High-dimensional Domains Research Articles

Multiplication operators on the Bergman space of bounded domains

In this paper we study multiplication operators on Bergman spaces of high dimensional bounded domains and those von Neumann algebras induced by them via the geometry of domains and function theory of their symbols. In particular, using local inverses and La2-removability, we show that for a holomorphic proper map Φ=(ϕ1,ϕ2,⋯,ϕd) on a bounded domain Ω in Cd, the dimension of the von Neumann algebra V⁎(Φ,Ω) consisting of bounded operators on the Bergman space La2(Ω), which commute with both Mϕj and its adjoint Mϕj⁎ for each j, equals the number of components of the complex manifold SΦ={(z,w)∈Ω2:Φ(z)=Φ(w),z∉Φ−1(Φ(Z))}, where Z is the zero variety of the Jacobian JΦ of Φ. This extends the main result in [14] in high dimensional complex domains. Moreover we show that the von Neumann algebra V⁎(Φ,Ω) may not be abelian in general although Douglas, Putinar and Wang [15] showed that V⁎(Φ,D) for the unit disk D is abelian.

C^0 Generalized Coons Patches for High-order Cage-based Deformation

Space deformations deform the ambient space and thus implicitly deform the embedded objects. Free-Form Deformation allows high-order deformation to the embedding space, yet the lattice may fail to conform to the object and involves many internal control points. Cage-based Deformation utilizes a cage space that conforms to the object, obviating the need for additional internal control points, but it is typically linear at edges. In this paper, we propose a simple and general method with both advantages while avoiding their drawbacks, allowing users to implement high-order cage-based deformation. To achieve this goal, we introduce a new parametric transfinite interpolation scheme based on generalized barycentric coordinates, which unifies and generalizes the rectangular and triangular Coons patch. This C 0 Generalized Coons patch can be defined not only over 2D domains but also 3D domains or even higher-dimensional domains, with arbitrary polytopes, even including non-manifold topologies. Moreover, the C 0 Generalized Coons patch has an elegant mathematical expression.

Long-time behavior of solutions for time-periodic reaction-diffusion equations and applications

This article concerns the asymptotic behavior of solutions for time-periodic reaction-diffusion equations with a drift term in one dimensional space. Assuming that drift term is decaying, we analyze the effect of the decaying rate of the drift term on the propagation speed of solutions. Also we show some applications of our results in high-dimensional domains. For more information see https://ejde.math.txstate.edu/Volumes/2024/69/abstr.html

Functional Data Analysis: An Introduction and Recent Developments.

Functional data analysis (FDA) is a statistical framework that allows for the analysis of curves, images, or functions on higher dimensional domains. The goals of FDA, such as descriptive analyses, classification, and regression, are generally the same as for statistical analyses of scalar-valued or multivariate data, but FDA brings additional challenges due to the high- and infinite dimensionality of observations and parameters, respectively. This paper provides an introduction to FDA, including a description of the most common statistical analysis techniques, their respective software implementations, and some recent developments in the field. The paper covers fundamental concepts such as descriptives and outliers, smoothing, amplitude and phase variation, and functional principal component analysis. It also discusses functional regression, statistical inference with functional data, functional classification and clustering, and machine learning approaches for functional data analysis. The methods discussed in this paper are widely applicable in fields such as medicine, biophysics, neuroscience, and chemistry and are increasingly relevant due to the widespread use of technologies that allow for the collection of functional data. Sparse functional data methods are also relevant for longitudinal data analysis. All presented methods are demonstrated using available software in R by analyzing a dataset on human motion and motor control. To facilitate the understanding of the methods, their implementation, and hands-on application, the code for these practical examples is made available through a code and data supplement and on GitHub.

Global solvability and asymptotic behavior of solutions for a fully parabolic nutrient taxis system

In this paper, we consider the fully parabolic nu’trient taxis system: ut = d1Δu − ∇ · (ϕ(u, v)∇v), vt = d2Δv − ξug(v) − μv + r(x, t), x ∈ Ω, t > 0 under homogeneous Neumann boundary conditions in a convex bounded domain with smooth boundary. We show that the system possesses a global bounded classical solution in domains of arbitrary dimension and at least one global generalized solution in high-dimensional domain. In addition, the asymptotic behavior of generalized solutions is discussed. Our results not only generalize and partly improve upon previously known findings but also introduce new insights.

Approximate averaged derivative moments and stochastic response of high-DOF quasi-non-integrable Hamiltonian systems under Poisson white noises

Stochastic response prediction is one of the principal topics in diverse branches of engineering, such as aerospace engineering, mechanical engineering, and vehicle engineering. However, the challenge of addressing high-dimensional partial differential equations or evaluating high-dimensional domain integrals in determining averaged derivative moments has limited the research. This study proposes a method to predict the stochastic response of high-degree-of-freedom (DOF) quasi-non-integrable Hamiltonian systems under Poisson white noises. The original finite DOF system is reduced to a one-dimensional one governing the Hamiltonian by applying the stochastic averaging method, and the associated reduced generalized Fokker–Planck–Kolmogorov (GFPK) equation is derived. Then, the two-step generalized elliptical coordinate transformation is extended to evaluate averaged derivative moments that are the key to obtaining the stationary probability density function (SPDF) for probabilistic information prediction. Moreover, the perturbation method is introduced to address the high-order GFPK equation. Finally, an 8-DOF strongly nonlinear system under purely additive Poisson white noises is given as an example to highlight the accuracy of the proposed method. Results from the proposed method show good agreement with those from Monte Carlo simulations, which indicates that the proposed method can be applied to assess the response of high-DOF quasi-non-integrable Hamiltonian systems under Poisson white noises. In addition, if the potential energy of an arbitrary finite n-DOF system is the same form as that in Appendix B, pertinent high-dimensional domain integrals are converted into explicit expressions when n is even.

Learning nonlinear operators in latent spaces for real-time predictions of complex dynamics in physical systems

Predicting complex dynamics in physical applications governed by partial differential equations in real-time is nearly impossible with traditional numerical simulations due to high computational cost. Neural operators offer a solution by approximating mappings between infinite-dimensional Banach spaces, yet their performance degrades with system size and complexity. We propose an approach for learning neural operators in latent spaces, facilitating real-time predictions for highly nonlinear and multiscale systems on high-dimensional domains. Our method utilizes the deep operator network architecture on a low-dimensional latent space to efficiently approximate underlying operators. Demonstrations on material fracture, fluid flow prediction, and climate modeling highlight superior prediction accuracy and computational efficiency compared to existing methods. Notably, our approach enables approximating large-scale atmospheric flows with millions of degrees, enhancing weather and climate forecasts. Here we show that the proposed approach enables real-time predictions that can facilitate decision-making for a wide range of applications in science and engineering.

Steady States with Jump Discontinuity in a Receptor-Based Model with Hysteresis in Higher-Dimensional Domains

Steady States with Jump Discontinuity in a Receptor-Based Model with Hysteresis in Higher-Dimensional Domains

Compressive Fourier collocation methods for high-dimensional diffusion equations with periodic boundary conditions

Abstract High-dimensional partial differential equations (PDEs) are a popular mathematical modelling tool, with applications ranging from finance to computational chemistry. However, standard numerical techniques for solving these PDEs are typically affected by the curse of dimensionality. In this work, we tackle this challenge while focusing on stationary diffusion equations defined over a high-dimensional domain with periodic boundary conditions. Inspired by recent progress in sparse function approximation in high dimensions, we propose a new method called compressive Fourier collocation. Combining ideas from compressive sensing and spectral collocation, our method replaces the use of structured collocation grids with Monte Carlo sampling and employs sparse recovery techniques, such as orthogonal matching pursuit and $\ell ^1$ minimization, to approximate the Fourier coefficients of the PDE solution. We conduct a rigorous theoretical analysis showing that the approximation error of the proposed method is comparable with the best $s$-term approximation (with respect to the Fourier basis) to the solution. Using the recently introduced framework of random sampling in bounded Riesz systems, our analysis shows that the compressive Fourier collocation method mitigates the curse of dimensionality with respect to the number of collocation points under sufficient conditions on the regularity of the diffusion coefficient. We also present numerical experiments that illustrate the accuracy and stability of the method for the approximation of sparse and compressible solutions.

VPP-ART: An Efficient Implementation of Fixed-Size-Candidate-Set Adaptive Random Testing Using Vantage Point Partitioning

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Adaptive random testing</i> (ART) is an enhancement of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">random testing</i> (RT), and aims to improve the RT failure-detection effectiveness by distributing test cases more evenly in the input domain. Many ART algorithms have been proposed, with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fixed-size-candidate-set</i> ART (FSCS-ART) being one of the most effective and popular. FSCS-ART ensures high failure-detection effectiveness by selecting as the next test case the candidate farthest from previously executed test cases. Although FSCS-ART has good failure-detection effectiveness, it also faces some challenges, including heavy computational overheads. In this article, we propose an enhanced version of FSCS-ART, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vantage point partitioning ART</i> (VPP-ART). VPP-ART addresses the FSCS-ART computational overhead problem using VPP, while maintaining the failure-detection effectiveness. VPP-ART partitions the input domain space using a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">modified vantage point tree</i> (VP-tree) and finds the approximate nearest executed test cases of a candidate test case in the partitioned subdomains—thereby significantly reducing the time overheads compared with the searches required for FSCS-ART. To enable the FSCS-ART dynamic insertion process, we modify the traditional VP-tree to support dynamic data. The simulation results show that VPP-ART has a much lower time overhead compared to FSCS-ART, but also delivers similar (or better) failure-detection effectiveness, especially in the higher dimensional input domains. According to statistical analyses, VPP-ART can improve on the FSCS-ART failure-detection effectiveness by approximately 50–58%. VPP-ART also compares favorably with the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">KD-tree-enhanced fixed-size-candidate-set ART</i> (KDFC-ART) algorithms (a series of enhanced ART algorithms based on the KD-tree). Our experiments also show that VPP-ART is more cost-effective than FSCS-ART and KDFC-ART.

Innovations in Computational Approaches for Nonlinear Problems and Complex System Simulations

Nonlinear problem solving and complex system simulation have become critical issues in many fields of science. The development of novel computational methods is crucial to understanding these complex systems. In this abstract, we explore the dynamic landscape of computational techniques, focusing on their uses in simulating complex systems and tackling nonlinear challenges. Creating complex algorithms that can deal with nonlinearity, chaos, and emergent behaviours is where it's at. Tools for modelling and comprehending such complex systems are few, but machine learning, artificial intelligence, and evolutionary computation are at the forefront. The way problems are solved has been completely rethought because of their nonlinearity-tolerance and ability to operate in high-dimensional domains. In addition, novel opportunities have arisen due to the combination of classical mathematical models with computer methods. The behaviour and emergent features of complex systems are best understood by hybrid approaches that combine differential equations, agent-based modelling, and cellular automata. These techniques provide a fine-grained comprehension of component interactions, illuminating emergent events. Moreover, the advent of high-performance computing has substantially expanded the breadth and resolution of simulations. Scientists are now able to probe increasingly complex systems, shedding light on their dynamics and behaviours. Computational capacities have been vastly improved by parallel computing, distributed systems, and cloud computing infrastructures, allowing for the study of systems that were once thought to be intractable. Nonlinear issues and complex system simulations can benefit greatly from the combination of cutting-edge computational approaches with domain-specific expertise. This abstract is a testament to the expanding significance and potential of these computational approaches in understanding complex systems and opening up new frontiers for research and solving problems.

H-Analysis and data-parallel physics-informed neural networks

We explore the data-parallel acceleration of physics-informed machine learning (PIML) schemes, with a focus on physics-informed neural networks (PINNs) for multiple graphics processing units (GPUs) architectures. In order to develop scale-robust and high-throughput PIML models for sophisticated applications which may require a large number of training points (e.g., involving complex and high-dimensional domains, non-linear operators or multi-physics), we detail a novel protocol based on h-analysis and data-parallel acceleration through the Horovod training framework. The protocol is backed by new convergence bounds for the generalization error and the train-test gap. We show that the acceleration is straightforward to implement, does not compromise training, and proves to be highly efficient and controllable, paving the way towards generic scale-robust PIML. Extensive numerical experiments with increasing complexity illustrate its robustness and consistency, offering a wide range of possibilities for real-world simulations.

A high resolution Physics-informed neural networks for high-dimensional convection–diffusion–reaction equations

In practical problems, some partial differential equations defined in high-dimensional domains or complex surfaces are difficult to calculate by traditional methods. In this paper, a novel data-driven deep learning algorithm is proposed to solve high-dimensional convection–diffusion–reaction equations. The main idea of the method is to use the neural network which combines the physical characteristics of the equation to get high accuracy numerical solution. The proposed method not only avoids the high cost of mesh generation, but also effectively reduces the numerical oscillation caused by the domination of the convection. In addition, two types of loss functions are designed to force physical properties, such as the positivity or maximum principle of the solution. Various numerical examples are performed to demonstrate the validity and accuracy of the proposed method.

Causal Effect Identification in Cluster DAGs

Reasoning about the effect of interventions and counterfactuals is a fundamental task found throughout the data sciences. A collection of principles, algorithms, and tools has been developed for performing such tasks in the last decades. One of the pervasive requirements found throughout this literature is the articulation of assumptions, which commonly appear in the form of causal diagrams. Despite the power of this approach, there are significant settings where the knowledge necessary to specify a causal diagram over all variables is not available, particularly in complex, high-dimensional domains. In this paper, we introduce a new graphical modeling tool called cluster DAGs (for short, C-DAGs) that allows for the partial specification of relationships among variables based on limited prior knowledge, alleviating the stringent requirement of specifying a full causal diagram. A C-DAG specifies relationships between clusters of variables, while the relationships between the variables within a cluster are left unspecified, and can be seen as a graphical representation of an equivalence class of causal diagrams that share the relationships among the clusters. We develop the foundations and machinery for valid inferences over C-DAGs about the clusters of variables at each layer of Pearl's Causal Hierarchy - L1 (probabilistic), L2 (interventional), and L3 (counterfactual). In particular, we prove the soundness and completeness of d-separation for probabilistic inference in C-DAGs. Further, we demonstrate the validity of Pearl's do-calculus rules over C-DAGs and show that the standard ID identification algorithm is sound and complete to systematically compute causal effects from observational data given a C-DAG. Finally, we show that C-DAGs are valid for performing counterfactual inferences about clusters of variables.

Sharp error estimates for spatial-temporal finite difference approximations to fractional sub-diffusion equation without regularity assumption on the exact solution

Finite difference method as a popular numerical method has been widely used to solve fractional diffusion equations. In the general spatial error analyses, an assumption $$u\in C^{4}({\bar{\varOmega }})$$ is needed to preserve $${\mathcal {O}}(h^{2})$$ convergence when using central finite difference scheme to solve fractional sub-diffusion equation with Laplace operator, but this assumption is somewhat strong, where u is the exact solution and h is the mesh size. In this paper, a novel analysis technique is proposed to show that the spatial convergence rate can reach $${\mathcal {O}}(h^{\min (\sigma +\frac{1}{2}-\epsilon ,2)})$$ in both $$l^{2}$$ -norm and $$l^{\infty }$$ -norm in one-dimensional domain when the initial value and source term are both in $${\dot{H}}^{\sigma }(\varOmega )$$ but without any regularity assumption on the exact solution, where $$\sigma \ge 0$$ and $$\epsilon >0$$ being arbitrarily small. After making slight modifications on the scheme, acting on the initial value and source term, the spatial convergence rate can be improved to $${\mathcal {O}}(h^{2})$$ in $$l^{2}$$ -norm and $${\mathcal {O}}(h^{\min (\sigma +\frac{3}{2}-\epsilon ,2)})$$ in $$l^{\infty }$$ -norm. It is worth mentioning that our spatial error analysis is applicable to high dimensional cube domain by using the properties of tensor product. Moreover, two kinds of averaged schemes are provided to approximate the Riemann–Liouville fractional derivative, and $${\mathcal {O}}(\tau ^{2})$$ convergence is obtained for all $$\alpha \in (0,1)$$ . Finally, some numerical experiments verify the effectiveness of the built theory.

Synthetic Model Combination: A new machine-learning method for pharmacometric model ensembling.

When aiming to make predictions over targets in the pharmacological setting, a data-focused approach aims to learn models based on a collection of labeled examples. Unfortunately, data sharing is not always possible, and this can result in many different models trained on disparate populations, leading to the natural question of how best to use and combine them when making a new prediction. Previous work has focused on global model selection or ensembling, with the result of a single final model across the feature space. Machine-learning models perform notoriously poorly on data outside their training domain, however, due to a problem known as covariate shift, and so we argue that when ensembling models the weightings for individual instances must reflect their respective domains-in other words, models that are more likely to have seen information on that instance should have more attention paid to them. We introduce a method for such an instance-wise ensembling of models called Synthetic Model Combination (SMC), including a novel representation learning step for handling sparse high-dimensional domains. We demonstrate the use of SMC on an example with dosing predictions for vancomycin, although emphasize the applicability of the method to any scenario involving the use of multiple models.

Classification of Working Memory Loads via Assessing Broken Detailed Balance of EEG-fNIRS Neurovascular Coupling Measures.

Human brain breaks the detailed balance to drive a variety of cognitive functions, such as memory. Recently, a promising classification framework of working memory loads has been proposed based on functional magnetic resonance imaging (fMRI) data with relative entropy (RE) measurement to quantify the broken detailed balance of human brain. However, there are limitations in previousely developed methods. First, single-modality fMRI can only detect the cerebral hemodynamics but not the neuronal activity, lacking detailed information of the neurovascular coupling process in the brain. Second, the RE measurement utilized to quantify the broken detailed balance may be biased and have no finite upper bound, limiting its application in high dimensional signal domains. In this study, a neurovascular coupling strategy based on concurrent electroencephalography (EEG) and functional near-infrared spectroscopy (fNIRS) recordings was proposed to take both the cerebral hemedynamics and neuronal activity into consideration in assessing broken detailed balance of the brain. Furthermore, the generalized relative entropy (GRE) was employed to reduce the bias associated with the conventional RE measure. Our results demonstrated that the proposed framework showed higher classification accuracy (82.48%) to identify different levels of working memory loads than conventional methods. In addition, our results revealed that the broken detailed balance was significantly stronger when subjects performed cognitively demanding tasks (P<0.05) and was highly correlated with the neurovascular coupling models derived from the EEG θ and α bands, respectively. In conclusion, our findings provide an advanced framework to accurately classify various levels of working memory with the broken detailed balance of human brain and can be extended to explore the underlying broken detailed balance related to other cognitive behaviors and diseases.

Neural-signature methods for structured EHR prediction

Models that can effectively represent structured Electronic Healthcare Records (EHR) are central to an increasing range of applications in healthcare. Due to the sequential nature of health data, Recurrent Neural Networks have emerged as the dominant component within state-of-the-art architectures. The signature transform represents an alternative modelling paradigm for sequential data. This transform provides a non-learnt approach to creating a fixed vector representation of temporal features and has shown strong performances across an increasing number of domains, including medical data. However, the signature method has not yet been applied to structured EHR data. To this end, we follow recent work that enables the signature to be used as a differentiable layer within a neural architecture enabling application in high dimensional domains where calculation would have previously been intractable. Using a heart failure prediction task as an exemplar, we provide an empirical evaluation of different variations of the signature method and compare against state-of-the-art baselines. This first application of neural-signature methods in real-world healthcare data shows a competitive performance when compared to strong baselines and thus warrants further investigation within the health domain.

Secondary bifurcations in semilinear ordinary differential equations

We consider the Neumann problem for the equation $$u_{xx}+{\uplambda }f(u)=0$$ in the punctured interval $$(-1,1) \setminus \{0\}$$ , where $${\uplambda }>0$$ is a bifurcation parameter and $$f(u)=u-u^3$$ . At $$x=0$$ , we impose the conditions $$u(-0)+au_x(-0)=u(+0)-au_x(+0)$$ and $$u_x(-0)=u_x(+0)$$ for a constant $$a>0$$ (the symbols $$+0$$ and $$-0$$ stand for one-sided limits). The problem appears as a limiting equation for a semilinear elliptic equation in a higher dimensional domain shrinking to the interval $$(-1,1)$$ . First we prove that odd solutions and even solutions form families of branches $$\{ {{\mathcal {C}}}^o_k\}_{k \in {{\mathbb {N}}}}$$ and $$\{ {{\mathcal {C}}}^e_k\}_{k \in {{\mathbb {N}}}}$$ , respectively. Both $${{\mathcal {C}}}^o_k$$ and $${{\mathcal {C}}}^e_k$$ bifurcate from the trivial solution $$u=0$$ . We then show that $${{\mathcal {C}}}^e_k$$ contains no other bifurcation point, while $${{\mathcal {C}}}^o_k$$ contains two points where secondary bifurcations occur. Finally we determine the Morse index of solutions on the branches. General conditions on f(u) for the same assertions to hold are also given.

Extending FEniCS to work in higher dimensions using tensor product finite elements

We present a method to extend the finite element library FEniCS to solve problems with domains in dimensions above three by constructing tensor product finite elements. This methodology only requires that the high dimensional domain is structured as a Cartesian product of two lower dimensional subdomains. In this study we consider Dirichlet problems for scalar linear partial differential equations, though the methodology can be extended to non-linear problems. The utilization of tensor product finite elements allows us to construct a global system of linear algebraic equations that only relies on the finite element infrastructure of the lower dimensional subdomains contained in FEniCS. We demonstrate the effectiveness of our methodology in four distinctive test cases. The first test case is a Poisson equation posed in a four dimensional domain which is a Cartesian product of two unit squares solved using the classical Galerkin finite element method. The second test case is the wave equation in space–time, where the computational domain is a Cartesian product of a two dimensional space grid and a one dimensional time interval. In this second case we also employ the Galerkin method. The third test case is an advection dominated advection–diffusion equation where the global domain is a Cartesian product of two one dimensional intervals in which the streamline upwind Petrov–Galerkin method is applied to ensure discrete stability. The final test case uses the Galerkin approach to solve a Poisson problem on a Cartesian product of two intervals with a spatially varying, non-separable diffusivity term In all cases, a p=1 basis is used and optimal L2 convergence rates of order hp+1 of the errors are achieved with respect to h refinement.