Interpolation Spaces Research Articles

The Constant of Interpolation in Bloch Type Spaces

It is known that there exists a constant 0<Delta _1 < 1 such that any Delta _1-separated sequence for the pseudohyperbolic distance in the open unit disk textbf{D} of mathbb {C} is interpolating for the classical Bloch space mathcal {B}. We will prove that 0.8114< Delta _1 < 0.9785 and we will also generalize this result for Bloch type spaces mathcal {B}_{v_p} for v_p(z)=(1-|z|^2)^p. In particular, we will provide a construction to calculate an estimate of the lower and upper bounds for the corresponding constant of separation Delta _p for these spaces. We also prove that Delta _p tends to 1 when p rightarrow infty .

DRIPS: A framework for dimension reduction and interpolation in parameter space

Reduced-order models are often used to describe the behavior of complex systems, whose simulation with a full model is too expensive, or to extract salient features from the full model's output. We introduce a new model-reduction framework DRIPS (dimension reduction and interpolation in parameter space) that combines the offline local model reduction with the online parameter interpolation of reduced-order bases (ROBs). The offline step of this framework relies on dynamic mode decomposition (DMD) to build a low-rank linear surrogate model, equipped with a local ROB, for quantities of interest derived from the training data generated by repeatedly solving the (nonlinear) high-fidelity model for multiple parameter points. The online step consists of the construction of a parametric reduced-order model for each target/test point in the parameter space, with the interpolation of ROBs done on a Grassman manifold and the interpolation of reduced-order operators done on a matrix manifold. The DMD component enables DRIPS to model (typically low-dimensional) quantities of interest directly, without having to access the (typically high-dimensional and possibly nonlinear) operators in a high-fidelity model that governs the dynamics of the underlying high-dimensional state variables, as required in projection-based reduced-order modeling. A series of numerical experiments suggests that DRIPS yields a model reduction, which is computationally more efficient than the commonly used projection-based proper orthogonal decomposition; it does so without requiring a prior knowledge of the governing equation for quantities of interest. Moreover, for the nonlinear systems considered, DRIPS is more accurate than Gaussian-process interpolation (Kriging).

Representations and Regularity of Vector-Valued Right-Shift Invariant Operators Between Half-Line Bessel Potential Spaces

Representation and boundedness properties of linear, right-shift invariant operators on half-line Bessel potential spaces (also known as fractional-order Sobolev spaces) as operator-valued multiplication operators in terms of the Laplace transform are considered. These objects are closely related to the input–output operators of linear, time-invariant control systems. Characterisations of when such operators map continuously between certain interpolation spaces and/or Bessel potential spaces are provided, including characterisations in terms of boundedness and integrability properties of the symbol, also known as the transfer function in this setting. The paper considers the Hilbert space case, and the theory is illustrated by a range of examples.

Hyperreduced-order modeling of thermally coupled flows

This article presents a full and reduced-order methodology based on the finite element method that allows the characterization of convective-dominant conjugate heat transfer flows. Reduced-order modeling (ROM) allows the description of several degrees of freedom problems, referred to as full-order modeling (FOM), through a reduced-order surrogate representation based on a Petrov-Galerkin projection onto a reduced subspace. The FOM and ROM problems are stabilized through the Variational Multi-Scale (VMS) method, which ensures stability in dominant convective problems and allows equal interpolation spaces for the pressure, velocity, and temperature. The proposed methodology is validated solving natural heat convection in a differentially heated square cavity, considering air as the working fluid and Rayleigh numbers in the interval 104≤Ra≤108. In this problem, the benefit of using a ROM approximation for the three unknown fields, i.e., velocity, pressure, and temperature, is compared against ROM approximation only in the velocity and pressure fields and a FOM type approximation for temperature. The benefit of using mesh-based hyperreduction and its computational performance and precision are also analyzed. Subsequently, the method simulates a conjugate heat transfer problem that includes natural heat convection, allowing the inclusion of different rheologies between the working fluids. The results verify the precision and speed up of the calculations by more than three hundred times with respect to the time necessary to solve a full-order formulation, emerging it as a great potential tool in the resolution of thermally coupled flows.

SHAPE DEPENDENT MOTION INTERPOLANTS FOR PLANAR OBJECTS.

Kinematics is most commonly about the motion of unbounded spaces. This paper deals with the kinematics of bounded shapes in a plane. This paper studies the problem of motion interpolation of a planar object with its shape taken into consideration. It applies and extends a shape dependent distance measure between two positions in the context of motion interpolation. Instead of using a fixed reference frame, a shape-dependent inertia frame of reference is used for formulating the distance between positions of a rigid object in a plane. The resulting distance function is then decomposed in two orthogonal directions and is used to formulate an interpolating function for the distance functions in these two directions. This leads to a shape dependent interpolation of translational components of a planar motion. In difference to the original concept of Kazerounian and Rastegar that comes with a shape dependent measure of the angular motion, it is assumed in this paper that the angular motion is shape independent as the angular metric is dimensionless. The resulting distance measure is not only a combination of translation and rotation parameters but also depends on the area moments of inertia of the object. It derives the explicit expressions for decomposing the shape dependent distance in two orthogonal directions, which is then used to obtain shape dependent motion interpolants in these directions. The resulting interpolants have similarities to the well-known spherical linear interpolants widely used in computer graphics in that they are defined using sinusoidal functions instead of linear interpolation in Euclidean space. The path of the interpolating motion can be adjusted by different choice of shape parameters. Examples are provided to illustrate the effect of object shapes on the resulting interpolating motions.

A penalty-free Shifted Boundary Method of arbitrary order

We introduce and analyze a penalty-free formulation of the Shifted Boundary Method (SBM), inspired by the asymmetric version of the Nitsche method. We prove its stability and convergence for arbitrary order finite element interpolation spaces and we test its performance with a number of numerical experiments. Moreover, while the SBM was previously believed to be only asymptotically consistent (in the sense of Galerkin orthogonality), we prove here that it is indeed exactly consistent.

Weighted fractional Sobolev spaces as interpolation spaces in bounded domains

AbstractWe characterize the real interpolation space between a weighted space and a weighted Sobolev space in arbitrary bounded domains in , with weights that are positive powers of the distance to the boundary.

Euclidean Structures and Operator Theory in Banach Spaces

We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators Γ \Gamma on a Hilbert space. Our assumption on Γ \Gamma is expressed in terms of α \alpha -boundedness for a Euclidean structure α \alpha on the underlying Banach space X X . This notion is originally motivated by R \mathcal {R} - or γ \gamma -boundedness of sets of operators, but for example any operator ideal from the Euclidean space ℓ n 2 \ell ^2_n to X X defines such a structure. Therefore, our method is quite flexible. Conversely we show that Γ \Gamma has to be α \alpha -bounded for some Euclidean structure α \alpha to be representable on a Hilbert space. By choosing the Euclidean structure α \alpha accordingly, we get a unified and more general approach to the Kwapień–Maurey factorization theorem and the factorization theory of Maurey, Nikišin and Rubio de Francia. This leads to an improved version of the Banach function space-valued extension theorem of Rubio de Francia and a quantitative proof of the boundedness of the lattice Hardy–Littlewood maximal operator. Furthermore, we use these Euclidean structures to build vector-valued function spaces. These enjoy the nice property that any bounded operator on L 2 L^2 extends to a bounded operator on these vector-valued function spaces, which is in stark contrast to the extension problem for Bochner spaces. With these spaces we define an interpolation method, which has formulations modelled after both the real and the complex interpolation method. Using our representation theorem, we prove a transference principle for sectorial operators on a Banach space, enabling us to extend Hilbert space results for sectorial operators to the Banach space setting. We for example extend and refine the known theory based on R \mathcal {R} -boundedness for the joint and operator-valued H ∞ H^\infty -calculus. Moreover, we extend the classical characterization of the boundedness of the H ∞ H^\infty -calculus on Hilbert spaces in terms of B I P BIP , square functions and dilations to the Banach space setting. Furthermore we establish, via the H ∞ H^\infty -calculus, a version of Littlewood–Paley theory and associated spaces of fractional smoothness for a rather large class of sectorial operators. Our abstract setup allows us to reduce assumptions on the geometry of X X , such as (co)type and UMD. We conclude with some sophisticated counterexamples for sectorial operators, with as a highlight the construction of a sectorial operator of angle 0 0 on a closed subspace of L p L^p for 1 &gt; p &gt; ∞ 1&gt;p&gt;\infty with a bounded H ∞ H^\infty -calculus with optimal angle ω H ∞ ( A ) &gt; 0 \omega _{H^\infty }(A) &gt;0 .

Novel interpolation spaces and maximal‐weighted Hölder regularity results for the fractional abstract Cauchy problem

AbstractIn this paper, the maximal‐weighted Hölder regularity for the fractional abstract Cauchy problem is presented. First, two classes of novel interpolation spaces constructed by resolvent families are portrayed. Afterward, by applying the resolvent method and the properties of interpolation spaces, the maximal‐weighted Hölder regularity of the mild solution is established in weighted Hölder space (, ). Eventually, an instance for the fractional differential equation of 2m‐order multidimensional parabolic type is provided to verify the reasonability of main conclusions.

Beurling-Type Density Criteria for System Identification

This paper addresses the problem of identifying a linear time-varying (LTV) system characterized by a (possibly infinite) discrete set of delay-Doppler shifts without a lattice (or other “geometry-discretizing”) constraint on the support set. Concretely, we show that a class of such LTV systems is identifiable whenever the upper uniform Beurling density of the delay-Doppler support sets, measured “uniformly over the class”, is strictly less than 1/2. The proof of this result reveals an interesting relation between LTV system identification and interpolation in the Bargmann-Fock space. Moreover, we show that the density condition we obtain is also necessary for classes of systems invariant under time-frequency shifts and closed under a natural topology on the support sets. We furthermore find that identifiability guarantees robust recovery of the delay-Doppler support set, as well as the weights of the individual delay-Doppler shifts, both in the sense of asymptotically vanishing reconstruction error for vanishing measurement error.

The Hölder Regularity for Abstract Fractional Differential Equation with Applications to Rayleigh–Stokes Problems

In this paper, we studied the Hölder regularities of solutions to an abstract fractional differential equation, which is regarded as an abstract version of fractional Rayleigh–Stokes problems, rising up to describing a non-Newtonian fluid with a Riemann–Liouville fractional derivative. The purpose of this article was to establish the Hölder regularities of mild solutions, classical solutions, and strict solutions. We introduced an interpolation space in terms of an analytic resolvent to lower the spatial regularity of initial value data. By virtue of the properties of analytic resolvent and the interpolation space, the Hölder regularities were obtained. As applications, the main conclusions were applied to the regularities of fractional Rayleigh–Stokes problems.

Random attractors for rough stochastic partial differential equations

This paper is devoted to the existence of random attractors for rough partial differential equations driven by nonlinear multiplicative Hölder rough paths with exponents in (1/3,1/2]. Our approach relies upon rough paths theory and stopping times analysis in a suitable scale of interpolation spaces. The core step is to derive the adequate algebraic and analytical properties of a sequence of stopping times, which allows us to establish the required compact tempered absorbing set. The existence of a pullback attractor for the generated random dynamical system is straightforward. An illustrative example is presented by reaction-diffusion equations subjected to fractional Brownian rough paths.

Description of K‐spaces by means of J‐spaces and the reverse problem in the limiting real interpolation

AbstractWe establish conditions under which K‐spaces in the limiting real interpolation involving slowly varying functions can be described by means of J‐spaces and we also solve the reverse problem. To this end, we prove several versions of the fundamental lemma of the real interpolation theory. We apply our results to obtain density theorems for the corresponding limiting interpolation spaces.

Interpolation synthesis of thecontrollers for a multi-motorelectric drive system containingan elastically linked element

Partial differential equations, integral, differential, or other equations describe multi-motor automatic electric drive systems containing elastic conveyor belts. Because of the elastic and distributive nature of the system parameters, the transfer function describing them is often a complex expression, containing not only the arguments as a linear system but also the inertial and transcendental components. This makes the precise control of tension and speed synchronously much more complicated than the centralized parameter system. A promising numerical solution based on the real interpolation method will simplify the procedure for synthesizing control loops while preserving the characteristic properties of objects with distributed parameters. The objective of the study is to propose a feasible solution for synthesizing the regulators based on the real interpolation method; it allows direct operation with the original transfer function containing the inertial and transcendental components. In this paper, we proposed an approach to synthesize the control system for objects with distributed parameters using the real interpolation method to reduce computational capacity and synthesis error while preserving the properties of this object class. Building an experimental model of the two-motor electric drive system containing an elastic conveyor to verify the effectiveness of the proposed algorithm. The simulation and experimental results indicate that the control system with the received regulators operating stably and meets the required quality criteria. It proves the efficiency of the synthesis algorithm based on the real interpolation method.

Recalibration of LBM Populations for Construction of Grid Refinement with No Interpolation

Grid refinement is used to reduce computing costs while maintaining the precision of fluid simulation. In the lattice Boltzmann method (LBM), grid refinement often uses interpolated values. Here, we developed a method in which interpolation in space and time is not required. For this purpose, we used the moment matching condition and rescaled the nonequilibrium part of the populations, thereby developing a recalibration procedure that allows for the transfer of information between different LBM stencils in the simulation domain. Then, we built a nonuniform lattice that uses stencils with different shapes on the transition. The resulting procedure was verified by performing benchmarks with the 2D Poisselle flow and the advected vortex. It is suggested that grids with adaptive geometry can be built with the proposed method.

Pathology Synthesis of 3D-Consistent Cardiac MR Images using 2D VAEs and GANs

We propose a method for synthesizing cardiac magnetic resonance (MR) images with plausible heart pathologies and realistic appearances for the purpose of generating labeled data for the application of supervised deep-learning (DL) training. The image synthesis consists of label deformation and label-to-image translation tasks. The former is achieved via latent space interpolation in a VAE model, while the latter is accomplished via a label-conditional GAN model. We devise three approaches for label manipulation in the latent space of the trained VAE model; i) intra-subject synthesis aiming to interpolate the intermediate slices of a subject to increase the through-plane resolution, ii) inter-subject synthesis aiming to interpolate the geometry and appearance of intermediate images between two dissimilar subjects acquired with different scanner vendors, and iii) pathology synthesis aiming to synthesize a series of pseudo-pathological synthetic subjects with characteristics of a desired heart disease. Furthermore, we propose to model the relationship between 2D slices in the latent space of the VAE prior to reconstruction for generating 3D-consistent subjects from stacking up 2D slice-by-slice generations. We demonstrate that such an approach could provide a solution to diversify and enrich an available database of cardiac MR images and to pave the way for the development of generalizable DL-based image analysis algorithms. We quantitatively evaluate the quality of the synthesized data in an augmentation scenario to achieve generalization and robustness to multi-vendor and multi-disease data for image segmentation. Our code is available at &lt;a href='https://github.com/sinaamirrajab/CardiacPathologySynthesis'&gt;https://github.com/sinaamirrajab/CardiacPathologySynthesis&lt;/a&gt;

Optimal behavior of weighted Hardy operators on rearrangement‐invariant spaces

AbstractThe behavior of certain weighted Hardy‐type operators on rearrangement‐invariant function spaces is thoroughly studied. Emphasis is put on the optimality of the obtained results. First, the optimal rearrangement‐invariant function spaces guaranteeing the boundedness of the operators from/to a given rearrangement‐invariant function space are described. Second, the optimal rearrangement‐invariant function norms being sometimes complicated, the question of whether and how they can be simplified to more manageable expressions is addressed. Next, the relation between optimal rearrangement‐invariant function spaces and interpolation spaces is investigated. Last, iterated weighted Hardy‐type operators are also studied.

A Surrogate Modeling Approach for Frequency-Reconfigurable Antennas

A novel generalizable surrogate modeling approach is specifically developed for frequency reconfigurable antennas. The generalizable modeling processes is based on the rigorous mathematical derivation, including the solution of a non-linear overdetermined system, the optimization in the complex field, and the interpolation in multi-dimensional continuous space. As a post-processing method, the approach can convert the discrete data of CAD simulation to a surrogate model. Subsequently, a reconfigurable UWB antenna with a tunable notch-band is taken as an example to demonstrate that the surrogate modeling approach is feasible, effective, and precise. It also has the flexible ability to adapt to strict requirements and complicated scenarios. The proposed surrogate model is a good candidate for the interface standard between a reconfigurable antenna and signal processing part in a Cognitive Radio system.

Adversarial MixUp with implicit semantic preservation for semi-supervised hyperspectral image classification

MixUp generates virtual samples by performing the identical convex combination in feature and semantic spaces simultaneously. However, convexly combining random pairs of images and their associated labels is likely to result in misunderstandings between the generated sample and its assigned pseudo label, particularly for hyperspectral images (HSI), leading to a confusing effect on the classifier. In this paper, adversarial MixUp with implicit semantic preservation (AdvMixup) is proposed for semi-supervised HSI classification (HSIc). The proposed AdvMixup exploits interpolation in the latent space for unlabeled samples. The virtual hyperspectral sample is generated by a deconvolutional network on the interpolated latent data. A semantic preservation network is performed on the newly generated virtual sample to recover the MixUp coefficient, where the semantic is preserved implicitly in the virtual sample. Meanwhile, an adversarial regularization is also conducted on the recovered mixup coefficient to ensure the authenticity of the virtual HSI. In such a way, the proposed AdvMixup sees the unseen HSI and knows the unknown semantic. Throughout the learning process, the feature extraction network is shared in both supervised and unsupervised branches to obtain a superior decision boundary. Experimental results on three HSI benchmark datasets demonstrate that AdvMixup achieves state-of-the-art performance in semi-supervised HSI classification.

Comprehensive Compensation Method for Motion Trajectory Error of End-Effector of Cable-Driven Parallel Mechanism

The accuracy of the end-effector motion trajectory is a critical performance indicator for cable-driven parallel mechanisms. This study aims to address the problem of trajectory error during the end-effector motion in a cable-driven parallel mechanism. It proposes a comprehensive compensation method based on the simultaneous application of the improved sparrow search algorithm and the cable length space error compensation algorithm, leveraging kinematic analysis. To compensate for the motion trajectory error of the end-effector caused by the geometric parameter error, the study establishes the kinematic model of the cable-driven parallel mechanism using the vector method. It creates the end-effector position error model and motion trajectory error model using the differential kinematic theory, analyzes the impact of the geometric parameter error on the motion trajectory error, constructs the kinematic parameter identification matrix, and uses an improved sparrow search algorithm to compensate for the position error of the motion trajectory interpolation point. For the motion trajectory error of the end-effector caused by non-geometric parameter error, the study analyzes the intrinsic correlation between the adjacent position error of the end-effector and the variation of the cable length using the error similarity theory. It then compensates for the position error of the interpolation point of the trajectory using a cable length space interpolation compensation method to enhance the motion trajectory accuracy of the end-effector. The study experimentally verifies the proposed comprehensive compensation method for end-effector motion trajectory error on a 4-cable-driven 2-DOF parallel mechanism, which reduces the motion trajectory error of the end-effector by 75%.