Coarse-scale Representation Research Articles

A Space-Time Multiscale Method for Parabolic Problems

We present a space-time multiscale method for a parabolic model problem with an underlying coefficient that may be highly oscillatory with respect to both the spatial and the temporal variables. The method is based on the framework of the Variational Multiscale Method in the context of a space-time formulation and computes a coarse-scale representation of the differential operator that is enriched by auxiliary space-time corrector functions. Once computed, the coarse-scale representation allows us to efficiently obtain well-approximating discrete solutions for multiple right-hand sides. We prove first-order convergence independently of the oscillation scales in the coefficient and illustrate how the space-time correctors decay exponentially in both space and time, making it possible to localize the corresponding computations. This localization allows us to define a practical and computationally efficient method in terms of complexity and memory, for which we provide a posteriori error estimates and present numerical examples.

Multilevel Graph Partitioning for Three-Dimensional Discrete Fracture Network Flow Simulations

We present a topology-based method for mesh-partitioning in three-dimensional discrete fracture network (DFN) simulations that takes advantage of the intrinsic multi-level nature of a DFN. DFN models are used to simulate flow and transport through low-permeability fractured media in the subsurface by explicitly representing fractures as discrete entities. The governing equations for flow and transport are numerically integrated on computational meshes generated on the interconnected fracture networks. Modern high-fidelity DFN simulations require high-performance computing on multiple processors where performance and scalability depends partially on obtaining a high-quality partition of the mesh to balance work-loads and minimize communication across all processors. The discrete structure of a DFN naturally lends itself to various graph representations, which can be thought of as coarse-scale representations of the computational mesh. Using this concept, we develop two applications of the multilevel graph partitioning algorithm to partition the mesh of a DFN. In the first, we project a partition of the graph based on the DFN topology onto the mesh of the DFN and in the second, this DFN-based projection is used as the initial condition for further partitioning refinement of the mesh. We compare the performance of these methods with standard multi-level graph partitioning using graph-based metrics (cut, imbalance, partitioning time), computational-based metrics (FLOPS, iterations, solver time), and total run time. The DFN-based and the mesh-based partitioning methods are comparable in terms of the graph-based metrics, but the time required to obtain the partition is several orders of magnitude faster using the DFN-based partitions. The computation-based metrics show comparable performance between both methods so, in combination, the DFN-based partitions are several orders of magnitude faster than the mesh-based partition. Moreover, the method which uses the DFN-partition solution as the initial condition of the mesh partition provided cut and imbalance values that were close to the mesh-based partition but in a fraction of the time. In turn, this hybrid method outperformed both of the other methods in terms of the total run time.

Spectral variational multiscale enrichment method—A new computational approach to transient dynamics of phononic crystals and acoustic metamaterials

Phononic crystals and acoustic metamaterials hold tremendous potential in controlling mechanical and acoustic waves, thereby enabling remarkable and novel engineering applications such as seismic wave mitigation, impact wave mitigation, elastic cloaking, and acoustic superlens. Given the potential complexity associated with the microstructure of these materials, direct numerical simulation of irregularly shaped and/or large engineering components could be computationally very costly and in many cases prohibitive. Multiscale computational modeling, coupled with model order reduction methodologies present a feasible strategy to simulate the dynamic response of phononic crystals and acoustic metamaterials. In this study, we present the spectral variational multiscale enrichment approach that allows the analysis of a broad range of material microstructures for phononic crystals and acoustic metamaterials. The spectral coarse-scale representation is proposed to capture the salient transient wave phenomena, such as wave dispersion and band gaps that occur in the short wavelength regime. A material phase based model order reduction strategy is devised at the fine-scale. By using the reduced basis for the fine-scale problem, significant numerical efficiency is achieved. Transient elastic wave propagation in two-dimensional phononic crystals and acoustic metamaterials are investigated and the proposed multiscale approach is verified against direct numerical simulations.

Spectral variational multiscale model for transient dynamics of phononic crystals and acoustic metamaterials

In this manuscript, a spectral multiscale model is developed for transient elastic wave propagation in periodic composites. The basis of the multiscale approach is the variational multiscale enrichment method that permits analysis of wave propagation in the scale-inseparable condition, i.e., short wave propagation. A spectral coarse-scale representation is proposed to capture the salient transient wave phenomena, such as wave dispersion and band gaps that occur in the short wavelength regime. A material-phase-based model basis reduction strategy is devised at the fine scale to achieve computational efficiency. An important feature of the proposed method is that it does not rely on the classical assumption of separation of scales, which permits its application to a broad range of architectured composites including phononic crystals and acoustic metamaterials. Transient elastic wave propagation in two-dimensional periodic structures is investigated. The proposed multiscale approach is verified against direct numerical simulations. The accuracy and computational efficiency of the approach are demonstrated for both phononic crystals and acoustic metamaterials.

Numerical upscaling of discrete network models

In this paper a numerical multiscale method for discrete networks is presented. The method gives an accurate coarse scale representation of the full network by solving sub-network problems. The method is used to solve problems with highly varying connectivity or random network structure, showing optimal order convergence rates with respect to the mesh size of the coarse representation. Moreover, a network model for paper-based materials is presented. The numerical multiscale method is applied to solve problems governed by the presented network model.

Direct current resistivity with steel-cased wells

SUMMARY The work in this paper is motivated by the increasing use of electrical and electromagnetic methods in geoscience problems where steel-cased wells are present. Applications of interest include monitoring carbon capture and storage and hydraulic fracturing operations. Also of interest is detecting flaws or breaks in degrading steel-casings—such wells pose serious environmental hazards. The general principles of electrical methods with steel-cased wells are understood and several authors have demonstrated that the presence of steel-cased wells can be beneficial for detecting signal due to targets at depth. However, the success of a direct current (DC) resistivity survey lies in the details. Secondary signals might only be a few per cent of the primary signal. In designing a survey, the geometry of the source and receivers, and whether the source is at the top of the casing, inside of it, or beneath the casing will impact measured responses. Also the physical properties and geometry of the background geology, target and casing will have a large impact on the measured data. Because of the small values of the diagnostic signals, it is important to understand the detailed physics of the problem and also to be able to carry out accurate simulations. This latter task is computationally challenging because of the extreme geometry of the wells, which extend kilometers in depth but have millimeter variations in the radial direction, and the extreme variation in the electrical conductivity which is typically 5–7 orders of magnitude larger than that of the background geology. In this paper, we adopt a cylindrical discretization for numerical simulations to investigate three important aspects of DC resistivity in settings with steel-cased wells. (1) We examine the feasibility of using a surface-based DC resistivity survey for diagnosing impairments along a well in a casing integrity experiment. This parameter study demonstrates the impact of the background conductivity, the conductivity of the casing, the depth of the flaw, and the proportion of the casing circumference that is compromised on amplitude of the secondary electric fields measured at the surface. (2) Next, we consider elements of survey design for exciting a conductive or resistive target at depth. We show that conductive targets generate stronger secondary responses than resistive targets, and that having an electrical connection between the target and well can significantly increase the measured secondary responses. (3) Finally, we examine common strategies for approximating the fine-scale structure of a steel cased well with a coarse-scale representation to reduce computational load. We show that for DC resistivity experiments, the product of the conductivity and the cross-sectional area of the casing is the important quantity for controlling the distribution of currents and charges along its length. To promote insight into the physics, we present results by plotting the currents, charges, and electric fields in each of the scenarios examined. All of the examples shown in this paper are built on open-source software and are available as Jupyter notebooks.

Deformable image registration by combining uncertainty estimates from supervoxel belief propagation.

Discrete optimisation strategies have a number of advantages over their continuous counterparts for deformable registration of medical images. For example: it is not necessary to compute derivatives of the similarity term; dense sampling of the search space reduces the risk of becoming trapped in local optima; and (in principle) an optimum can be found without resorting to iterative coarse-to-fine warping strategies. However, the large complexity of high-dimensional medical data renders a direct voxel-wise estimation of deformation vectors impractical. For this reason, previous work on medical image registration using graphical models has largely relied on using a parameterised deformation model and on the use of iterative coarse-to-fine optimisation schemes. In this paper, we propose an approach that enables accurate voxel-wise deformable registration of high-resolution 3D images without the need for intermediate image warping or a multi-resolution scheme. This is achieved by representing the image domain as multiple comprehensive supervoxel layers and making use of the full marginal distribution of all probable displacement vectors after inferring regularity of the deformations using belief propagation. The optimisation acts on the coarse scale representation of supervoxels, which provides sufficient spatial context and is robust to noise in low contrast areas. Minimum spanning trees, which connect neighbouring supervoxels, are employed to model pair-wise deformation dependencies. The optimal displacement for each voxel is calculated by considering the probabilities for all displacements over all overlapping supervoxel graphs and subsequently seeking the mode of this distribution. We demonstrate the applicability of this concept for two challenging applications: first, for intra-patient motion estimation in lung CT scans; and second, for atlas-based segmentation propagation of MRI brain scans. For lung registration, the voxel-wise mode of displacements is found using the mean-shift algorithm, which enables us to determine continuous valued sub-voxel motion vectors. Finding the mode of brain segmentation labels is performed using a voxel-wise majority voting weighted by the displacement uncertainty estimates. Our experimental results show significant improvements in registration accuracy when using the additional information provided by the registration uncertainty estimates. The multi-layer approach enables fusion of multiple complementary proposals, extending the popular fusion approaches from multi-image registration to probabilistic one-to-one image registration.

Variational multiscale enrichment method with mixed boundary conditions for elasto-viscoplastic problems

This manuscript presents the formulation and implementation of the variational multiscale enrichment (VME) method for the analysis of elasto-viscoplastic problems. VME is a global---local approach that allows accurate fine scale representation at small subdomains, where important physical phenomena are likely to occur. The response within far-fields is idealized using a coarse scale representation. The fine scale representation not only approximates the coarse grid residual, but also accounts for the material heterogeneity. A one-parameter family of mixed boundary conditions that range from Dirichlet to Neumann is employed to study the effect of the choice of the boundary conditions at the fine scale on accuracy. The inelastic material behavior is modeled using Perzyna type viscoplasticity coupled with flow stress evolution idealized by the Johnson---Cook model. Numerical verifications are performed to assess the performance of the proposed approach against the direct finite element simulations. The results of verification studies demonstrate that VME with proper boundary conditions accurately model the inelastic response accounting for material heterogeneity.

Group sparse coding with a collection of winner-take-all networks

Degeneracy is a ubiquitous feature of computation and coding in biological systems. Degenerate codes—codes in which multiple code words have the same meaning or interpretation—arise in a wide range of biological processes, from the many-to-one mapping of codons to amino acids to the numerous instances of degenerate coding in the nervous system, both in the periphery and in the cortex. There are a number of reasons why neural systems might seek some degree of degeneracy; by enabling a number of inputs (stimuli) to be mapped to the same output code, the system is endowed with a certain level of robustness to noise and to cell death. Furthermore, degenerate codes provide invariance to certain types of variability in the raw sensory input. This is particularly useful in object recognition for instance, where despite photometric variability, the representation of an object under different illumination conditions should be equivalent at some level of processing even if the neural representations for these states are slightly different. In tandem with nature's desire to produce degeneracy and invariance in the brain’s representation of sensory inputs, sparsity also appears to also play an important role in neural coding; a sparse code is one that requires a small number of active neurons relative to the size of the population. Sparse codes are also associated with a high degree of specificity: each cell in the population only responds to a limited number of inputs. Sparse coding has been observed in the visual cortex of macaque, mushroom body in locusts, and auditory cortex in grasshoppers. Experimental studies suggest that neural systems might seek to strike an appropriate balance between the degeneracy and the sparsity of the neural code. Here, we describe a framework in which one can tradeoff between these two objectives in a natural way. In contrast to models for global sparse coding such as the locally competitive architectures in LCA [1] which find a population code that captures the primary features of the stimulus while minimizing the number of active neurons in the population, our goal is to find a representation of the sensory input that minimizes the number of ‘groups’ that must be active to encode the input. The idea is that when the excitatory cells are grouped in a meaningful way, the activation of a small set of groups might have a particular meaning to the organism. If we view the activation of these ‘functional groups’ as a high-level or coarse scale representation of the stimulus, the resulting code is stable to a wide-range of perturbations in the input space. This two-level representation (the coarse and the fine) provides a robust and degenerate mapping of the input space that is also sparse. After motivating the utility of group sparse coding in neural systems, we show how one can implement these sparse coding networks by coupling a collection of winner-take-all (WTAs) networks that were first introduced for state-dependent computation in [2]. WTAs are particularly attractive as they represent neurally plausible microcircuits that have been hypothesized to serve as a computational primitive in complex cortical networks. An interesting property of these microcircuits is the fact that a single inhibitory unit delivers a common inhibitory signal to all of the excitatory units within the WTA. This architecture is in stark contrast the highly specified point-to-point inhibition structure required to faithfully implement a LCA, i.e., the inhibition between every pair of cells can be different. To solve sparse coding problems with collections of WTAs, we adapt the analytical approach in [1] and show how one can couple a collection of WTAs to descend an energy function that promotes grouped sparse representations. In addition to building in invariances into the code, we demonstrate that there are a number of computational advantages to employing this type of grouped network model for sparse coding. First, we demonstrate that we require fewer “long-range” connections to produce group sparse codes with collections of WTAs than that required for a global LCA. Second, the lateral and recurrent inhibition in the network modulates the threshold of excitatory units collectively: this means that fewer interneurons are required to produce a representation, the number of long range messages which must be sent is reduced, and the network converges to a solution faster than the global LCA.

Coarse-scale representations and smoothed Wigner transforms

Smoothed Wigner transforms have been used in signal processing, as a regularized version of the Wigner transform, and have been proposed as an alternative to it in the homogenization and/or semiclassical limits of wave equations. We derive explicit, closed formulations for the coarse-scale representation of the action of pseudodifferential operators. The resulting “smoothed operators” are in general of infinite order. The formulation of an appropriate framework, resembling the Gelfand–Shilov spaces, is necessary. Similarly we treat the “smoothed Wigner calculus”. In particular this allows us to reformulate any linear equation, as well as certain nonlinear ones (e.g., Hartree and cubic nonlinear Schrödinger), as coarse-scale phase-space equations (e.g., smoothed Vlasov), with spatial and spectral resolutions controlled by two free parameters. Finally, it is seen that the smoothed Wigner calculus can be approximated, uniformly on phase-space, by differential operators in the semiclassical regime. This improves the respective weak-topology approximation result for the Wigner calculus.

Probabilistic equivalence and stochastic model reduction in multiscale analysis

This paper presents a probabilistic upscaling of mechanics models. A reduced-order probabilistic model is constructed as a coarse-scale representation of a specified fine-scale model whose probabilistic structure can be accurately determined. Equivalence of the fine- and coarse-scale representations is identified such that a reduction in the requisite degrees of freedom can be achieved while accuracy in certain quantities of interest is maintained. A significant stochastic model reduction can a priori be expected if a separation of spatial and temporal scales exists between the fine- and coarse-scale representations. The upscaling of probabilistic models is subsequently formulated as an optimization problem suitable for practical computations. An illustration in stochastic structural dynamics is provided to demonstrate the proposed framework.

A multiscale projection method for the analysis of carbon nanotubes

The main objective of this paper is to develop a multiscale method for the analysis of carbon nanotube systems. The multiscale coupled governing equations are first derived based on a multiscale partition, which consists of a coarse scale component represented by meshfree approximation, and a fine scale component to be resolved from molecular dynamics. It is noted that the coarse scale representation is present in the entire domain of the problem, and coexists with the fine scale representation in the enriched region. Using the projection properties of the partition, the decoupled non-linear set of equations are obtained and solved iteratively using Newton's method. In this multiscale analysis, a virtual atom cluster (VAC) model is also proposed in the coarse scale treatment. This model does not involve a stress update scheme employing the Cauchy–Born hypothesis, which has been widely used in the crystal elasticity theory. Finally, to approximate a curved surface at the nanoscale, meshfree approximations are introduced to interpolate a single layer of atoms. Unlike traditional shell or continuum elements, the geometric constraint is only imposed in two dimensions. The high order continuity property of the meshfree shape functions guarantees an accurate description of the geometry and thus the energy of the atomic bond. The accuracy and efficiency of the method are illustrated in the post-buckling analysis of carbon nanotube structures. To our knowledge, this is the first multiscale post-buckling analysis presented for such a nanoscale system.