Interpolation Spaces Research Articles

SYSTEMATIZATION OF INTERPOLATION METHODS IN GEOMETRIC MODELING

This article proposes to show the connections between the practical problems of geometric modeling, for the solution of which there are no adequate interpolation methods. Interpolation methods in mathematics first emerged as part of mathematical modeling, and eventually became a separate area of mathematics. The classical definition of interpolation is the constructive reproduction of a function of a certain class according to its known values or the values of its derivatives at given points.Interpolation problems arise in spaces of different dimensions when solving engineering problems. Elements of such spaces can be not only points but also other geometric shapes. The dimension of space is always equal to the parametric number of the element of space. In applied geometry, interpolation problems have been studied since the 1960s. Since there are so many publications related to interpolation problems, systematization is proposed for their analysis. The largest number of existing publications is devoted to the problems of one-dimensional interpolation in two-dimensional space, both continuous and discrete. A much smaller number of publications are devoted to two-dimensional continuous and discrete interpolation. The variety of interpolation methods is due, first, to different sets of initial geometric conditions. Only in some publications the choice of initial conditions depending on the purpose of interpolation problems in specific sectors of the economy. In many publications, the choice of interpolation method is related to the features of the basic mathematical apparatus. Among the considered publications there is none where the influence of parameters of the set points, namely distances from them to the current point of an interpolant, on its parameter is directly considered. Such interpolation can be the basis for modeling various energy fields, where given points are point sources of energy.

Reiteration Formulae for the Real Interpolation Method Including L or R Limiting Spaces

We consider a real interpolation method defined by means of slowly varying functions. We present some reiteration formulae including so-called L or R limiting interpolation spaces. These spaces arise naturally in reiteration formulae for the limiting cases θ = 0 or θ = 1 . Applications to grand and small Lorentz spaces are given.

Evaluation of Simple Space Interpolation Methods for the Depth of Precipitation: Application for Boyacá, Colombia

Evaluation of Simple Space Interpolation Methods for the Depth of Precipitation: Application for Boyacá, Colombia

Interpolation of martingale Orlicz–Hardy spaces

We investigate the real interpolation spaces between martingale Orlicz–Hardy spaces $$H_{\Phi}^s$$ and martingale Hardy spaces $$H_{\infty}^s$$ via atomic decomposition. Moreover, its corresponding weak version is also established.

Latent-space time evolution of non-intrusive reduced-order models using Gaussian process emulation

Non-intrusive reduced-order models (ROMs) have recently generated considerable interest for constructing computationally efficient counterparts of nonlinear dynamical systems emerging from various domain sciences. They provide a low-dimensional emulation framework for systems that may be intrinsically high-dimensional. This is accomplished by utilizing a construction algorithm that is purely data-driven. It is no surprise, therefore, that the algorithmic advances of machine learning have led to non-intrusive ROMs with greater accuracy and computational gains. However, in bypassing the utilization of an equation-based evolution, it is often seen that the interpretability of the ROM framework suffers. This becomes more problematic when black-box deep learning methods are used which are notorious for lacking robustness outside the physical regime of the observed data. In this article, we propose the use of a novel latent-space interpolation algorithm based on Gaussian process regression. Notably, this reduced-order evolution of the system is parameterized by control parameters to allow for interpolation in space. The use of this procedure also allows for a continuous interpretation of time which allows for temporal interpolation. The latter aspect provides information, with quantified uncertainty, about full-state evolution at a finer resolution than that utilized for training the ROMs. We assess the viability of this algorithm for an advection-dominated system given by the inviscid shallow water equations.

Interpolation in model spaces

In this paper we consider interpolation in model spaces, H 2 ⊖ B H 2 H^2 \ominus B H^2 with B B a Blaschke product. We study unions of interpolating sequences for two sequences that are far from each other in the pseudohyperbolic metric as well as two sequences that are close to each other in the pseudohyperbolic metric. The paper concludes with a discussion of the behavior of Frostman sequences under perturbations.

The Calderón construction for a couple of global Morrey spaces

We employ a new approach to show that the Calderón construction for a couple of global Morrey spaces coincides with the Morrey space with appropriate parameters only under rather strong assumptions on the couples of ideal spaces that parameterize the original Morrey spaces. We show that, in the case of classical examples of global Morrey spaces, these assumptions are necessary and sufficient. Applying a well-known reduction, we use the Calderón construction for a couple of global Morrey spaces to describe the spaces given by the complex interpolation method and also to prove new interpolation theorems for global Morrey spaces.

P.220 Theory of mind brain network works differently during interaction with real and anonymous opponents

This paper investigates for the first time a goal-based angular adaptivity method for thermal radiation transport, suitable for non grey media when the radiation field is coupled with an unsteady flow field through an energy balance. Anisotropic angular adaptivity is achieved by using a Haar wavelet finite element expansion that forms a hierarchical angular basis with compact support and does not require any angular interpolation in space. The novelty of this work lies in (1) the definition of a target functional to compute the goal-based error measure equal to the radiative source term of the energy balance, which is the quantity of interest in the context of coupled flow-radiation calculations; (2) the use of different optimal angular resolutions for each absorption coefficient class, built from a global model of the radiative properties of the medium. The accuracy and efficiency of the goal-based angular adaptivity method is assessed in a coupled flow-radiation problem relevant for air pollution modelling in street canyons. Compared to a uniform Haar wavelet expansion, the adapted resolution uses 5 times fewer angular basis functions and is 6.5 times quicker, given the same accuracy in the radiative source term.

Compressible lattice Boltzmann methods with adaptive velocity stencils: An interpolation-free formulation

Adaptive lattice Boltzmann methods (LBMs) are based on velocity discretizations that self-adjust to local macroscopic conditions such as velocity and temperature. While this feature improves the accuracy and the stability of LBMs for large velocity and temperature variations, it also strongly impacts the efficiency of the algorithm due to space interpolations that are required to get populations at grid nodes. To avoid this defect, the present work proposes new formulations of adaptive LBMs that do not rely anymore on space interpolations, hence drastically improving their parallel efficiency for the simulation of high-speed compressible flows. To reach this goal, the adaptive phase discretization is restricted to particular states that are compliant with the efficient “collide-and-stream” algorithm, and as a consequence, it does not require additional interpolation steps. The development of proper state-adaptive solvers with on-grid propagation imposes new restrictions and challenges on the discrete stencils, namely, the need for an extended operability range allowing for the transition between two phase discretizations. Achieving the minimum operability range for discrete polynomial equilibria requires rather large stencils (e.g., D2Q81, D2Q121) and is therefore not competitive for compressible flow simulations. However, as shown in this article, the use of numerical equilibria can provide for overlaps in the operability ranges of neighboring discrete shifts at acceptable cost using the D2Q21 lattice. Through several numerical validations, the present approach is shown to allow for an efficient realization of discrete state-adaptive LBMs for high Mach number flows even in the low-viscosity regime.

Interpolation of Spaces of Functions of Positive Smoothness on a Domain

An interpolation theorem for spaces of functions of positive smoothness on a domain with flexible cone condition is established.

MULTIVARIATE AFFINE FRACTAL INTERPOLATION

Fractal interpolation functions capture the irregularity of some data very effectively in comparison with the classical interpolants. They yield a new technique for fitting experimental data sampled from real world signals, which are usually difficult to represent using the classical approaches. The affine fractal interpolants constitute a generalization of the broken line interpolation, which appears as a particular case of the linear self-affine functions for specific values of the scale parameters. We study the [Formula: see text] convergence of this type of interpolants for [Formula: see text] extending in this way the results available in the literature. In the second part, the affine approximants are defined in higher dimensions via product of interpolation spaces, considering rectangular grids in the product intervals. The associate operator of projection is considered. Some properties of the new functions are established and the aforementioned operator on the space of continuous functions defined on a multidimensional compact rectangle is studied.

Parallel multi-view polygon rasterization for 3D light field display.

Three-dimensional (3D) light field displays require samples of image data captured from a large number of regularly spaced camera images to produce a 3D image. Generally, it is inefficient to generate these images sequentially because a large number of rendering operations are repeated in different viewpoints. The current 3D image generation algorithm with traditional single viewpoint computer graphics techniques is not sufficiently well suited to the task of generating images for the light field displays. A highly parallel multi-view polygon rasterization (PMR) algorithm for 3D multi-view image generation is presented. Based on the coherence of the triangular rasterization calculation among different viewpoints, the related rasterization algorithms including primitive setup, plane function, and barycentric coordinate interpolation in the screen space are derived. To verify the proposed algorithm, a hierarchical soft rendering pipeline with GPU is designed and implemented. Several groups of images of 3D objects are used to verify the performance of the PMR method, and the correct 3D light field image can be achieved in real time.

Real Interpolation of Hardy-Type Spaces an Announcement with some Remarks

We consider couples (XA, YA) of Hardy-type spaces defined for quasi-Banach lattices of measurable functions on 𝕋×Ω. Under certain fairly general assumptions, the following conditions are shown to be equivalent: (XA, YA) is K-closed in (X, Y), this couple is stable with respect to the real interpolation in the sense that (XA, YA)θ,p = (XA + YA) ∩ (X, Y)θ,p, the inclusion (X1−θYθ)A ⊂ (XA, YA)θ,∞ holds true, and lattices (L1, (Xr)′Yr)δ,q are BMO-regular for some values of the parameters. The last property is weaker than the BMO-regularity of (X, Y), and it requires further study. Some new (compared to the main article) results are given concerning the characterization of this property in terms of the boundedness of the standard harmonic analysis operators such as the Hilbert transform and the Hardy-Littlewood maximal operator.

Latent Timbre Synthesis

We present the Latent Timbre Synthesis (LTS), a new audio synthesis method using Deep Learning. The synthesis method allows composers and sound designers to interpolate and extrapolate between the timbre of multiple sounds using the latent space of audio frames. We provide the details of two Variational Autoencoder architectures for LTS, and compare their advantages and drawbacks. The implementation includes a fully working application with graphical user interface, called \textit{interpolate\_two}, which enables practitioners to explore the timbre between two audio excerpts of their selection using interpolation and extrapolation in the latent space of audio frames. Our implementation is open-source, and we aim to improve the accessibility of this technology by providing a guide for users with any technical background.

The calculation of the sum of the spaces of the K‐method of real interpolation

AbstractIt is shown that the calculation of the sum of the spaces of the K‐Peetre interpolation method can be reduced to the calculation of the sum of the cones of the concave functions included by the parameter in the definition of the spaces of the K‐Peetre interpolation method. As an example, a new extrapolation theorem for operators in Lipschitz spaces is obtained.

On slow decay of Peetre's K-functional

We characterize when Peetre's K-functional slowly decays to zero and we use this characterization to demonstrate certain strict inclusions between real interpolation spaces.

General reiteration theorems for [formula omitted] and [formula omitted] classes: Case of left [formula omitted]-spaces and right [formula omitted]-spaces

Given E0,E1,E,F rearrangement invariant spaces, a,b,b0,b1 slowly varying functions and 0≤θ0<θ1≤1, we characterize the interpolation spaces(X‾θ0,b0,E0,a,FR,X‾θ1,b1,E1)θ,b,E,(X‾θ0,b0,E0,X‾θ1,b1,E1,a,FL)θ,b,E, for all possible values of θ∈[0,1]. As application of our results we prove interpolation formulas for grand and small Lebesgue spaces.

Stability of Fredholm properties on interpolation Banach spaces

The main aim of this paper is to prove novel results on stability of the semi-Fredholm property of operators on interpolation spaces generated by interpolation functors. The methods are based on some general ideas we develop in the paper. This allows us to extend some previous work in literature to the abstract setting. We show an application to interpolation methods introduced by Cwikel–Kalton–Milman–Rochberg which includes, as special cases, the real and complex methods up to equivalence of norms and also some other well known methods of interpolation. A by-product of these results get the stability of isomorphisms on Calderón products of Banach function lattices. We also study the important characteristics in operator Banach space theory, the so-called modules of injection and surjection, and we prove interpolation estimates of these modules of operators on scales of the Calderón complex interpolation spaces.

Initial Boundary Value Problems of Semi-Linear Sub-Diffusion with Gradient Terms

We consider the existence and uniqueness of the saturated classical solutions and the positive classical solutions to initial boundary value problems of semi-linear sub-diffusion with gradient terms. Applying this to the fractional power of the sectorial operator theory and the imbedding theory in the interpolation spaces, where the nonlinear term satisfies more general conditions, we obtain the existence and uniqueness of the saturated classical solutions. The results obtained generalize the recent conclusions on this topic. Finally, an example is given to illustrate the feasibility of our main results.

Complex interpolation of families of Orlicz sequence spaces

We study the complex interpolation and derivation process induced by a family of Orlicz sequence spaces. We present a concrete example of an interpolation family of three spaces inducing a centralizer that cannot be obtained from complex interpolation of two spaces.