Interpolation Spaces Research Articles

DIFFERENTIATION OF SURFACE AND INTERNAL DEFECTS BY USING TAIL FIELD LEAKAGE DATA AT HIGH-SPEED MFL TESTING

In the practice of magnetic flux leakage non-destructive testing the problem of distinguishing signals generated by dangerous internal defects from signals generated by surface flaws is very acute. In this paper on the basis of three-dimensional computer simulation it was shown that at significant inspection speed the area of the maximum magnetization of the most remote from the surface metal layers under the influence of emerging eddy currents is shifted behind the rear pole of the magnetizing system. Studies of magnetic induction distributions in the testing object in the form of a rail from speed, from the distance of the section under consideration relative to the rear pole and from the depth of the layer are carried out. Also, the distributions of the field in the presence of two types of defects – surface and internal – are studied. It was found that the magnetic field distribution behind the rear pole differs significantly from the distribution in the interpole space. Two contrary magnetic fluxes are formed behind the rear pole: in the bulk of the object the magnetic flux moves in one direction and near the surface magnetic flux flows in the opposite one. Since the flux directions are opposite, the signals from surface and internal defects have different polarity, which can be used to differentiate them.

Existence and Uniqueness of Nonmonotone Solutions in Porous Media Flow

Existence and uniqueness of solutions for a simplified model of immiscible two-phase flow in porous media are obtained in this paper. The mathematical model is a simplified physical model with hysteresis in the flux functions. The resulting semilinear hyperbolic-parabolic equation is expected from numerical work to admit non-monotone imbibition-drainage fronts. We prove the local existence of imbibition-drainage fronts. The uniqueness, global existence, maximal regularity and boundedness of the solutions are also discussed. Methodically, the results are established by means of semigroup theory and fractional interpolation spaces.

Adversarial Adaptive Interpolation in Autoencoders for Dually Regularizing Representation Learning

Linear interpolation in the latent space may induce mismatch between the constructed data and the distribution a model was trained on. In this article, we propose an Adversarial Adaptive Interpolation-based AutoEncoder (AdvAI-AE). To constrain the interpolation path on the underlying manifold, an additional interpolation correction module is trained to offset the deviation between the linearly interpolated data points and the statistics of real ones in latent space. Furthermore, we apply prior matching to control the characteristics of the representation. Toward this end, the maximum mean discrepancy-based and adversarial regularizers are incorporated into the model. The synthesized data from random variables are in turn leveraged to regularize the interpolation process. The proposed improvement strategies lead to significant performance gains in downstream classification, clustering, and synthesis tasks on multiple benchmark datasets.

Autonomous construction of parameterizable 3D leaf models from scanned sweet pepper leaves with deep generative networks

Abstract Visible traits can be criteria for selecting a suitable crop. Three-dimensional (3D)-scanned plant models can be used to extract visible traits; however, collecting scanned data and physically manipulating point-cloud structures of the scanned models are difficult. Recently, deep generative models have shown high performance in learning and creating target data. Deep generative models can improve the versatility of scanned models. The objectives of this study were to generate sweet pepper (Capsicum annuum) leaf models and to extract their traits by using deep generative models. The leaves were scanned, preprocessed and used to train the deep generative models. The variational autoencoder, generative adversarial network (GAN) and latent space GAN were used to generate the desired leaves. The optimal number of latent variables in the model was selected via the Jensen–Shannon divergence (JSD). The generated leaves were evaluated by using the JSD, coverage and minimum matching distance to determine the best model for leaf generation. Among the deep generative models, a modified GAN showed the highest performance. Sweet pepper leaves with various shapes were generated from eight latent variables following a normal distribution, and the morphological traits of the leaves were controlled through linear interpolation and simple arithmetic operations in latent space. Simple arithmetic operations and gradual changes in the latent space modified the leaf traits. Deep generative models can parameterize and generate morphological traits in digitized 3D plant models and add realism and diversity to plant phenotyping studies.

Rewriting geometric rules of a GAN

Deep generative models make visual content creation more accessible to novice users by automating the synthesis of diverse, realistic content based on a collected dataset. However, the current machine learning approaches miss a key element of the creative process - the ability to synthesize things that go far beyond the data distribution and everyday experience. To begin to address this issue, we enable a user to "warp" a given model by editing just a handful of original model outputs with desired geometric changes. Our method applies a low-rank update to a single model layer to reconstruct edited examples. Furthermore, to combat overfitting, we propose a latent space augmentation method based on style-mixing. Our method allows a user to create a model that synthesizes endless objects with defined geometric changes, enabling the creation of a new generative model without the burden of curating a large-scale dataset. We also demonstrate that edited models can be composed to achieve aggregated effects, and we present an interactive interface to enable users to create new models through composition. Empirical measurements on multiple test cases suggest the advantage of our method against recent GAN fine-tuning methods. Finally, we showcase several applications using the edited models, including latent space interpolation and image editing.

Trajectory planning of 3-CRU parallel robot with linear kinematics equation

With advantages of linear kinematics equation, no singular pose, regular workspace, the best motion/force transmission performance, 3-CRU parallel robot is potential to perform better when rapid and repetitive grasping/handling in packaging and medicine industry. For this reason, this paper deals with trajectory planning of the 3-CRU parallel robot. The trajectory planning method of the 3-CRU parallel robot is proposed for the first time whose advantage is the robot can arbitrarily increase the interpolation period in the Cartesian space without a twice interpolation in the joint space. This method simplifies the interpolation process of trajectory planning, reduces the occupation of computational resources, improves the computational efficiency of the controller, and benefits the real-time online control. For the problem of residual vibration when the robot stops, the input shaping technology is adopted and time delay compensation strategy is proposed, which improves the positioning accuracy of the moving platform and ensures fixed running time. This paper lays a theoretical and practical foundation for the application of the 3-CRU parallel robot in the future.

The Kalton-Peck space is the complexification of the real Kalton-Peck space

The Kalton-Peck Z2 space is the derived space obtained from the scale of ℓp spaces by complex interpolation at 1/2. If we denote by Z2real the derived space obtained from the scale of ℓp spaces by real interpolation at (1/2,1/2), we show that Z2 is the complexification of Z2real. We also show that Z2real shares the most important properties of Z2: it is isomorphic to its dual, it is singular and contains no complemented copies of ℓ2.

Improvement of the Constructive A Priori Error Estimates for a Fully Discretized Periodic Solution of Heat Equation

Abstract In this study, we present new sharper constructive a priori error estimates for a full-discrete numerical solution of the heat equation that refines our previous work. The full discretization is given using the finite element method in space and linear interpolation in time, similar to that in the previous work. In particular, we adopt an approach based on the effective use of the properties both for exact and discretized time-periodic solutions to establish the error estimates simpler and sharper than the previous work. Furthermore, we derive some time-stepwise error estimates in the space direction. Finally, we present several numerical examples that confirm the actual refinements of the convergence.

Wavelet Characterization of Local Muckenhoupt Weighted Sobolev Spaces with Variable Exponents

The goal of this paper is to define local weighted variable Sobolev spaces of fractional and negative order and their characterization by wavelets. We first consider local weighted variable Sobolev spaces by means of weak derivatives and obtain a wavelet characterization for these spaces. Using the Bessel potentials, we next define local weighted variable Sobolev spaces of fractional order. We show that Sobolev spaces obtained by weak derivatives and those by the Bessel potentials coincide. Finally, using duality, we define local weighted variable Sobolev spaces with negative order. We also show that local weighted variable Sobolev spaces are closed under complex interpolation. Some examples are given including the applications to weighted uniformly local Lebesgue spaces with variable exponents and periodic function spaces as a by-product, although the exponent is constant.

Strong Haagerup inequalities on non-Kac free orthogonal quantum groups

We present natural analogues of strong Haagerup inequalities on non-Kac free orthogonal quantum groups OF+ in which Lp-analytic problems are harder due to their non-tracial nature. Furthermore, we prove optimality of the inequalities, and apply the obtained results to compute the optimal time for ultracontractivity of the heat semigroup and to distinguish the complex interpolation space Lp(OF+) and the real interpolation space Lp,p(OF+).

Weak Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Decompositions, Real Interpolation, and Calderón–Zygmund Operators

Let $$({\mathbb {X}},d,\mu )$$ be a space of homogeneous type in the sense of Coifman and Weiss, and $$X({\mathbb {X}})$$ a ball quasi-Banach function space on $${\mathbb {X}}$$ . In this article, the authors introduce the weak Hardy space $$WH_X({\mathbb {X}})$$ associated with $$X({\mathbb {X}})$$ via the grand maximal function, and characterize $$WH_X({\mathbb {X}})$$ by other maximal functions and atoms. The authors then apply these characterizations to obtain the real interpolation and the boundedness of Calderón–Zygmund operators in the critical case. The main novelties of this article exist in that the authors use the Aoki–Rolewicz theorem and both the dyadic system and the exponential decay of approximations of the identity on $${\mathbb {X}}$$ , which closely connect with the geometrical properties of $${\mathbb {X}}$$ , to overcome the difficulties caused by the deficiency of both the triangle inequality of $$\Vert \cdot \Vert _{X({\mathbb {X}})}$$ and the reverse doubling assumption of the measure $$\mu $$ under consideration, and also use the relation between the convexification of $$X({\mathbb {X}})$$ and the weak ball quasi-Banach function space $$WX({\mathbb {X}})$$ associated with $$X({\mathbb {X}})$$ to prove that the infinite summation of atoms converges in the space of distributions on $${\mathbb {X}}$$ . Moreover, all these results have a wide range of generality and, particularly, even when they are applied to the weighted Lebesgue space, the Orlicz space, and the variable Lebesgue space, the obtained results are also new and, actually, some of them are new even on RD-spaces (namely, spaces of homogeneous type satisfying the additional reverse doubling condition).

The high-order Shifted Boundary Method and its analysis

The Shifted Boundary Method (SBM) is an approximate domain method for boundary value problems, in the broader class of unfitted/embedded/immersed methods, that has proven efficient in handling partial differential equation problems with complex geometries. The key feature of the SBM is a shift in the location where boundary conditions are applied – from the true to a surrogate boundary – and an appropriate modification (again, a shift) of the value of the boundary conditions, in order to reduce the consistency error. This paper presents the high-order version of the method, its mathematical analysis, and numerical experiments. The proposed method retains optimal accuracy for any order of the finite element interpolation spaces despite the surrogate boundary is piecewise linear. As such, the proposed approach bypasses the problematic issue of meshing complex geometries with high-order body-fitted boundary representations, without the need of complex data structures for the integration on cut cells.

Maximal L1-Regularity of Generators for Bounded Analytic Semigroups in Banach Spaces

In this paper, we prove that the generator of any bounded analytic semigroup in (θ, 1)-type real interpolation of its domain and underlying Banach space has maximal L1-regularity, using a duality argument combined with the result of maximal continuous regularity. As an application, we consider maximal L1-regularity of the Dirichlet-Laplacian and the Stokes operator in inhomogeneous B sq,1 -type Besov spaces on domains of ℝn, n ≥ 2.

Stability and large-time behavior of the inviscid Boussinesq system for the micropolar fluid with damping

In this paper, we consider the three-dimensional inviscid Boussinesq system for the micropolar fluid in porous media. We proved the global well-posedness and large time behavior of solutions in the whole space R3. Precisely, when the H3-norm of initial data is small, but the higher-order derivatives can be arbitrarily large, the system is globally well-posed by the pure energy method. Moreover, by a set of mature negative Sobolev and Besov space interpolation methods, the Lp − L2 (1 ≤ p ≤ 2) type of the optimal time decay rates are obtained without any smallness assumption on the Lp norm of the initial data. Our results mathematically explain the stability of the system in an unbounded domain.

Simulating Oil and Water Production in Reservoirs with Generative Deep Learning

Summary This study investigated the ability to produce accurate multiphase flow profiles simulating the response of producing reservoirs, using generative deep learning (GDL) methods. Historical production data from numerical simulators were used to train a variational autoencoder (VAE) algorithm that was then used to predict the output of new wells in unseen locations. This work describes a procedure in which data analysis techniques can be applied to existing historical production profiles to gain insight into field-level reservoir flow behavior. The procedure includes clustering, dimensionality reduction, correlation, in addition to novel interpretation methodologies that synthesize the results from reservoir simulation output, characterizing flow conditions. The insight was then used to build and select samples to train a VAE algorithm that reproduces the multiphase reservoir behavior for unseen operational conditions with high accuracy. Furthermore, using deep feature space interpolation, the trained algorithm can be used to further generate new predictions of the reservoir response under operational conditions for which we do not have previous examples in the training data set. It is found that VAE can be used as a robust multiphase flow simulator. Applying the methodology to the problem of determining multiphase production rate from new producing wells in undrilled locations showed positive results. The methodology was tested successfully in predicting multiphase production under different scenarios including multiwell channelized and heterogeneous reservoirs. Comparison with other shallow supervised algorithms demonstrated improvements realized by the proposed methodology. The study developed a novel methodology to interpret both data and GDL algorithms, geared toward improving reservoir management. The method was able to predict the performance of new wells in previously undrilled locations, potentially without using a reservoir simulator.

3D Stratum Interpolation Algorithm of Metro Tunnel Based on BIM and Data Fusion

The three-dimensional stratum space interpolation algorithm plays a vital role in promoting the safety, management, and decision-making of subway tunnels and is a current research hotspot. The purpose of full text is to study the 3D stratum space interpolation algorithm of subway tunnels and to study the 3D stratum space interpolation algorithm based on building information modeling (BIM) and data fusion technology. This article first uses computer three-dimensional visualization technology to simulate the surface of the ground-fall tunnel based on known data. It is to do the interpolation processing on the two-dimensional space of the subway tunnel to create an accurate surface model of the subway tunnel. It then uses the block model method to divide the internal data of the tunnel according to actual needs. In order to create a model of the internal data of the tunnel, three-dimensional spatial interpolation is performed. After the three-dimensional interpolation is completed, it is convenient to quickly estimate the taste and reserves according to the interpolation results. The experiments in the text show that the average error value of the algorithm in this paper is 38.98 and the performance is improved by 23.4%. The algorithm in the text has obvious advantages in interpolation effect, and the interpolation efficiency is faster. A bird’s eye view of the current study is shown in Figure 1.

On interpolation of reflexive variable Lebesgue spaces on which the Hardy–Littlewood maximal operator is bounded

Abstract We show that if the Hardy–Littewood maximal operator M is bounded on a reflexive variable exponent space L p ⁢ ( ⋅ ) ⁢ ( ℝ d ) {L^{p(\,\cdot\,)}(\mathbb{R}^{d})} , then for every q ∈ ( 1 , ∞ ) {q\in(1,\infty)} , the exponent p ⁢ ( ⋅ ) {p(\,\cdot\,)} admits, for all sufficiently small θ > 0 {\theta>0} , the representation 1 p ⁢ ( x ) = θ q + 1 - θ r ⁢ ( x ) {\frac{1}{p(x)}=\frac{\theta}{q}+\frac{1-\theta}{r(x)}} , x ∈ ℝ d {x\in\mathbb{R}^{d}} , such that the operator M is bounded on the variable Lebesgue space L r ⁢ ( ⋅ ) ⁢ ( ℝ d ) {L^{r(\,\cdot\,)}(\mathbb{R}^{d})} . This result can be applied for transferring properties like compactness of linear operators from standard Lebesgue spaces to variable Lebesgue spaces by using interpolation techniques.

Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic Approach

The problems of polynomial interpolation with several variables present more difficulties than those of one-dimensional interpolation. The first problem is to study the regularity of the interpolation schemes. In fact, it is well-known that, in contrast to the univariate case, there is no universal space of polynomials which admits unique Lagrange interpolation for all point sets of a given cardinality, and so the interpolation space will depend on the set Z of interpolation points. Techniques of univariate Newton interpolating polynomials are extended to multivariate data points by different generalizations and practical algorithms. The Newton basis format, with divided-difference algorithm for coefficients, generalizes in a straightforward way when interpolating at nodes on a grid within certain schemes. In this work, we propose a random algorithm for computing several interpolating multivariate Lagrange polynomials, called RLMVPIA (Random Lagrange Multivariate Polynomial Interpolation Algorithm), for any finite interpolation set. We will use a Newton-type polynomials basis, and we will introduce a new concept called Z , z -partition. All the given algorithms are tested on examples. RLMVPIA is easy to implement and requires no storage.

Impact of Meteorological Uncertainties in the Methane Retrieval Ground Segment of the MERLIN Lidar Mission

MERLIN (MEthane Remote sensing LIdar missioN) is a Franco-German space mission designed to provide weighted columns of atmospheric methane through an inversion of the lidar signal using a priori information on the atmospheric state. Uncertainties about the meteorological parameters of the observed scene used in the ground segment contribute to the error budget on the retrieved methane column. With the LIDSIM (LIDar SIMulator) data simulator and the PROLID (PROcessor LIDar) inversion processor developed for MERLIN, we perform an impact experiment using ECMWF (European Centre for Medium Weather Range Forecast) ensemble forecast data. In addition, we estimate the standard deviation of the error in the methane column due to the meteorological uncertainties to be about 0.6 ppb. In addition, we innovate by discussing the impact of interpolations both in time and space, focusing on vertical extrapolations under the topography by using state-of-the-art methods to determine from the scatter between these methods the range in which the actual profile should be. We conclude that, in areas where the topography variations exceed 10 m over 10 km, an additional random error of 0.1 ppb is due to our lack of knowledge of the adjustment of atmospheric profiles to terrain. Finally, we point out that further work needs to be performed on temporal interpolation. Indeed, the 3 h time interpolation of atmospheric tides can create regional biases of up to 2 ppm (which is a major problem for models trying to identify methane sinks and sources).

3D generation and reconstruction of the fuel cell catalyst layer using 2D images based on deep learning

The catalyst layer (CL) being the site of electrochemical reactions, is the core subunit of the membrane electrode assembly (MEA) in polymer electrolyte fuel cells (PEFCs). Thus, the porous structure of the CL has a significant influence on oxygen transfer resistance and affects the charge/discharge performance. In this study, the three-dimensional (3D) porous structure of the catalyst layer is reconstructed based on the deep convolutional generative adversarial network (DCGAN) deep learning method, utilizing focused ion beam scanning electron microscopy (FIB-SEM) microstructure graphs as training data. Each set of spatial-continuous microstructure graphs, generated by DCGAN with interpolation in latent space, is applied to build a unique 3D microstructure of the CL without the use of real FIB-SEM data. Meanwhile, distinct interpolation conditions in the DCGAN are discussed to optimize the ultimate structure by approaching the structural information to real data, including that of porosity, particle size distribution, and tortuosity. Moreover, the comparison of real and generated structural data reveal that the data generated by DCGAN shows an adjacency relationship with real data, indicating its potential applicability in the field of electrochemical simulation with reduced situational costs. • Application of DCGAN-based method in 3D microstructure generation of catalyst layer • Optimization of interpolation methods by morphology and mass transport properties • Reduction of the computation cost for catalyst layer design in PEFC