Interpolation Spaces Research Articles

Stiefel manifold interpolation for non-intrusive model reduction of parameterized fluid flow problems

Many engineering problems are parameterized. In order to minimize the computational cost necessary to evaluate a new operating point, the interpolation of singular matrices representing the data seems natural. Unfortunately, interpolating such data by conventional methods usually leads to unphysical solutions, as the data live on manifolds and not vector spaces. An alternative is to perform the interpolation in the tangent space to the Grassmann manifold to obtain interpolated spatial modes. Temporal modes are afterwards determined via the Galerkin projection of the high-fidelity model onto the interpolated spatial basis. This method, which is known for some fifteen years, is intrusive. Recently, Oulghelou and Allery (JCP, 2021) have proposed a non-intrusive approach (equation-free), but requiring the resolution of two low-dimensional optimization problems after interpolation. In this paper, a non-intrusive alternative based on Interpolation on the Tangent Space of the Stiefel Manifold (ITSSM) is presented. This approach has the advantage of not requiring a calibration phase after interpolation. To assess the method, we compare our results with those obtained using global POD on the one hand, and two methods based on Grassmann interpolation on the other. These comparisons are performed for two classical configurations encountered in fluid dynamics. The first corresponds to the one-dimensional non-linear Burgers' equation. The second example is the two-dimensional cylinder wake flow. We show that the proposed strategy can accurately reconstruct the physical quantities associated with a new operating point. Moreover, the estimation is fast enough to allow real-time computation.

Choreographing multi-degree of freedom behaviors in large-scale crowd simulations

This study introduces a novel framework for choreographing multi-degree of freedom (MDoF) behaviors in large-scale crowd simulations. The framework integrates multi-objective optimization with spatio-temporal ordering to effectively generate and control diverse MDoF crowd behavior states. We propose a set of evaluation criteria for assessing the aesthetic quality of crowd states and employ multi-objective optimization to produce crowd states that meet these criteria. Additionally, we introduce time offset functions and interpolation progress functions to perform complex and diversified behavior state interpolations. Furthermore, we designed a user-centric interaction module that allows for intuitive and flexible adjustments of crowd behavior states through sketching, spline curves, and other interactive means. Qualitative tests and quantitative experiments on the evaluation criteria demonstrate the effectiveness of this method in generating and controlling MDoF behaviors in crowds. Finally, case studies, including real-world applications in the Opening Ceremony of the 2022 Beijing Winter Olympics, validate the practicality and adaptability of this approach.

AlphaFold Meets De Novo Drug Design: Leveraging Structural Protein Information in Multitarget Molecular Generative Models.

Recent advancements in deep learning and generative models have significantly expanded the applications of virtual screening for drug-like compounds. Here, we introduce a multitarget transformer model, PCMol, that leverages the latent protein embeddings derived from AlphaFold2 as a means of conditioning a de novo generative model on different targets. Incorporating rich protein representations allows the model to capture their structural relationships, enabling the chemical space interpolation of active compounds and target-side generalization to new proteins based on embedding similarities. In this work, we benchmark against other existing target-conditioned transformer models to illustrate the validity of using AlphaFold protein representations over raw amino acid sequences. We show that low-dimensional projections of these protein embeddings cluster appropriately based on target families and that model performance declines when these representations are intentionally corrupted. We also show that the PCMol model generates diverse, potentially active molecules for a wide array of proteins, including those with sparse ligand bioactivity data. The generated compounds display higher similarity known active ligands of held-out targets and have comparable molecular docking scores while maintaining novelty. Additionally, we demonstrate the important role of data augmentation in bolstering the performance of generative models in low-data regimes. Software package and AlphaFold protein embeddings are freely available at https://github.com/CDDLeiden/PCMol.

On interpolation spaces of piecewise polynomials on mixed meshes

We consider fractional Sobolev spaces Hθ, θ ∈ (0,1), on 2D domains and H1-conforming discretizations by globally continuous piecewise polynomials on a mesh consisting of shaperegular triangles and quadrilaterals. We prove that the norm obtained from interpolating between the discrete space equipped with the L2-norm on the one hand and the H1-norm on the other hand is equivalent to the corresponding continuous interpolation Sobolev norm, and the norm-equivalence constants are independent of meshsize and polynomial degree. This characterization of the Sobolev norm is then used to show an inverse inequality between H1 and Hθ.

Existence and regularity of strict solutions for a class of fractional evolution equations

AbstractWe study the existence and Hölder regularity of solutions for fractional evolution equations of order . By means of an analytic resolvent, we construct an interpolation space, which can effectively lower the regularity of initial data. By virtue of the interpolation space and some properties of the analytic resolvent, we derive the existence and Hölder regularity of strict solutions for an inhomogeneous problem, as well as the existence and Hölder regularity of a nonlinear problem.

Generating Protein Structures for Pathway Discovery Using Deep Learning.

Resolving the intricate details of biological phenomena at the molecular level is fundamentally limited by both length- and time scales that can be probed experimentally. Molecular dynamics (MD) simulations at various scales are powerful tools frequently employed to offer valuable biological insights beyond experimental resolution. However, while it is relatively simple to observe long-lived, stable configurations of, for example, proteins, at the required spatial resolution, simulating the more interesting rare transitions between such states often takes orders of magnitude longer than what is feasible even on the largest supercomputers available today. One common aspect of this challenge is pathway discovery, where the start and end states of a scientific phenomenon are known or can be approximated, but the mechanistic details in between are unknown. Here, we propose a representation-learning-based solution that uses interpolation and extrapolation in an abstract representation space to synthesize potential transition states, which are automatically validated using MD simulations. The new simulations of the synthesized transition states are subsequently incorporated into the representation learning, leading to an iterative framework for targeted path sampling. Our approach is demonstrated by recovering the transition of a RAS-RAF protein domain (CRD) from membrane-free to interacting with the membrane using coarse-grain MD simulations.

Black holes, complex curves, and graph theory: Revising a conjecture by Kasner

The ratios 8/9=22/3≈0.9428 and 3/2≈0.866 appear in various contexts of black hole physics, as values of the charge-to-mass ratio Q/M or the rotation parameter a/M for Reissner-Nordström and Kerr black holes, respectively. In this work, in the Reissner-Nordström case, I relate these ratios with the quantization of the horizon area, or equivalently of the entropy. Furthermore, these ratios are related to a century-old work of Kasner, in which he conjectured that certain sequences arising from complex analysis may have a quantum interpretation. These numbers also appear in the case of Kerr black holes, but the explanation is not as straightforward. The Kasner ratio may also be relevant for understanding the random matrix and random graph approaches to black hole physics, such as fast scrambling of quantum information, via a bound related to Ramanujan graph. Intriguingly, some other pure mathematical problems in complex analysis, notably complex interpolation in the unit disk, appear to share some mathematical expressions with the black hole problem and thus also involve the Kasner ratio.

Max-Product Sampling Kantorovich Operators: Quantitative Estimates in Functional Spaces

In this paper, we study the order of approximation for max-product sampling Kantorovich operators based upon generalized kernels in the setting of Orlicz spaces. We establish a quantitative estimate for the considered family of sampling-type operators using the Orlicz-type modulus of smoothness, which involves the modular functional of the space. From this result, it is possible to obtain the qualitative order of convergence when functions belonging to suitable Lipschitz classes are considered. On the other hand, in the compact case, we exploit a suitable definition of K-functional in Orlicz spaces in order to provide an upper bound for the approximation error of the involved operators. The treatment in the general framework of Orlicz spaces allows one to obtain a unifying theory on the rate of convergence, as the proved results can be deduced for a wide range of functional spaces, such as L p -spaces, interpolation spaces and exponential spaces.

Feature-Based Interpolation and Geodesics in the Latent Spaces of Generative Models.

Interpolating between points is a problem connected simultaneously with finding geodesics and study of generative models. In the case of geodesics, we search for the curves with the shortest length, while in the case of generative models, we typically apply linear interpolation in the latent space. However, this interpolation uses implicitly the fact that Gaussian is unimodal. Thus, the problem of interpolating in the case when the latent density is non-Gaussian is an open problem. In this article, we present a general and unified approach to interpolation, which simultaneously allows us to search for geodesics and interpolating curves in latent space in the case of arbitrary density. Our results have a strong theoretical background based on the introduced quality measure of an interpolating curve. In particular, we show that maximizing the quality measure of the curve can be equivalently understood as a search of geodesic for a certain redefinition of the Riemannian metric on the space. We provide examples in three important cases. First, we show that our approach can be easily applied to finding geodesics on manifolds. Next, we focus our attention in finding interpolations in pretrained generative models. We show that our model effectively works in the case of arbitrary density. Moreover, we can interpolate in the subset of the space consisting of data possessing a given feature. The last case is focused on finding interpolation in the space of chemical compounds.

Efficient Computation of Geodesics in Color Space.

Although scientists agree that a perceptual color space is not Euclidean and color difference measures, such as CIELAB's ∆E2000, model these aspects of color perception, colormaps are still mostly evaluated through piecewise linear interpolation in a Euclidean color space. In a non-Euclidean setting, the piecewise linear interpolation of a colormap through control points translates to finding shortest paths. Alternatively, a smooth interpolation can be generalized to finding the straightest path. Both approaches are difficult to solve and are compute intensive. We compare the 11 most promising optimization algorithms for the computation of a geodesic either as the shortest or as the straightest path to find the most efficient one to use for colormap interpolation in real-world applications. For two control points, the zero curvature algorithms excelled, especially the 2D relaxation method. For multiple control points, only the mimimal curvature algorithms can produce smooth curves, amongst which the 1D relaxation method performed best.

Analysis of Polynomial Interpolation Characteristics Based on 6DOF Manipulator Trajectory Planning

Aiming at the problems of choosing and using polynomials of different times and mixed polynomials in the process of manipulator trajectory planning, such as difficulty, wide range of light and low accuracy, in this paper, the application of different polynomials in manipulator trajectory planning is systematically analyzed by using MATLAB Robot Toolbox simulation platform and Polynomial interpolation algorithm. Using a 6-dof robotic arm Puma560 as a simulation object, the effects of three-, five-, and seven-Polynomial interpolation on joint position and pose in joint space trajectory planning were analyzed, a preliminary understanding of the application of different polynomials, and secondly, the more complex application of the fifth and seventh degree polynomials in Descartes space for the trajectory planning of the manipulator, the influence of trajectory planning with different degree polynomials on the position and pose of manipulator joint is clarified. The results show that the higher the number of polynomials, the smoother the transition of the joint at the critical path points, but the greater the maximum angular velocity and angular acceleration of the joint, the Polynomial interpolation of joint space and Descartes space trajectory planning are 14% and 23% lower than the average of absolute maximum angular velocity and absolute maximum angular acceleration of the Polynomial interpolation, respectively. Based on the analysis of the research results, this paper presents the applicable conditions and applications of Polynomial interpolation method in trajectory planning, and further summarizes the application conditions and methods of mixed polynomials.

Best Scanline Determination of Pushbroom Images for a Direct Object to Image Space Transformation Using Multilayer Perceptron

Working with pushbroom imagery in photogrammetry and remote sensing presents a fundamental challenge in object-to-image space transformation. For this transformation, accurate estimation of Exterior Orientation Parameters (EOPs) for each scanline is required. To tackle this challenge, Best Scanline Search or Determination (BSS/BSD) methods have been developed. However, the current BSS/BSD methods are not efficient for real-time applications due to their complex procedures and interpolations. This paper introduces a new non-iterative BSD method specifically designed for line-type pushbroom images. The method involves simulating a pair of sets of points, Simulated Control Points (SCOPs), and Simulated Check Points (SCPs), to train and test a Multilayer Perceptron (MLP) model. The model establishes a strong relationship between object and image spaces, enabling a direct transformation and determination of best scanlines. This proposed method does not rely on the Collinearity Equation (CE) or iterative search. After training, the MLP model is applied to the SCPs for accuracy assessment. The proposed method is tested on ten images with diverse landscapes captured by eight sensors, exploiting five million SCPs per image for statistical assessments. The Root Mean Square Error (RMSE) values range between 0.001 and 0.015 pixels across ten images, demonstrating the capability of achieving the desired sub-pixel accuracy within a few seconds. The proposed method is compared with conventional and state-of-the-art BSS/BSD methods, indicating its higher applicability regarding accuracy and computational efficiency. These results position the proposed BSD method as a practical solution for transforming object-to-image space, especially for real-time applications.

Stepping into high-order interpolation in space

Stepping into high-order interpolation in space

P. Jones' interpolation theorem for noncommutative martingale Hardy spaces II

AbstractLet be a semifinite von Neumann algebra equipped with an increasing filtration of (semifinite) von Neumann subalgebras of . For , let denote the noncommutative column martingale Hardy space constructed from column square functions associated with the filtration and the index . We prove the following real interpolation identity: If and , then This is new even for classical martingale Hardy spaces as it is previously known only under the assumption that the filtration is regular. We also obtain an analogous result for noncommutative column martingale Orlicz–Hardy spaces.

Fractional operators with homogeneous kernel on the Calderón product of Morrey spaces

AbstractWe investigate fractional operators with homogeneous kernel in Morrey spaces. In particular, we prove that fractional integral operators and fractional maximal operators with homogeneous kernel are bounded from the Calderón product of Morrey spaces to certain Morrey spaces. Our results can be seen as a generalization of a recent result on the relation between the boundedness of (classical) fractional operators and interpolation of Morrey spaces. What is new about this paper is not only the passage from the classical fractional integral operators to the rough integral operators. Even the case of fractional integral operators, handled in earlier papers, is significantly simplified.

CoMix: Confronting with Noisy Label Learning with Co-training Strategies on Textual Mislabeling

The existence of noisy labels is inevitable in real-world large-scale corpora. As deep neural networks are notably vulnerable to overfitting on noisy samples, this highlights the importance of the ability of language models to resist noise for efficient training. However, little attention has been paid to alleviating the influence of label noise in natural language processing. To address this problem, we present CoMix, a robust Noise-Against training strategy taking advantage of Co-training that deals with textual annotation errors in text classification tasks. In our proposed framework, the original training set is first split into labeled and unlabeled subsets according to a sample partition criteria and then applies label refurbishment on the unlabeled subsets. We implement textual interpolation in hidden space between samples on the updated subsets. Meanwhile, we employ peer diverged networks simultaneously leveraging co-training strategies to avoid the accumulation of confirm bias. Experimental results on three popular text classification benchmarks demonstrate the effectiveness of CoMix in bolstering the network’s resistance to label mislabeling under various noise types and ratios, which also outperforms the state-of-the-art methods.

Randomized Least-Squares with Minimal Oversampling and Interpolation in General Spaces

Randomized Least-Squares with Minimal Oversampling and Interpolation in General Spaces

Projective tensor products of approximation spaces associated with positive operators

In this paper the projective tensor products of approximation spaces associated with positive operators in Banach spaces are characterized. We show that the tensor products of approximation spaces can be considered as the interpolation spaces generated by $K$-method of real interpolation. The inequalities that provide a sharp estimates of best approximations by analytic vectors of positive operators on projective tensor products are established. Application to spectral approximations of the regular elliptic operators on projective tensor products of Lebesgue spaces is shown.

A national-scale hybrid model for enhanced streamflow estimation – consolidating a physically based hydrological model with long short-term memory (LSTM) networks

Abstract. Accurate streamflow estimation is essential for effective water resource management and adapting to extreme events in the face of changing climate conditions. Hydrological models have been the conventional approach for streamflow interpolation and extrapolation in time and space for the past few decades. However, their large-scale applications have encountered challenges, including issues related to efficiency, complex parameterization, and constrained performance. Deep learning methods, such as long short-term memory (LSTM) networks, have emerged as a promising and efficient approach for large-scale streamflow estimation. In this study, we have conducted a series of experiments to identify optimal hybrid modeling schemes to consolidate physically based models with LSTM aimed at enhancing streamflow estimation in Denmark. The results show that the hybrid modeling schemes outperformed the Danish National Water Resources Model (DKM) in both gauged and ungauged basins. While the standalone LSTM rainfall–runoff model outperformed DKM in many basins, it faced challenges when predicting the streamflow in groundwater-dependent catchments. A serial hybrid modeling scheme (LSTM-q), which used DKM outputs and climate forcings as dynamic inputs for LSTM training, demonstrated higher performance. LSTM-q improved the mean Nash–Sutcliffe efficiency (NSE) by 0.22 in gauged basins and 0.12 in ungauged basins compared to DKM. Similar accuracy improvements were achieved with alternative hybrid schemes, i.e., by predicting the residuals between DKM-simulated streamflow and observations using LSTM. Moreover, the developed hybrid models enhanced the accuracy of extreme events, which encourages the integration of hybrid models within an operational forecasting framework. This study highlights the advantages of synergizing existing physically based hydrological models (PBMs) with LSTM models, and the proposed hybrid schemes hold the potential to achieve high-quality large-scale streamflow estimations.

Stress Wave Hybrid Imaging for Detecting Wood Internal Defects under Sparse Signals

Stress wave technology is very suitable for detecting internal defects of standing trees, logs, and wood and has gradually become the mainstream technology in this research field. Usually, 12 sensors are positioned equidistantly around the cross-section of tree trunks in order to obtain enough stress wave signals. However, the arrangement of sensors is time-consuming and laborious, and maintaining the accuracy of stress wave imaging under sparse signals is a challenging problem. In this paper, a novel stress wave hybrid imaging method based on compressive sensing and elliptic interpolation is proposed. The spatial structure of the defective area is reconstructed by using the advantages of compressive sensing in sparse signal representation and solution of stress waves, and the healthy area is reconstructed by using the elliptic space interpolation method. Then, feature points are selected and mixed for imaging. The comparative experimental results show that the overall imaging accuracy of the proposed method reaches 89.7%, and the high-quality imaging effect can be guaranteed when the number of sensors is reduced to 10, 8, or even 6.