Interpolation Spaces Research Articles

A Comparison of U-Net with Conditional Generative Adversarial Networks and Cycle-Consistent Adversarial Networks for real seismic data interpolation: Tupi Field

Deep learning models have been used to improve seismic trace interpolation in recent years. Encode-Decode models, such as U-Net, have been implemented to solve interpolation problems. The success of U-Net in seismic interpolation has inspired us to test the U-net also as an image generator for further Generative Adversarial Network (GAN) interpolation models. The objective of the present paper is to compare the performance of U-Net inside the GAN models for seismic interpolation: the U-Net alone and two GAN models, the conditional GAN (cGAN) and the Cycle Consistent GAN (CycleGAN), both using the U-Net as a generator inside their workflow. We test the methodologies for two scenarios: regular and irregular interpolation. All tests were performed in a real dataset from the Tupi Field which belongs to the Brazilian pre-salt region. A comparison of the statistical metrics shows that cGAN performs better than CycleGAN and the U-Net alone in most cases. The computational training time of the cGAN model, for all interpolation scenarios, is better than the CycleGAN. Finally, the cGAN training time is comparable to the training of the U-Net alone.

A Theory for Interpolation of Metric Spaces

In this work, we develop an interpolation theory for metric spaces inspired by the real method of interpolation. These interpolation spaces preserve Lipschitz operators under certain conditions. We also show that this method, valid in metrics spaces, still holds in normed spaces without any algebraic structure required. Furthermore, this interpolation method for metric spaces when applied to normed spaces is equivalent to the K-method, which has been widely studied in the literature. As an application, we interpolate Fréchet sequence spaces.

Complex interpolation between noncommutative martingale BMO spaces and Hardy–Orlicz spaces

Complex interpolation between noncommutative martingale BMO spaces and Hardy–Orlicz spaces

Non-autonomous fractional nonlocal evolution equations with superlinear growth nonlinearities

We carry out an analysis of the existence of solutions for a class of nonlinear fractional partial differential equations of parabolic type with nonlocal initial conditions. Sufficient conditions for the solvability of the desired problem are presented by transforming it into an abstract non-autonomous fractional evolution equation, and constructing two families of solution operators based on the Mittag-Leffler function, the Mainardi Wright-type function and the analytic semigroup generated by the closed densely defined operator −A(⋅). The discussions are based on the fractional power theory as well as the Banach fixed point theorem in the interpolation space Xpυ (0⩽υ<1, 1<p<∞).

On spaces of integrable functions associated to vector measures and limiting real interpolation

We investigate which spaces are obtained when considering the limiting class of real interpolation spaces (0,q;J) for ordered Banach couples of spaces of (scalar) integrable functions with respect to a vector measure m, defined on a σ-algebra, with values in a Banach space. If m is in particular a finite positive scalar measure, previous known results are derived from ours. Furthermore, we study the interpolation of p-th power factorable operators by the extreme real interpolation method (1,q;K). We also deduce interpolation results for the (1,q;K)-method that apply to other related classes of operators to p-th power factorable operators, such as bidual (p,q)-power-concave operators and q-concave operators.

The Kalton-Peck space as a spreading model

The so-called Kalton-Peck space $Z_2$ is a twisted Hilbert space induced, using complex interpolation, by $c_0$ or $\ell _p$ for any $1\leq p\neq 2&lt;\infty $. Kalton and Peck developed a scheme of results for $Z_2$ showing that it is a very rigid space. For example, every normalized basic sequence in $Z_2$ contains a subsequence which is equivalent to either the Hilbert copy $\ell _2$ or the Orlicz space $\ell _M$. Recently, new examples of twisted Hilbert spaces, which are induced by asymptotic $\ell _p$-spaces, have appeared on the stage. Thus, our aim is to extend the Kalton-Peck theory of $Z_2$ to twisted Hilbert spaces $Z(X)$ induced by asymptotic $c_0$ or $\ell _p$-spaces $X$ for $1\leq p&lt;\infty $. One of the novelties is to use spreading models to gain information on the isomorphic structure of the subspaces of a twisted Hilbert space. As a sample of our results, the only spreading models of $Z(X)$ are $\ell _2$ and $\ell _M$, whenever $X$ is as above and $p\neq 2$.

Interpolation and moduli spaces of vector bundles on very general blowups of the projective plane

In this paper, we study certain moduli spaces of vector bundles on the blowup of the projective plane in at least 10 very general points. Moduli spaces of sheaves on general type surfaces may be nonreduced, reducible and even disconnected. In contrast, moduli spaces of sheaves on minimal rational surfaces and certain del Pezzo surfaces are irreducible and smooth along the locus of stable bundles. We find examples of moduli spaces of vector bundles on more general blowups of the projective plane that are disconnected and have components of different dimensions. In fact, assuming the SHGH Conjecture, we can find moduli spaces with arbitrarily many components of arbitrarily large dimension.

An interpolation result for A_1 weights with applications to fractional Poincaré inequalities

We characterize the real interpolation space between weighted \(L^1\) and \(W^{1,1}\) spaces on arbitrary domains different from \(\mathbb{R}^n\), when the weights are positive powers of the distance to the boundary multiplied by an \(A_1\) weight. As an application of this result we obtain weighted fractional Poincaré inequalities with sharp dependence on the fractional parameter \(s\) (for \(s\) close to 1) and show that they are equivalent to a weighted Poincaré inequality for the gradient.

Real interpolation for variable martingale Hardy–Lorentz–Karamata spaces

In this paper, we study the real interpolation theory for variable Lorentz–Karamata spaces as well as for the corresponding martingale Hardy spaces. As a consequence, the generalization of Doob’s maximal inequality will be proved. Moreover, the atomic decompositions of the variable martingale Hardy–Lorentz–Karamata spaces are also presented. The results obtained here are new even for martingale Hardy–Lorentz–Karamata spaces.

Core decreasing functions

Given a measure space and a totally ordered collection of measurable sets, called an ordered core, the notion of a core decreasing function is introduced and used to define the down space of a Banach function space. This is done using a variant of the Köthe dual restricted to core decreasing functions. To study down spaces, the least core decreasing majorant construction and the level function construction, already known for functions on the real line, are extended to this general setting. These are used to give concrete descriptions of the duals of the down spaces and, in the case of universally rearrangement invariant (u.r.i.) spaces, of the down spaces themselves.The down spaces of L1 and L∞ are shown to form an exact Calderón couple with divisibility constant 1; a complete description of the exact interpolation spaces for the couple is given in terms of level functions; and the down spaces of u.r.i. spaces are shown to be precisely those interpolation spaces that have the Fatou property. The dual couple is also an exact Calderón couple with divisibility constant 1; a complete description of the exact interpolation spaces for the couple is given in terms of least core decreasing majorants; and the duals of down spaces of u.r.i. spaces are shown to be precisely those interpolation spaces that have the Fatou property.

The K-functional and reiteration theorems for Left and Right spaces, Part II

We consider the K-interpolation methods involving slowly varying functions. Let A‾θ,⁎L and A‾θ,⁎R (0≤θ≤1) be the so-called L or R limiting interpolation spaces which arise naturally in reiteration formulae for the limiting cases. We characterize the interpolation spaces (X,Y)θ,r,a (0≤θ≤1), where X=A‾θ0,⁎L or X=A‾θ0,⁎R and Y=A‾θ1,⁎L or Y=A‾θ1,⁎R, provided that 0≤θ0<θ1≤1 and at least one of the parameters θ0 or θ1 has the limiting value θ0=0 or θ1=1. This supplements the first part of the paper, where one of the operands is standard interpolation spaces involving slowly varying functions. The proofs of most reiteration theorems are based on Holmstedt-type formulae. Applications to grand and small Lorentz spaces are given.

Integrating CEDGAN and FCNN for Enhanced Evaluation and Prediction of Plant Growth Environments in Urban Green Spaces

Conducting precise evaluations and predictions of the environmental conditions for plant growth in green spaces is crucial for ensuring their health and sustainability. Yet, assessing the health of urban greenery and the plant growth environment represents a significant and complex challenge within the fields of urban planning and environmental management. This complexity arises from two main challenges: the limitations in acquiring high-density, high-precision data, and the difficulties traditional methods face in capturing and modeling the complex nonlinear relationships between environmental factors and plant growth. In light of the superior spatial interpolation capabilities of CEDGAN (conditional encoder–decoder generative adversarial neural network), notwithstanding its comparative lack of robustness across different subjects, and the excellent ability of FCNN (fully connected neural network) to fit multiple nonlinear equation models, we have developed two models based on these network structures. One model performs high-precision spatial attribute interpolation for urban green spaces, and the other predicts and evaluates the environmental conditions for plant growth within these areas. Our research has demonstrated that, following training with various samples, the CEDGAN network exhibits satisfactory performance in interpolating soil pH values, with an average pixel error below 0.03. This accuracy in predicting both spatial distribution and feature aspects improves with the increase in sample size and the number of controlled sampling points, offering an advanced method for high-precision spatial attribute interpolation in the planning and routine management of urban green spaces. Similarly, FCNN has shown commendable performance in predicting and evaluating plant growth environments, with prediction errors generally less than 0.1. Comparing different network structures, models with fewer hidden layers and nodes yielded superior training outcomes.

Multi-task convex combination interpolation for meta-learning with fewer tasks

Meta-learning methods try to enhance the generalization of the meta-learning model by various tasks. Diverse tasks can provide sufficient knowledge to assist the model in understanding and capturing different types of information. The sufficient number of meta-training tasks is a crucial factor in ensuring the generalization of meta-learning algorithms. However, obtaining a sufficient number of labeled meta-training tasks is often a challenging endeavor in real-world scenarios. Some scholars have addressed the issue of inadequate tasks only by interpolating between two tasks, but they do not take into account the problem of task diversity. To resolve this concern, we propose a Multi-task Convex Combination Interpolation (MCCI) method that simultaneously considers both task quantity and task diversity issues. First, we increase the number of tasks based on interpolation techniques, which provides the possibility to extend it from 2-task interpolation to n-task interpolation. Second, we randomly select multiple tasks and perform convex combination interpolation in the n-task space, in order to increase the task diversity. Third, we prove the positive effect of the convex combination interpolation on the generalization ability and noise resistance of meta-learning algorithms in theory. Finally, we verify the generalization ability and noise resistance of the proposed MCCI on seven datasets. The extensive experimental results show the superior performance of the proposed method compared to the state-of-art methods.

Convergence results in Orlicz spaces for sequences of max-product sampling Kantorovich operators

In this paper, we provide a unifying theory concerning the convergence properties of the so-called max-product sampling Kantorovich operators based upon generalized kernels in the setting of Orlicz spaces. The approximation of functions defined on both bounded intervals and on the whole real axis has been considered. Here, under suitable assumptions on the kernels, considered in order to define the operators, we are able to establish a modular convergence theorem for these sampling-type operators. As a direct consequence of the main theorem of this paper, we obtain that the involved operators can be successfully used for approximation processes in a wide variety of functional spaces, including the well-known interpolation and exponential spaces. This makes the Kantorovich variant of max-product sampling operators suitable for reconstructing not necessarily continuous functions (signals) belonging to a wide range of functional spaces. Finally, several examples of Orlicz spaces and of kernels for which the above theory can be applied, are presented.

Controllability of Mild Solution to Hilfer Fuzzy Fractional Differential Inclusion with Infinite Continuous Delay

This work investigates the solvability of the generalized Hilfer fractional inclusion associated with the solution set of a controlled system of minty type–fuzzy mixed quasi-hemivariational inequality (FMQHI). We explore the assumed inclusion via the infinite delay and the semi-group arguments in the area of solid continuity that sculpts the compactness area. The conformable Hilfer fractional time derivative, the theory of fuzzy sets, and the infinite delay arguments support the solution set’s controllability. We explain the existence due to the convergence properties of Mittage–Leffler functions (Eα,β), that is, hatching the existing arguments according to FMQHI and the continuity of infinite delay, which has not been presented before. To prove the main results, we apply the Leray–Schauder nonlinear alternative thereom in the interpolation of Banach spaces. This problem seems to draw new extents on the controllability field of stochastic dynamic models.

New pipe element based on Hermite-Jackson interpolation

A new pipe finite element is proposed for piping analysis within the framework of linear shell theory. The approach involves the use of a mixed interpolation of classical polynomial and trigonometric polynomial spaces, with trigonometric interpolation performed via Hermite-Jackson polynomials along the mid-surface section. Error estimates are provided and a convergence analysis is performed under specific assumptions on the regularity of the solution. The proposed element is validated through several numerical examples, which demonstrate its accuracy and efficiency in terms of computational cost. This method represents a promising approach for addressing the challenges of piping analysis.

Data-driven models of nonautonomous systems

Nonautonomous dynamical systems are characterized by time-dependent inputs, which complicates the discovery of predictive models describing the spatiotemporal evolution of the state variables of quantities of interest from their temporal snapshots. When dynamic mode decomposition (DMD) is used to infer a linear model, this difficulty manifests itself in the need to approximate the time-dependent Koopman operators. Our approach is to approximate the original nonautonomous system with a modified system derived via a local parameterization of the time-dependent inputs. The modified system comprises a sequence of local parametric systems, which are subsequently approximated by a parametric surrogate model using the DRIPS (dimension reduction and interpolation in parameter space) framework. The offline step of DRIPS relies on DMD to build a linear surrogate model, endowed with reduced-order bases for the observables mapped from training data. The online step interpolates on suitable manifolds to construct a sequence of iterative parametric surrogate models; the target/test parameter points on these manifolds are specified by a local parameterization of the test time-dependent inputs. We use numerical experimentation to demonstrate the robustness of our method and compare its performance with that of deep neural networks.

Mixup-Induced Domain Extrapolation for Domain Generalization

Domain generalization aims to learn a well-performed classifier on multiple source domains for unseen target domains under domain shift. Domain-invariant representation (DIR) is an intuitive approach and has been of great concern. In practice, since the targets are variant and agnostic, only a few sources are not sufficient to reflect the entire domain population, leading to biased DIR. Derived from PAC-Bayes framework, we provide a novel generalization bound involving the number of domains sampled from the environment (N) and the radius of the Wasserstein ball centred on the target (r), which have rarely been considered before. Herein, we can obtain two natural and significant findings: when N increases, 1) the gap between the source and target sampling environments can be gradually mitigated; 2) the target can be better approximated within the Wasserstein ball. These findings prompt us to collect adequate domains against domain shift. For seeking convenience, we design a novel yet simple Extrapolation Domain strategy induced by the Mixup scheme, namely EDM. Through a reverse Mixup scheme to generate the extrapolated domains, combined with the interpolated domains, we expand the interpolation space spanned by the sources, providing more abundant domains to increase sampling intersections to shorten r. Moreover, EDM is easy to implement and be plugged-and-played. In experiments, EDM has been plugged into several methods in both closed and open set settings, achieving up to 5.73% improvement.

Uniform homeomorphisms between spheres induced by interpolation methods

M. Daher [Canad. Math. Bull. 38 (1995), pp. 286–294] showed that if ( X 0 , X 1 ) (X_0, X_1) is a regular couple of uniformly convex spaces then the unit spheres of the complex interpolation spaces X θ X_{\theta } and X η X_{\eta } are uniformly homeomorphic for every 0 &gt; θ , η &gt; 1 0 &gt; \theta , \eta &gt; 1 . We show that this is a rather general phenomenon of the interpolation methods described by the discrete framework of interpolation of Lindemulder and Lorist [A discrete framework for the interpolation of Banach spaces, https://arxiv.org/abs/2105.08373, 2021].

Complex interpolation of function spaces with general weights

Complex interpolation of function spaces with general weights