Coarsest Partition Research Articles

A mutual information statistic for assessing state space partitions of dynamical systems.

We propose a mutual information statistic to quantify the information encoded by a partition of the state space of a dynamical system. We measure the mutual information between each point's symbolic trajectory history under a coarse partition (one with few unique symbols) and its partition assignment under a fine partition (one with many unique symbols). When applied to a set of test cases, this statistic demonstrates predictable and consistent behavior. Empirical results and the statistic's formulation suggest that partitions based on trajectory history, such as the ordinal partition, perform best. As an application, we introduce the weighted ordinal partition, an extension of the popular ordinal partition with parameters that can be optimized using the mutual information statistic, and demonstrate improvements over the ordinal partition in time series analysis. We also demonstrate the weighted ordinal partition's applicability to real experimental datasets.

FBSM: Foveabox-based boundary-aware segmentation method for green apples in natural orchards

Image segmentation of target fruits is an essential part of machine vision systems, aiming to facilitate more accurate early fruit measurement and machine harvesting in natural orchard environments. Given that instance-level pooling and down-sampling operations in conventional segmentation models often lead to the loss of detailed information, resulting in coarse partitioning masks, we endeavor to restore the boundary information of the masks. To achieve high-quality fruit segmentation with clear boundaries and refined masks, a Foveabox-based boundary-aware segmentation model (FBSM) is constructed by adding a multi-stage mask prediction head incorporating fine-grained features to the anchor-free detection model FoveaBox to realize accurate segmentation of green apple fruits. At each stage, a bi-layer fusion structure (BFS) is employed to fully fuse the fine-grained features in a double-fusion manner to guide the subsequent instance mask prediction. Finally, the boundary recovery module (BRM) is leveraged to recover the lost boundary detail information and obtain more accurate boundaries for the continuous optimization process of the fruit instance mask. Experimental results demonstrate that the proposed FBSM model achieves a mean average precision (mAP) of 60.7 % for green apple instance segmentation in unstructured natural orchard environments, surpassing traditional instance segmentation models. Moreover, it enhances base model detection accuracy by 1.3 % while striking a better balance between detection and segmentation performance.

Robust methods for multiscale coarse approximations of diffusion models in perforated domains

For the Poisson equation posed in a domain containing a large number of polygonal perforations, we propose a low-dimensional coarse approximation space based on a coarse polygonal partitioning of the domain. Similarly to other multiscale numerical methods, this coarse space is spanned by locally discrete harmonic basis functions. Along the subdomain boundaries, the basis functions are piecewise polynomial. The main contribution of this article is an error estimate regarding the H1-projection over the coarse space; this error estimate depends only on the regularity of the solution over the edges of the coarse partitioning. For a specific edge refinement procedure, the error analysis establishes superconvergence of the method even if the true solution has a low general regularity. Additionally, this contribution numerically explores the combination of the coarse space with domain decomposition (DD) methods. This combination leads to an efficient two-level iterative linear solver which reaches the fine-scale finite element error in few iterations. It also bodes well as a preconditioner for Krylov methods and provides scalability with respect to the number of subdomains.

Algebraic multiscale grid coarsening using unsupervised machine learning for subsurface flow simulation

Subsurface flow simulation is vital for many geoscience applications, including geoenergy extraction and gas (energy) storage. Reservoirs are often highly heterogeneous and naturally fractured. Therefore, scalable simulation strategies are crucial to enable efficient and reliable operational strategies. One of these scalable methods, which has also been recently deployed in commercial reservoir simulators, is algebraic multiscale (AMS) solvers. AMS, like all multilevel schemes, is found to be highly sensitive to the types (geometries and size) of coarse grids and local basis functions. Commercial simulators benefit from a graph-based partitioner; e.g., METIS to generate the multiscale coarse grids. METIS minimizes the amount of interfaces between coarse partitions, while keeping them of similar size which may not be the requirement to create a coarse grid. In this work, we employ a novel approach to generate the multiscale coarse grids, using unsupervised learning methods which is based on optimizing different parameter. We specifically use the Louvain algorithm and Multi-level Markov clustering. The Louvain algorithm optimizes modularity, a measure of the strength of network division while Markov clustering simulates random walks between the cells to find clusters. It is found that the AMS performance is improved when compared with the existing METIS-based partitioner on several field-scale test cases. This development has the potential to enable reservoir engineers to run ensembles of thousands of detailed models at a much faster rate.

The MHM Method for Linear Elasticity on Polytopal Meshes

Abstract The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a new family of finite elements for the linear elasticity equation defined on coarse polytopal partitions of the domain. The finite elements rely on face degrees of freedom associated with multiscale bases obtained from local Neumann problems with piecewise polynomial interpolations on faces. We establish sufficient conditions on the fine-scale interpolations such that the MHM method is well-posed and optimally convergent under local regularity conditions. Also, a multi-level error analysis demonstrates that the MHM method achieves convergence without refining the coarse partition. The upshot is that the Poincaré and Korn’s inequalities do not degenerate, and then the convergence arises on general meshes. Two- and three-dimensional numerical tests assess theoretical results and verify the robustness of the method on a multi-layer media case.

Sensitivity of the Euler–Poinsot Tensor Values to the Choice of the Body Surface Triangulation Mesh

The inertial characteristics of celestial bodies can be calculated using their triangle partitions based on photometric observations. Such partitions can be refined along with the accumulation of necessary information. In this regard, the question arises to what extent the approximations of the inertial characteristics of celestial bodies, in particular, the approximations of the components of the Euler–Poinsot tensor of different orders, are susceptible to the choice of such partitions. Such components enter into the expansion of the gravitational potential in harmonic polynomials. In this paper, for some small celestial bodies, a comparison of such coefficients is carried out as coarse partitions are replaced with finer ones.

New geographic model of care to manage the post-COVID-19 elective surgery aftershock in England: a retrospective observational study

ObjectivesThe suspension of elective surgery during the COVID-19 pandemic is unprecedented and has resulted in record volumes of patients waiting for operations. Novel approaches that maximise capacity and efficiency of...

Oriented to Multi-Branched Structure Unsupported 3D Printing Method Research

For the traditional three-axis (3D) configuration of the additive manufacturing (AM) platform, when printing the target model with a multi-branched structure, it is imperative to construct adequate support structures. To eliminate the use of support during the printing process, a non-directional unsupported 3D printing method for five-axis AM is proposed in this paper. By carrying out the K-means clustering algorithm, the coarse partition of the model is obtained, and then the fine decomposition represented by a sequence of separating planes is determined by a local dynamic search adjustment algorithm according to manufacturing constraints. The multi-branched structure of the model is divided into simple subparts so that the general model can be built in different directions and be printed with its own parts as the support. Two case studies were carried out for verification. The experimental results showed that the branch-model can be printed without support using the non-directional unsupported 3D printing method, and the non-directional unsupported 3D printing can save 18.72–20.60% of materials and 20.60–23.33% of time compared to conventional 3D printing.

Dynamic Coarsening and Local Reordered Nonlinear Solvers for Simulating Transport in Porous Media

Summary We present a robust and flexible sequential solution approach in which the flow equation is solved on the original grid, whereas the transport equations are solved with a new dynamic coarsening method that adapts the grid resolution locally to reduce the number of cells as much as possible. The resulting grid is formed by combining precomputed coarse partitions of an underlying fine model. Our approach is flexible and makes very few assumptions on cell geometries and the topology of the grid. To further accelerate the transport step, we combine dynamic coarsening with a local nonlinear solver that permutes the discrete transport equations into an optimal block-triangular form so that these can be solved very efficiently using a nonlinear back-substitution method. Efficiency and utility of the overall approach are assessed through a number of conceptual test cases, including the Olympus field model.

Preserving vorticity improves the accuracy of multiscale solvers

Within reservoir management, there is a growing demand for highly accurate and fast reservoir simulators. So-called multiscale methods are designed to preserve fine-scale information accurately in an accelerated coarse-scale computation of fluid flow. In many cases, this requires that coarse partitions are meticulously adapted to prominent geological features that determine flow paths. Partitions that seek to preserve the vorticity from a representative fine-scale flow simulation can be particularly effective to this end. We describe how to incorporate such partitions in the state-of-the-art multiscale restriction-smoothed basis (MsRSB) method, and present a series of two-phase test cases to validate and demonstrate the effectiveness of the resulting method. Our results show that using a nonuniform vorticity-based partition improves the accuracy of the MsRSB solver compared with uniform partitions and nonuniform partitions with a similar number of coarse blocks that adapt to permeability or velocity.

An evolutionary explanation of assassins and zealots in peer review

The peer review system aims to be effective in separating unacceptable from acceptable manuscripts. However, a reviewer can distinguish them or not. If reviewers distinguish unacceptable from acceptable manuscripts they use a fine partition of categories. But, if reviewers do not distinguish them they use a coarse partition in the evaluation of manuscripts. Most reviewers learned how to evaluate a manuscript from good and bad experiences, and they have been characterized as zealots (who uncritically favor a manuscript), assassins (who advise rejection much more frequently than the norm), and mainstream referees. In this paper we use the quasi-species model to describe the evolution of recommendation profiles in peer review. A recommendation profile is composed of a reviewer recommendation for each manuscript category under a particular categorization of manuscripts (fine or coarse). We see the reviewer mind as being built up with recommendation profiles. Assassins, zealots and mainstream reviewers are “ecologically” interrelated species whose progeny tend to mutate through errors made in the process of reviewer training. We define the recommendation profile as replicator, and selection arises because different types of recommendation profiles tend to replicate at different rates. Our results help to explain why assassins and zealots evolutionary appear in peer review because of the evolutionary success of reviewers who do not distinguish acceptable and unacceptable manuscripts.

Computations of permeability of large rock images by dual grid domain decomposition

Digital rock physics (DRP) is an eminent technology that facilitates and repeatable core analysis and multi-physics simulation of rock properties. One of the challenges in this field is the scalability of the problem size, whereby large micro-CT images over the order of 10003 voxels incur a high computational demand on performance. We estimate the of permeability in large digital samples of rocks imaged by micro-CT by using a fast and efficient Dual Grid Domain Decomposition technique based on the Schwarz Alternating Method (slow, low memory) with Algebraic Multigrid (AMG) solvers (fast, high memory) to solve on an otherwise unfeasible shared-memory machine. The comparisons and differences to other methods commonly used have been added in the introduction. The method applies a scalable parallel simulation algorithm to solve pressure and velocity fields using the Semi Analytical Pore Scale Finite Volume Solver (PFVS) within real 3D pore-scale micro-CT images. The domain is split into non-overlapping coarse partitions and also split into a set of dual coarse partitions of varying width. The governing equations are then solved iteratively between the partition sets by updating the pressure and flux at the relevant boundaries before each step. The method is validated and shown to converge to flux continuous conditions requisite of the governing equations. Permeability estimation is within 5–10% of the fine scale solution and significantly reduces memory limitations and computational time for solving problems in micro-CT images, allowing ordinary workstations to solve images over 10003 within the magnitude of 1–100 h of CPU time. The permeability of a 2520 × 2520 × 7250 sample was calculated with a workstation to within 9% error of LBM calculated with a supercomputer within a similar timeframe.

A substructuring preconditioner with vertex-related interface solvers for elliptic-type equations in three dimensions

In this paper, we propose a variant of the substructuring preconditioner for solving three-dimensional elliptic-type equations with strongly discontinuous coefficients. In the proposed preconditioner, we use the simplest coarse solver associated with the finite element space induced by the coarse partition and construct inexact interface solvers based on overlapping domain decomposition with small overlaps. This new preconditioner has an important merit: its construction and efficiency do not depend on the concrete form of the considered elliptic-type equations. We apply the proposed preconditioner to solve the linear elasticity problems and Maxwell’s equations in three dimensions. Numerical results show that the convergence rate of PCG method with the preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficients in the considered equations.

GPU optimization of material point methods

The Material Point Method (MPM) has been shown to facilitate effective simulations of physically complex and topologically challenging materials, with a wealth of emerging applications in computational engineering and visual computing. Borne out of the extreme importance of regularity, MPM is given attractive parallelization opportunities on high-performance modern multiprocessors. Parallelization of MPM that fully leverages computing resources presents challenges that require exploring an extensive design-space for favorable data structures and algorithms. Unlike the conceptually simple CPU parallelization, where the coarse partition of tasks can be easily applied, it takes greater effort to reach the GPU hardware saturation due to its many-core SIMT architecture. In this paper we introduce methods for addressing the computational challenges of MPM and extending the capabilities of general simulation systems based on MPM, particularly concentrating on GPU optimization. In addition to our open-source high-performance framework, we also conduct performance analyses and benchmark experiments to compare against alternative design choices which may superficially appear to be reasonable, but can suffer from suboptimal performance in practice. Our explicit and fully implicit GPU MPM solvers are further equipped with a Moving Least Squares MPM heat solver and a novel sand constitutive model to enable fast simulations of a wide range of materials. We demonstrate that more than an order of magnitude performance improvement can be achieved with our GPU solvers. Practical high-resolution examples with up to ten million particles run in less than one minute per frame.

A locally conservative Multiscale Finite Element Method for multiphase flow simulation through heterogeneous and fractured porous media

A Multiscale Locally Conservative Galerkin (MsLCG) method is proposed to accurately simulate multiphase flow in heterogeneous and fractured porous media. MsLCG employs a coarse partition of the fine grids and multiscale basis function for mapping the fine-scale information to the coarse-scale unknowns. Different from standard Multiscale Finite Element Method (MsFEM), the main improvement of our MsLCG is to use the property of local conservation at steady state conditions to define a numerical flux at element boundaries. MsLCG provides a way to extend standard MsFEM to handle challenging multiphase flow problems in heterogeneous and fractured porous media. MsLCG preserves all the advantages of the standard MsFEM while it provides explicitly conservative fluxes through each element. We present a number of representative numerical examples to demonstrate that our method is efficient and accurate for simulating multiphase flow through heterogeneous and fractured porous media.

Compositional abstraction refinement for control synthesis

This paper presents a compositional approach to specification-guided abstraction refinement for control synthesis of a nonlinear system associated with a method to over-approximate its reachable sets. Given an initial coarse partition of the state space, the control specification is given as a sequence of the cells of this partition to visit at each sampling time. The dynamics are decomposed into subsystems where some states and inputs are not observed, some states are observed but not controlled and where assume-guarantee obligations are used on the uncontrolled states of each subsystem. A finite abstraction is created for each subsystem through a refinement procedure starting from a coarse partition of the state space, then proceeding backwards on the specification sequence to iteratively split the elements of the partition whose coarseness prevents the satisfaction of the specification. Each refined abstraction is associated with a controller and it is proved that combining these local controllers can enforce the specification on the original system. The efficiency of the proposed approach compared to other abstraction methods is illustrated in a numerical example.

A mixed-finite volume element coupled with the method of characteristic fractional step difference for simulating transient behavior of semiconductor device of heat conductor and its numerical analysis

The mathematical system is formulated by four partial differential equations combined with initialboundary value conditions to describe transient behavior of three-dimensional semiconductor device with heat conduction. The first equation of an elliptic type is defined with respect to the electric potential, the successive two equations of convection dominated diffusion type are given to define the electron concentration and the hole concentration, and the fourth equation of heat conductor is for the temperature. The electric potential appears in the equations of electron concentration, hole concentration and the temperature in the formation of the intensity. A mass conservative numerical approximation of the electric potential is presented by using the mixed finite volume element, and the accuracy of computation of the electric intensity is improved one order. The method of characteristic fractional step difference is applied to discretize the other three equations, where the hyperbolic terms are approximated by a difference quotient in the characteristics and the diffusion terms are discretized by the method of fractional step difference. The computation of three-dimensional problem works efficiently by dividing it into three one-dimensional subproblems and every subproblem is solved by the method of speedup in parallel. Using a pair of different grids (coarse partition and refined partition), piecewise threefold quadratic interpolation, variation theory, multiplicative commutation rule of differential operators, mathematical induction and priori estimates theory and special technique of differential equations, we derive an optimal second order estimate in L2-norm. This numerical method is valuable in the simulation of semiconductor device theoretically and actually, and gives a powerful tool to solve the international problem presented by J. Douglas, Jr.

Abstraction refinement and plan revision for control synthesis under high level specifications

This paper presents a novel framework combining abstraction refinement and plan revision for control synthesis problems under temporal logic specifications. The control problem is first solved on a simpler nominal model in order to obtain a satisfying plan to be followed by the real system. A controller synthesis is then attempted for an abstraction of the real system to follow this plan. Upon failure of this synthesis, cost functions are defined to guide towards either refining the initially coarse partition to obtain a finer abstraction, or looking for an alternative plan using the nominal model as above. This tentative synthesis is then repeated until a plan and an abstraction of the real system able to follow this plan are found. The obtained controller also ensures that the real system satisfies the initial specification.

Multiscale hybrid-mixed method for the Stokes and Brinkman equations—The method

The multiscale hybrid-mixed (MHM) method is extended to the Stokes and Brinkman equations with highly heterogeneous coefficients. The approach is constructive. We first propose an equivalent dual-hybrid formulation of the original problem using a coarse partition of the heterogeneous domain. Faces may be not aligned with jumps in the data. Then, the exact velocity and the pressure are characterized as the solution of a global face problem and the solutions of local independent Stokes (or Brinkman) problems at the continuous level. Owing to this decomposition, the one-level MHM method stems from the standard Galerkin approach for the Lagrange multiplier space. Basis functions are responsible for upscaling the unresolved scales of the medium into the global formulation. They are the exact solution of the local problems with prescribed Neumann boundary conditions on faces driven by the Lagrange multipliers. We make the MHM method effective by adopting the unusual stabilized finite element method to solve the local problems approximately. As such, equal-order interpolation turns out to be an option for the velocity, the pressure and the Lagrange multipliers. The numerical solutions share the important properties of the continuum, such as local equilibrium with respect to external forces and local mass conservation. Several academic and highly heterogeneous tests infer that the method achieves super-convergence for the velocity as well optimal convergence for the pressure and also for the stress tensor in their natural norms.

Successful application of multiscale methods in a real reservoir simulator environment

For the past 10 years or so, a number of so-called multiscale methods have been developed as an alternative approach to upscaling and to accelerate reservoir simulation. The key idea of all these methods is to construct a set of prolongation operators that map between unknowns associated with cells in a fine grid holding the petrophysical properties of the geological reservoir model and unknowns on a coarser grid used for dynamic simulation. The prolongation operators are computed numerically by solving localized flow problems, much in the same way as for flow-based upscaling methods, and can be used to construct a reduced coarse-scale system of flow equations that describe the macro-scale displacement driven by global forces. Unlike effective parameters, the multiscale basis functions have subscale resolution, which ensures that fine-scale heterogeneity is correctly accounted for in a systematic manner. Among all multiscale formulations discussed in the literature, the multiscale restriction-smoothed basis (MsRSB) method has proved to be particularly promising. This method has been implemented in a commercially available simulator and has three main advantages. First, the input grid and its coarse partition can have general polyhedral geometry and unstructured topology. Secondly, MsRSB is accurate and robust when used as an approximate solver and converges relatively fast when used as an iterative fine-scale solver. Finally, the method is formulated on top of a cell-centered, conservative, finite-volume method and is applicable to any flow model for which one can isolate a pressure equation. We discuss numerical challenges posed by contemporary geomodels and report a number of validation cases showing that the MsRSB method is an efficient, robust, and versatile method for simulating complex models of real reservoirs.