling and Interpolation in the Classical Fock Space

We study multiple sampling, interpolation and uni- queness for the classical Fock space in the case of unbounded mul- tiplicities. Sampling and interpolating sequences in Fock spaces were character- ized by Seip and Seip-Wallsten in (3, 4) by means of a certain Beurling- type asymptotic uniform density. The case of uniformly bounded mul- tiplicities was considered by Brekke and Seip (1) who gave a complete description in this situation. Their conditions show that it is not pos- sible that a sequence is simultaneously sampling and interpolating. Brekke and Seip also asked whether there exist sequences which are simultaneously sampling and interpolating when the multiplicities are unbounded. In this research note we formulate some conditions (of geometric nature) for sampling and interpolation. They show that the answer to this question is negative when the multiplicities tend to infinity. The detailed version of this work will be published elsewhere.