Irregular Grid Research Articles

Consistent grid operators with the cell-nodal definition of grid functions

A principle of consistency for grid operators that ensures grid-operator inhomogeneous boundary-value problems are posed well is considered. Grid analogs of first-order differential operators and boundary operators that are consistent in the sense of fulfilling the grid analogs of integral relations are constructed on an irregular triangular grid. These relations are corollaries to the divergence theorem for vector fields that are the product of a scalar by a vector, the vector product of vectors, or the interior product of a vector by a dyadic. In each grid relation, one function is defined at the nodes; the other, in cells. Construction is performed using a grid-operator interpretation of the corollaries to the integral relations, which hold if one of the functions is a piecewise-linear interpolant to the nodal grid function, and the other is the piecewise-constant interpolant of the cell grid function.

ACCURATE SOLUTION AND GRADIENT COMPUTATION FOR ELLIPTIC INTERFACE PROBLEMS WITH VARIABLE COEFFICIENTS.

A new augmented method is proposed for elliptic interface problems with a piecewise variable coefficient that has a finite jump across a smooth interface. The main motivation is not only to get a second order accurate solution but also a second order accurate gradient from each side of the interface. The key of the new method is to introduce the jump in the normal derivative of the solution as an augmented variable and re-write the interface problem as a new PDE that consists of a leading Laplacian operator plus lower order derivative terms near the interface. In this way, the leading second order derivatives jump relations are independent of the jump in the coefficient that appears only in the lower order terms after the scaling. An upwind type discretization is used for the finite difference discretization at the irregular grid points near or on the interface so that the resulting coefficient matrix is an M-matrix. A multi-grid solver is used to solve the linear system of equations and the GMRES iterative method is used to solve the augmented variable. Second order convergence for the solution and the gradient from each side of the interface has also been proved in this paper. Numerical examples for general elliptic interface problems have confirmed the theoretical analysis and efficiency of the new method.

ON THE DEFINITION OF HEIGHT REFERENCE SURFACES OVER AN ARBITRARY SELECTED AREA BY MEANS OF DFHRS APPROACH

The aim of this study is to analyze the best fitting geoid model for an arbitrary selected area of investigation. The Digital Finite Element Height Reference Surface (DFHRS) method, developed by Hochschule Karlsruhe has been chosen to define the height reference surface for the territory of Kosovo, which should be defined as national vertical datum. This approach allows the conversion of ellipsoidal heights determined by GPS into the standard heights, which refer to the height reference surface (HRS) of an orthometric, or normal surface system. The DFHRS is defined as continues HRS in arbitrary large areas by bivariate polynomials over an irregular grid (Jager, Schneid 2001). The DFHRS approach uses wide range of the input data (Geometric and Physical) and in our case there were 30 GPS/leveling height data as well as physical derivatives from different global geopotential models. The proposed approach has been successfully applied and results are compared to actual normal heights and in selected profiles of digital elevation reference surface calculated from the national control network. Special attention has been given to the choice of the geopotential model and the selection of the pass points in source and target surface.

UNPLANNED SETTLEMENT FORM IN POLONIA VILLAGE MEDAN INDONESIA

Urbanization results in the increase of urban facilities and infrastructure needs in urban areas, especially the need for a place to stay. As a result, unplanned settlement is formed. The people living in Polonia Village for example look shabby and some are still living there illegally. The purpose of this research is to discover the form of settlement, the form of the distribution of community residences, the underlying terms and the process of the settlement formation in PoloniaVillage located on the banks of the Babura River. The research methods are qualitative approach in the form of interviews and quantitative approach in the form of a questionnaire. The results of this study are (1) Basically, the process of formation or the history of a settlement affects the settlement itself that it can develop sustainably, (2) The spreading pattern of the residential areas in every neighborhood in Polonia Village is an organic pattern and the road pattern is generally an irregular grid pattern, (3) The local communities’ thoughts play an important role in shaping the physical environment of Polonia Village, and (4) Polonia Village Medan has six patterns of unplanned settlement such as organic pattern, linear pattern, cluster pattern, central corridor pattern, grid pattern, and leapfrogging development pattern. The results of this study can be useful for North Sumatera Department of Settlements and Spatial Planning and for the science development in the field of socio-cultural and morphology

A general endogenous grid method for multi-dimensional models with non-convexities and constraints

The endogenous grid method (EGM) significantly speeds up the solution of stochastic dynamic programming problems by simplifying or completely eliminating root-finding. We propose a general and parsimonious EGM extended to handle (1) multiple continuous states and choices, (2) multiple occasionally binding constraints, and (3) non-convexities such as discrete choices. Our method enjoys the speed gains of the original one-dimensional EGM, while avoiding expensive interpolation on multi-dimensional irregular endogenous grids. We explicitly define a broad class of models for which our solution method is applicable, and illustrate its speed and accuracy using a consumption–saving model with both liquid assets and illiquid pension assets and a discrete retirement choice.

Spatio-Temporal Analysis of Water Quality Parameters in Machángara River with Nonuniform Interpolation Methods

Water quality measurements in rivers are usually performed at intervals of days or months in monitoring campaigns, but little attention has been paid to the spatial and temporal dynamics of those measurements. In this work, we propose scrutinizing the scope and limitations of state-of-the-art interpolation methods aiming to estimate the spatio-temporal dynamics (in terms of trends and structures) of relevant variables for water quality analysis usually taken in rivers. We used a database with several water quality measurements from the Machángara River between 2002 and 2007 provided by the Metropolitan Water Company of Quito, Ecuador. This database included flow rate, temperature, dissolved oxygen, and chemical oxygen demand, among other variables. For visualization purposes, the absence of measurements at intermediate points in an irregular spatio-temporal sampling grid was fixed by using deterministic and stochastic interpolation methods, namely, Delaunay and k-Nearest Neighbors (kNN). For data-driven model diagnosis, a study on model residuals was performed comparing the quality of both kinds of approaches. For most variables, a value of k = 15 yielded a reasonable fitting when Mahalanobis distance was used, and water quality variables were better estimated when using the kNN method. The use of kNN provided the best estimation capabilities in the presence of atypical samples in the spatio-temporal dynamics in terms of leave-one-out absolute error, and it was better for variables with slow-changing dynamics, though its performance degraded for variables with fast-changing dynamics. The proposed spatio-temporal analysis of water quality measurements provides relevant and useful information, hence complementing and extending the classical statistical analysis in this field, and our results encourage the search for new methods overcoming the limitations of the analyzed traditional interpolators.

A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier–Stokes equations

In this work we present a mimetic spectral element discretization for the 2D incompressible Navier–Stokes equations that in the limit of vanishing dissipation exactly preserves mass, kinetic energy, enstrophy and total vorticity on unstructured triangular grids. The essential ingredients to achieve this are: (i) a velocity–vorticity formulation in rotational form, (ii) a sequence of function spaces capable of exactly satisfying the divergence free nature of the velocity field, and (iii) a conserving time integrator. Proofs for the exact discrete conservation properties are presented together with numerical test cases on highly irregular triangular grids.

The role of soil flux and soil gas monitoring in the characterisation of a CO2 surface leak: A case study in Qinghai, China

Following the drilling of a shallow natural CO2 reservoir at the Qinghai research site, west of Haidong, China, it was discovered that CO2 was continuously leaking from the wellbore due to well-failure. The site has become a useful research facility in China for studying CO2 leakage and monitoring technologies for application to geological storage sites of CO2. During an eight day period in 2014, soil gas and soil flux surveys were conducted to characterise the distribution, magnitude and likely source of the leaking CO2.Two different sampling patterns were utilised during soil flux surveys. A regular sampling grid was used to spatially map out the two high-flux zones which were located 20–50m away from the wellhead. An irregular sampling grid, with higher sampling density in the high-flux zones, allowed for more accurate mapping of the leak distribution and estimation of total field emission rate using cubic interpolation. The total CO2 emission rate for the site was estimated at 649-1015kgCO2/d and there appeared to be some degree of spatial correlation between observed CO2 fluxes and elevated surface H2O fluxes.Sixteen soil gas wells were installed across the field to test the real-time application of Romanak et al.’s (2012) process-based approach for soil gas measurements (using ratios of major soil gas components to identify the CO2 source) using a portable multi-gas analyser. Results clearly identified CO2 as being derived from one exogenous source, and are consistent with gas samples collected for laboratory analysis. Carbon-13 isotopes in the centre of each leak zone (−0.21‰ and −0.22‰) indicate the underlying CO2 is likely sourced from the thermal decomposition of marine carbonates.Surface soil mineralisation (predominantly calcite) can be used to infer prior distribution of the CO2 hotspots and as a consequence highlighted plume migration of 20m in 11 years. The broadening of the affected area beyond the wellbore at the Qinghai research site markedly increases the area that needs surveying at sufficient density to detect a leak. This challenges the role of soil gas and soil flux in a CCS monitoring and verification program for leak detection, suggesting that these techniques may be better applied for characterising the source and emission rate of a CO2 leak, respectively.

Endogenous Grids in Higher Dimensions: Delaunay Interpolation and Hybrid Methods

This paper investigates extensions of the method of endogenous gridpoints (ENDGM) introduced by Carroll (Econ Lett 91(3):312–320, 2006) to higher dimensions with more than one continuous endogenous state variable. We compare three different categories of algorithms: (i) the conventional method with exogenous grids (EXOGM), (ii) the pure method of endogenous gridpoints (ENDGM) and (iii) a hybrid method (HYBGM). ENDGM comes along with Delaunay interpolation on irregular grids. Comparison of methods is done by evaluating speed and accuracy by using a specific model with two endogenous state variables. We find that HYBGM and ENDGM both dominate EXOGM. In an infinite horizon model, ENDGM also always dominates HYBGM. In a finite horizon model, the choice between HYBGM and ENDGM depends on the number of gridpoints in each dimension. With less than 150 gridpoints in each dimension ENDGM is faster than HYBGM, and vice versa. For a standard choice of 25–50 gridpoints in each dimension, ENDGM is 1.4–1.7 times faster than HYBGM in the finite horizon version and 2.4–2.5 times faster in the infinite horizon version of the model.

YAC 1.2.0: new aspects for coupling software in Earth system modelling

Abstract. A lightweight software library has been developed to realise the coupling of Earth system model components. The software provides parallelised two-dimensional neighbourhood search, interpolation, and communication for the coupling between any two model components. The software offers flexible coupling of physical fields defined on regular and irregular grids on the sphere without a priori assumptions about grid structure or grid element types. All supported grids can be combined with any of the supported interpolations. We describe the new aspects of our approach and provide an overview of the implemented functionality and of some algorithms we use. Preliminary performance measurements for a set of realistic use cases are presented to demonstrate the potential performance and scalability of our approach. YAC 1.2.0 is now used for the coupling of the model components in the Icosahedral Nonhydrostatic (ICON) general circulation model.

Implicit finite volume and discontinuous Galerkin methods for multicomponent flow in unstructured 3D fractured porous media

We present a new implicit higher-order finite element (FE) approach to efficiently model compressible multicomponent fluid flow on unstructured grids and in fractured porous subsurface formations. The scheme is sequential implicit: pressures and fluxes are updated with an implicit Mixed Hybrid Finite Element (MHFE) method, and the transport of each species is approximated with an implicit second-order Discontinuous Galerkin (DG) FE method. Discrete fractures are incorporated with a cross-flow equilibrium approach. This is the first investigation of all-implicit higher-order MHFE-DG for unstructured triangular, quadrilateral (2D), and hexahedral (3D) grids and discrete fractures. A lowest-order implicit finite volume (FV) transport update is also developed for the same grid types. The implicit methods are compared to an Implicit-Pressure-Explicit-Composition (IMPEC) scheme. For fractured domains, the unconditionally stable implicit transport update is shown to increase computational efficiency by orders of magnitude as compared to IMPEC, which has a time-step constraint proportional to the pore volume of discrete fracture grid cells. However, when lowest-order Euler time-discretizations are used, numerical errors increase linearly with the larger implicit time-steps, resulting in high numerical dispersion. Second-order Crank–Nicolson implicit MHFE-DG and MHFE-FV are therefore presented as well. Convergence analyses show twice the convergence rate for the DG methods as compared to FV, resulting in two to three orders of magnitude higher computational efficiency. Numerical experiments demonstrate the efficiency and robustness in modeling compressible multicomponent flow on irregular and fractured 2D and 3D grids, even in the presence of fingering instabilities.

Use of mobile phone traffic data for the assessment of air pollution exposure of urban population.

Environmental epidemiological studies use air pollutants concentrations to relate exposure with living population for estimation of health effects. This relationship is normally inferred by matching concentrations with addresses of residence of population. However, a large part of population spent most of the time away from its living place, introducing an incorrect assessment of exposure and of related health effects. Consequently, mobility of population should be considered for a proper assessment of exposure. Mobile phone traffic data (TIM Big Data Challenge 2015) were collected for more than a million of persons living/working in Rome, Italy, during March and April 2015 and used to derive 15 minutes time resolved population maps over an irregular grid with a minimum resolution of 0.26x0.34 Km2. Two datasets were available providing the number of persons as total and gender/age/postal code separated respectively. The latter data were matched in time and space with air pollution model results to produce time resolved cumulative population exposures during the studied period. Strong variability of exposures has been detected as an effect of both mobility and time/spatial changing in pollutants concentrations. Quartiles values are found to be up to ±50 and ±30% wider of their median values for NO2 and PM2.5 respectively. A comparison with the stationary approach based on census data allowed detecting the inability of latter in estimating the actual variability of population exposure. Significant underestimations of the amount population exposed to daily WHO guideline level for PM2.5 has been detected, with 25% and 80% on April 2015 for the census and the mobile phone based methods respectively. Very small differences (up to a few ug/m3) on exposure are detected for gender, age ranges and postal codes selected population.

Accurate reaction-diffusion operator splitting on tetrahedral meshes for parallel stochastic molecular simulations.

Spatial stochastic molecular simulations in biology are limited by the intense computation required to track molecules in space either in a discrete time or discrete space framework, which has led to the development of parallel methods that can take advantage of the power of modern supercomputers in recent years. We systematically test suggested components of stochastic reaction-diffusion operator splitting in the literature and discuss their effects on accuracy. We introduce an operator splitting implementation for irregular meshes that enhances accuracy with minimal performance cost. We test a range of models in small-scale MPI simulations from simple diffusion models to realistic biological models and find that multi-dimensional geometry partitioning is an important consideration for optimum performance. We demonstrate performance gains of 1-3 orders of magnitude in the parallel implementation, with peak performance strongly dependent on model specification.

On the consistency of inversion-free parameter estimation for Gaussian random fields

Gaussian random fields are a powerful tool for modeling environmental processes. For high dimensional samples, classical approaches for estimating the covariance parameters require highly challenging and massive computations, such as the evaluation of the Cholesky factorization or solving linear systems. Recently, Anitescu, Chen and Stein \cite{M.Anitescu} proposed a fast and scalable algorithm which does not need such burdensome computations. The main focus of this article is to study the asymptotic behavior of the algorithm of Anitescu et al. (ACS) for regular and irregular grids in the increasing domain setting. Consistency, minimax optimality and asymptotic normality of this algorithm are proved under mild differentiability conditions on the covariance function. Despite the fact that ACS's method entails a non-concave maximization, our results hold for any stationary point of the objective function. A numerical study is presented to evaluate the efficiency of this algorithm for large data sets.

Benchmarking different clustering algorithms on functional data

Theoretical knowledge of clustering functions is still scarce and only few models are available in form of applicable code. In literature, most methods are based on the projection of the functions onto a basis and building fixed or random effects models of the basis coefficients. They involve various parameters, among them number of basis functions, projection dimension, number of iterations etc. They usually work well on the data presented in the articles, but their performance has in most cases not been tested objectively on other data sets, nor against each other. The purpose of this paper is to give an overview of several existing methods to cluster functional data. An outline of their theoretic concepts is given and the meaning of their hyperparameters is explained. A simulation study was set up to analyze the parameters’ efficiency and sensitivity on different types of data sets, that were registered on regular and on irregular grids. For each method, a linear model of the clustering results was evaluated with different parameter levels as predictors. Later, the methods’ performances were compared to each other with the help of a visualization tool, to identify which method works the best on a specific kind of data.

Geochemical pattern of soils in Bobovdol valley, Bulgaria. Assessment of Cd and Co contents

The chemical composition of soils spread in the Bobov dol valley was studied in order to reveal the natural and anthropogenic patterns of Cd and Co spatial distribution. A sampling procedure based on the irregular grid of points and validated analytical methods were used in the field and laboratory studies. It is found that Cd content varies from 0.21 to 0.90 mg kg -1 in studied soils and the average value of 0.55 mg kg -1 coincides with concentration demarcating soil pollution (0.5 mg kg -1 ). Co content ranges from 2.22 to 15.76 mg kg -1 and in 70 % of sampled points exceeds the natural background content of 7.8 mg kg -1 found in local rocks. Still, Cd enrichment of studied soils is more significant than Co’s with coefficient of Clarke concentration of 3.67. Hence, the secondary deposition of studied elements as a result of the Bobov dol Thermal power plant air emissions is verified by results obtained. The spatial distribution of Cd and Co is featured with an altitudinal gradient in deposition and a trend of quantitative depletion in the South of Plant. Soil organic matter and pH have no influence on the content and spatial distribution of studied elements. Elements iron affinity governs their geochemical linkage in soils although cobalt occurs allied with aluminum and titanium.

STANDARDS-BASED SERVICES FOR BIG SPATIO-TEMPORAL DATA

Abstract. With the unprecedented availability of continuously updated measured and generated data there is an immense potential for getting new and timely insights – yet, the value is not fully leveraged as of today. The quest is up for high-level service interfaces for dissecting datasets and rejoining them with other datasets – ultimately, to allow users to ask "any question, anytime, on any size" enabling them to "build their own product on the go". With OGC Coverages, a concrete, interoperable data model has been established which unifies n-D spatio-temporal regular and irregular grids, point clouds, and meshes. The Web Coverage Service (WCS) suite provides versatile streamlined coverage functionality ranging from simple access to flexible spatio-temporal analytics. Flexibility and scalability of the WCS suite has been demonstrated in practice through massive services run by large-scale data centers. We present the current status in OGC Coverage data and service models, contrast them to related work, and describe a scalable implementation based on the rasdaman array engine.

STANDARDS-BASED SERVICES FOR BIG SPATIO-TEMPORAL DATA

With the unprecedented availability of continuously updated measured and generated data there is an immense potential for getting new and timely insights – yet, the value is not fully leveraged as of today. The quest is up for high-level service interfaces for dissecting datasets and rejoining them with other datasets – ultimately, to allow users to ask "any question, anytime, on any size" enabling them to "build their own product on the go". <br><br> With OGC Coverages, a concrete, interoperable data model has been established which unifies n-D spatio-temporal regular and irregular grids, point clouds, and meshes. The Web Coverage Service (WCS) suite provides versatile streamlined coverage functionality ranging from simple access to flexible spatio-temporal analytics. Flexibility and scalability of the WCS suite has been demonstrated in practice through massive services run by large-scale data centers. We present the current status in OGC Coverage data and service models, contrast them to related work, and describe a scalable implementation based on the rasdaman array engine.

Analysis of Reinforcement Based Adaptive Routing in MANET

<p class="Abstract">This paper describes and evaluates the performance of various reinforcement learning algorithms with shortest path algorithms that are widely used for routing packets through the network. Shortest path routing is the simplest policy used for routing the packets along the path having minimum number of hops. In high traffic or high mobility conditions, the shortest path get flooded with huge number of packets and congestions occurs, So such shortest path does not provides the shortest path and increases delay for reaching the packets to the destination. Reinforcement learning algorithms are adaptive algorithms where the path is selected based on the traffic present on the network at real time. Thus they guarantee the least delivery time to reach the packets to the destination. Analysis done on a 6 by 6 irregular grid and sample ad hoc network shows that performance parameters used for judging the network - packet delivery ratio and delay provides optimum results using reinforcement learning algorithms. </p>

Off-the-Grid Low-Rank Matrix Recovery and Seismic Data Reconstruction

Matrix sensing problems capitalize on the knowledge that a data matrix of interest exhibits low rank properties. This low dimensional structure often arises because the data matrix is obtained by sampling a smooth function on a regular (or structured) grid. However, in many practical situations the measurements are taken on an irregular grid (that is accurately known). This results in an “unstructured data matrix” that is less fit for the low rank model in comparison to its regular counterpart and therefore subject to degraded reconstruction via rank penalization techniques. In this paper, we propose and analyze a modified low-rank matrix recovery work-flow that admits unstructured observations. By incorporating a regularization operator which accurately maps structured data to unstructured data, into the nuclear-norm minimization problem, we are able to compensate for data irregularity. Furthermore, by construction our formulation yields output that is supported on a structured grid. We establish recovery error bounds for our methodology and offer matrix sensing and matrix completion numerical experiments including applications to seismic trace interpolation to demonstrate the potential of the approach.