Unfolding Operator Research Articles

A fast Lanczos-based hierarchical algorithm for tensor ring decomposition

Tensor ring (TR) decomposition has made remarkable achievements in numerous high-order data processing tasks. However, the current alternating least squares (ALS)- and singular value decomposition (SVD)-based algorithms for TR decomposition, i.e., TR-ALS and TR-SVD, especially the former, are computationally expensive, making them unfriendly for large-scale data processing. This paper adopts three strategies to propose a novel fast TR decomposition algorithm: (1) Use a more efficient Lanczos bidiagonalization algorithm than SVD to generate the TR core tensors. (2) Exploit the hierarchical strategy to generate the TR core tensors in parallel. (3) Employ new reshaping and unfolding operations to reduce the dimensionality of the data used to generate TR core tensors. By incorporating these three strategies, we propose the TR-HLanczos algorithm for fast TR decomposition. This algorithm seamlessly produces the TR core tensors through the Lanczos bidiagonalization algorithm in a hierarchical manner. The effectiveness of the proposed TR-HLanczos algorithm is demonstrated through experimental results on both highly oscillatory functions and real-world datasets. For instance, when dealing with data of size 505, TR-HLanczos is nearly 561 times and 18 times faster than algorithms based on ALS and SVD, respectively.

Dimension reduction and homogenization of composite plate with matrix pre-strain

This paper focuses on the simultaneous homogenization and dimension reduction of periodic composite plates within the framework of non-linear elasticity. The composite plate in its reference (undeformed) configuration consists of a periodic perforated plate made of stiff material with holes filled by a soft matrix material. The structure is clamped on a cylindrical part. Two cases of asymptotic analysis are considered: one without pre-strain and the other with matrix pre-strain. In both cases, the total elastic energy is in the von-Kármán (vK) regime ( ε 5 ). A new splitting of the displacements is introduced to analyze the asymptotic behavior. The displacements are decomposed using the Kirchhoff–Love (KL) plate displacement decomposition. The use of a re-scaling unfolding operator allows for deriving the asymptotic behavior of the Green St. Venant’s strain tensor in terms of displacements. The limit homogenized energy is shown to be of vK type with linear elastic cell problems, established using the Γ-convergence. Additionally, it is shown that for isotropic homogenized material, our limit vK plate is orthotropic. The derived results have practical applications in the design and analysis of composite structures.

Constructing cospectral graphs by unfolding non-bipartite graphs

In 2010, Butler introduced the unfolding operation on a bipartite graph to produce two bipartite graphs, which are cospectral for the adjacency and the normalized Laplacian matrices. In this article, we describe how the idea of unfolding a bipartite graph with respect to another bipartite graph can be extended to nonbipartite graphs. In particular, we describe how unfoldings involving reflexive bipartite, semi-reflexive bipartite, and multipartite graphs are used to obtain cospectral nonisomorphic graphs for the adjacency matrix.

Homogenization of a nonlinear reaction‐diffusion problem in a perforated domain with different size obstacles

AbstractIn this paper we consider a nonlinear reaction‐diffusion problem with low diffusion scales of order in a porous domain which periodically perforated by two different size obstacles. A nonlinear Neumann condition is prescribed on the boundaries of smaller obstacles, while a homogeneous Neumann condition is imposed on the other boundaries. Our aim is to derive the homogenized model as the small parameters tend to zero by using the method of three‐scale convergence and periodic reiterated unfolding operator. The final homogenized macroscopic model depends on three scales.

Effective medium theory for second-gradient elasticity with chirality

We derive effective models for a heterogeneous second-gradient elastic material taking into account chiral scale-size effects. Our classification of the effective equations depends on the hierarchy of four characteristic lengths: The size of the heterogeneities ℓ, the intrinsic lengths of the constituents ℓ SG and ℓ chiral , and the overall characteristic length of the domain L. Depending on the different scale interactions between ℓ SG , ℓ chiral , ℓ, and L we obtain either an effective Cauchy continuum or an effective second-gradient continuum. The working technique combines scaling arguments with the periodic homogenization asymptotic procedure. Both the passage to the homogenization limit and the unveiling of the correctors’ structure rely on a suitable use of the periodic unfolding operator.

Homogenization of distributive optimal control problem governed by Stokes system in an oscillating domain

The present article deals with the homogenization of a distributive optimal control problem (OCP) subjected to the more generalized stationary Stokes equation involving unidirectional oscillating coefficients posed in a two-dimensional oscillating domain. The cost functional considered is of the Dirichlet type involving a unidirectional oscillating coefficient matrix. We characterize the optimal control and study the homogenization of this OCP with the aid of the unfolding operator. Due to the presence of oscillating matrices both in the governing Stokes equations and the cost functional, one obtains the limit OCP involving a perturbed tensor in the convergence analysis.

Syntactic ASP forgetting with forks

Answer Set Programming (ASP) constitutes nowadays one of the most successful paradigms for practical Knowledge Representation and declarative problem solving. The formal analysis of ASP programs is essential for a rigorous treatment of specifications, the correct construction of solvers and the extension with other representational features. In this paper, we present a syntactic transformation, called the unfolding operator, that allows forgetting an atom in a logic program (under ASP semantics). The main advantage of unfolding is that, unlike other syntactic operators, it is always applicable and guarantees strong persistence, that is, the result preserves the same stable models with respect to any context where the forgotten atom does not occur. The price for its completeness is that the result is an expression that may contain the fork operator. Yet, we illustrate how, in some cases, the application of fork properties may allow us to reduce the fork to a logic program.

Homogenization of elliptic PDE with L 1 source term in domains with boundary having very general oscillations

In the present article, we study the homogenization of a second-order elliptic PDE with oscillating coefficients in two different domains, namely a standard rectangular domain with very general oscillations and a circular type oscillating domain. Further, we consider the source term in L 1 and hence the solutions are interpreted as renormalized solutions. In the first domain, oscillations are in horizontal directions, while that of the second one is in the angular direction. To take into account the type of oscillations, we have used two different types of unfolding operators and have studied the asymptotic behavior of the renormalized solution of a second-order linear elliptic PDE with a source term in L 1 . In fact, we begin our study in oscillatory circular domain with oscillating coefficients and L 2 data which is also new in the literature. We also prove relevant strong convergence (corrector) results. We present the complete details in the context of circular domains, and sketch the proof in other domain.

Microscopic tridomain model of electrical activity in the heart with dynamical gap junctions. Part 2 – Derivation of the macroscopic tridomain model by unfolding homogenization method

We study the homogenization of a novel microscopic tridomain system, allowing for a more detailed analysis of the properties of cardiac conduction than the classical bidomain and monodomain models. In (Acta Appl.Math. 179 (2022) 1–35), we detail this model in which gap junctions are considered as the connections between adjacent cells in cardiac muscle and could serve as alternative or supporting pathways for cell-to-cell electrical signal propagation. Departing from this microscopic cellular model, we apply the periodic unfolding method to derive the macroscopic tridomain model. Several difficulties prevent the application of unfolding homogenization results, including the degenerate temporal structure of the tridomain equations and a nonlinear dynamic boundary condition on the cellular membrane. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding operator and a Kolmogorov–Riesz compactness’s result.

On a hierarchy of effective models for the biomechanics of human compact bone tissue

Abstract We derive a hierarchy of effective models that can be used to model the biomechanics of human compact bone taking into account scale-size effects observed experimentally. The classification of the effective models depends on the hierarchy of four characteristic lengths: the size of the heterogeneities, two intrinsic lengths of the constituents and the overall characteristic length of the domain. Depending on the different scale interactions between the size of the heterogeneities, the two intrinsic lengths of the constituents, and the characteristic length of the domain we obtain either an effective Cauchy continuum or an effective Cosserat continuum. The passage to the limit relies on suitable use of the periodic unfolding operator. Moreover, we perform numerical simulations to validate our results.

Tensor slow feature analysis and its applications for batch process monitoring

How to establish efficient monitoring models for complex batch processes is a challenging problem because of the three-dimensional (3-D) data array and serious dynamics. The unfolding operation in most methods would destroy the batch data structure and increase the dimensions of modeling data obviously. Meanwhile, it is more practical to perform within-batch detection rather than end-of-batch detection. In this work, a novel high-order tensor slow feature analysis model is proposed to handle the 3-D and dynamical issues simultaneously. The proposed model can be solved by iteratively tackling two sub-optimal problems. Within-batch detection is performed based on the properly defined monitoring statistics to timely detect the abnormal situation. The multi-phase property is further considered in this work to improve detection accuracy. A simulated case and an industrial case are taken to show the merits of the proposed method.

Homogenization theory of ion transportation in multicellular tissue

The transport of ions in biological tissues plays a crucial role in studying many biological and pathological problems. Certain multi-cellular structures, such as smooth muscles on vessel walls, can be considered as periodic bi-domain structures consisting of intracellular and extracellular spaces separated by semipermeable membranes. To model these structures, macro-scale models are proposed based on an electro-neutral (EN) microscale model with nonlinear interface conditions, using an unfolding operator. The membranes are treated as combinations of capacitors and resistors, while also taking into account the connectivity of the intracellular space. If the intracellular space is fully connected and forms a syncytium, the macroscale model is a bidomain nonlinear coupled partial differential equations system. Otherwise, if the intracellular cells are not connected, the macroscale model for the intracellular space is an ordinary differential system with source/sink terms from the connected extracellular space. The first-order error estimates for these models are achieved with proper regularity assumptions.

Homogenization of a nonlinear stochastic model with nonlinear random forces for chemical reactive flows in porous media

In this study, we establish new homogenization results for chemical reactive fluxes in porous media using a nonlinear stochastic model with nonlinear random forces. This model is mathematically represented by a linear stochastic response equation, a nonlinear boundary condition, a nonlinear stochastic diffusion equation, and external nonlinear random forces that affect the solute concentration and diffusion in the fluid phase. A homogenized model made up of a nonlinear stochastic diffusion equation with extra terms reflecting the impact of the adsorption and chemical reactions occurring on the boundaries of the perforations and a stochastic differential equation with extra nonlinear term representing the limit for the surface problem is derived using the periodic unfolding operator method and probabilistic compactness results. To guarantee that the entire sequence converges, we also show the uniqueness result for the limit problem. Finally, we go over some applications for the interaction of chemical fluxes.

A supervised functional Bayesian inference model with transfer-learning for performance enhancement of monitoring target batches with limited data

To increase the monitoring performance of the batch process with serious nonlinearity, uneven-length, and limited-data issues, a supervised transfer-learning based functional Bayesian inference method is developed in this study. The raw uneven-length batch data are firstly transformed into functional data by choosing appropriate orthogonal wavelet basis functions (WBFs). Using the approximation coefficients representing the latent features that are inherent to the raw data, a Gaussian process model and a Bayesian inference method are applied based on the coefficients in each source batch process and the established models are transferred to enhance modeling performance for the target process with limited batches. In the proposed functional method, the unfolding operation is not needed for preprocessing, avoiding distortion of the raw data structure. With the compact support property of WBFs, within-batch detection can be implemented effectively to recognize faults earlier. The advantages of the proposed model are verified using a numerical case and an industrial polytetrafluoroethylene process.

Variable-Friction-Based In-Hand Manipulation of Fabrics Applied to Unfolding Operations

The automatic state recognition and handling of fabrics, which are typical deformable objects, is difficult to realize. The development of related technologies for fabric manipulation remains impractical. This research was inspired by the process by which humans handle fabrics using their fingers. We proposed an in-hand manipulation method for fabrics based on variable friction and verified its effectiveness by sliding a single sheet of fabric and unfolding using in-hand manipulation functions. Experimental results demonstrate that the in-hand manipulation functions proposed in this study have the potential to solve complex problems in fabric handling.

Asymptotic analysis of an interior optimal control problem governed by Stokes equations

This article introduces an interior optimal control problem (OCP) in a two‐dimensional domain with a highly oscillatory boundary governed by the stationary Stokes equations. We consider the periodic controls in the oscillating region of the domain and use the unfolding operators to characterize the optimal controls. We establish the convergences of optimal control, state, and pressure in a suitable space to the ones of the limit system in a fixed domain.

Comparative study on wavelet functional partial least squares soft sensor for complex batch processes

Conventional data-driven models for batch processes conduct unfolding operations and neglect the continuous property. A novel soft sensor method is proposed based on the wavelet functional partial least squares (WFPLS). The three-dimensional (3D) batch data is transferred into an even two-dimensional function matrix no matter whether it has even-length or uneven-length. The orthogonal wavelet basis functions are selected by the proposed active algorithm to approximate the variables' trajectories. Two versions of WFPLS are developed to predict quality based on different assumptions of the loading function. By comparing one of the proposed methods and the classical multi-way partial least squares (MPLS), the root means square errors are relatively descended by 20.6566% and 63.9925%, R2 are improved by 2.5813% and 83.2518%, and the corresponding computation time of the proposed method is only 5.7% and 41% of the time used by MPLS in the numerical case and in the industrial case, respectively.

Homogenization of a coupled incompressible Stokes–Cahn–Hilliard system modeling binary fluid mixture in a porous medium

A phase-field model for two-phase immiscible, incompressible porous media flow with surface tension effects is considered. The pore-scale model consists of a strongly coupled system of Stokes–Cahn–Hilliard equations. The fluids are separated by an evolving diffuse interface of a finite width depending on the scale parameter ɛ in the considered model. At first, the existence of solution of a coupled system of partial differential equations at micro scale is investigated. We obtained the homogenized equations for the microscopic model via unfolding operator and two-scale convergence approach.

Symbolic Specialization of Rewriting Logic Theories withPresto

AbstractThis paper introduces$\tt{{Presto}}$, a symbolic partial evaluator for Maude’s rewriting logic theories that can improve system analysis and verification. In$\tt{{Presto}}$, the automated optimization of a conditional rewrite theory$\mathcal{R}$(whose rules define the concurrent transitions of a system) is achieved by partially evaluating, with respect to the rules of$\mathcal{R}$, an underlying, companion equational logic theory$\mathcal{E}$that specifies the algebraic structure of the system states of$\mathcal{R}$. This can be particularly useful for specializing an overly general equational theory$\mathcal{E}$whose operators may obey complex combinations of associativity, commutativity, and/or identity axioms, when being plugged into a host rewrite theory$\mathcal{R}$as happens, for instance, in protocol analysis, where sophisticated equational theories for cryptography are used.$\tt{{Presto}}$implements different unfolding operators that are based onfolding variant narrowing(the symbolic engine of Maude’s equational theories). When combined with an appropriate abstraction algorithm, they allow the specialization to be adapted to the theory termination behavior and bring significant improvement while ensuring strong correctness and termination of the specialization. We demonstrate the effectiveness of$\tt{{Presto}}$in several examples of protocol analysis where it achieves a significant speed-up. Actually, the transformation provided by$\tt{{Presto}}$may cut down an infinite folding variant narrowing space to a finite one, and moreover, some of the costly algebraic axioms and rule conditions may be eliminated as well. As far as we know, this is the first partial evaluator for Maude that respects the semantics of functional, logic, concurrent, and object-oriented computations.

Three Scale Unfolding Homogenization Method Applied to Cardiac Bidomain Model

In this paper, we are dealing with a rigorous homogenization result at two different levels for the bidomain model of cardiac electro-physiology. The first level associated with the mesoscopic structure such that the cardiac tissue consists of extracellular and intracellular domains separated by an interface (the sarcolemma). The second one related to the microscopic structure in such a way that the intracellular medium can only be viewed as a periodical layout of unit cells (mitochondria). At the interface between intra-and extracellular media, the fluxes are given by nonlinear functions of ionic and applied currents. A rigorous homogenization process based on unfolding operators is applied to derive the macroscopic (homogenized) model from our meso-microscopic bidomain model. We apply a three-scale unfolding method in the intracellular problem to obtain its homogenized equation at two levels. The first level upscaling of the intracellular structure yields the mesoscopic equation. The second step of the homogenization leads to obtain the intracellular homogenized equation. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding method and a Kolmogorov-Riesz compactness's result. Next, we use the standard unfolding method to homogenize the extracellular problem. Finally, we obtain, at the limit, a reaction-diffusion system on a single domain (the superposition of the intracellular and extracellular media) which contains the homogenized equations depending on three scales. Such a model is widely used for describing the macroscopic behavior of the cardiac tissue, which is recognized to be an important messengers between the cytoplasm (intracellular) and the other extracellular inside the biological cells. Contents 2020 Mathematics Subject Classification. 65N55 and 35A01 and 35B27 and 35K57 and 65M..