Numerical Benchmark Research Articles

Sequential Kalman Tuning of the t-preconditioned Crank-Nicolson algorithm: efficient, adaptive and gradient-free inference for Bayesian inverse problems

Abstract Ensemble Kalman Inversion (EKI) has been proposed as an efficient method for the approximate solution of Bayesian inverse problems with expensive forward models. However, when applied to the Bayesian inverse problem EKI is only exact in the regime of Gaussian target measures and linear forward models. In this work we propose embedding EKI and Flow Annealed Kalman Inversion (FAKI), its normalizing flow (NF) preconditioned variant, within a Bayesian annealing scheme as part of an adaptive implementation of the $t$-preconditioned Crank-Nicolson (tpCN) sampler. The tpCN sampler differs from standard pCN in that its proposal is reversible with respect to the multivariate $t$-distribution. The more flexible tail behaviour allows for better adaptation to sampling from non-Gaussian targets. Within our Sequential Kalman Tuning (SKT) adaptation scheme, EKI is used to initialize and precondition the tpCN sampler for each annealed target. The subsequent tpCN iterations ensure particles are correctly distributed according to each annealed target, avoiding the accumulation of errors that would otherwise impact EKI. We demonstrate the performance of SKT for tpCN on three challenging numerical benchmarks, showing significant improvements in the rate of convergence compared to adaptation within standard SMC with importance weighted resampling at each temperature level, and compared to similar adaptive implementations of standard pCN. The SKT scheme applied to tpCN offers an efficient, practical solution for solving the Bayesian inverse problem when gradients of the forward model are not available. Code implementing the SKT schemes for tpCN is available at \url{https://github.com/RichardGrumitt/KalmanMC}.

CartoMark: a benchmark dataset for map pattern recognition and map content retrieval with machine intelligence

Maps are fundamental medium to visualize and represent the real word in a simple and philosophical way. The emergence of the big data tide has made a proportion of maps generated from multiple sources, significantly enriching the dimensions and perspectives for understanding the characteristics of the real world. However, a majority of these map datasets remain undiscovered, unacquired and ineffectively used, which arises from the lack of numerous well-labelled benchmark datasets, which are of significance to implement the deep learning techniques into identifying complicated map content. To address this issue, we develop a large-scale benchmark dataset involving well-labelled datasets to employ the state-of-the-art machine intelligence technologies for map text annotation recognition, map scene classification, map super-resolution reconstruction, and map style transferring. Furthermore, these well-labelled datasets would facilitate map feature detection, map pattern recognition and map content retrieval. We hope our efforts would provide well-labelled data resources for advancing the ability to recognize and discover valuable map content.

A micropolar phase-field model for size-dependent electro-mechanical fracture

This article proposes a micropolar phase-field model for size-dependent brittle fracture in solids under electro-mechanical loading conditions. Considering displacement, micro-rotation, electric potential, and phase-field variable as the kinematic descriptors and employing the virtual power principle, we derive a set of coupled governing partial differential equations (PDEs) for size-dependent solids. Invoking the first and second laws of thermodynamics, we determine the constitutive relations for the thermodynamic fluxes. Carrying out the finite element implementation of the derived governing PDEs using the open-source Gridap package in Julia, we demonstrate the efficacy of the proposed phase-field model through a few representative numerical examples. Especially the importance of the proposed model in incorporating the effect of relative rotation, i.e., the difference between macro- and micro-rotation, on the response of solids under electro-mechanical loading is shown that may not be possible with the existing non-local models such as strain-gradient or couple-stress approaches. To capture the experimentally observed size effects in solids under electro-mechanical loading, the proposed model does not demand higher-order continuity of finite element shape functions, unlike a typically used strain gradient model. To demonstrate the efficacy of the proposed model, we have compared our results against demanding experimental and numerical benchmark results available in the literature. We provide a parametric study to unravel the effect of different micropolar material parameters on the electro-mechanical response of a brittle solid. Interestingly, the proposed micropolar model is less sensitive to the phase-field length scale than the conventional non-polar phase-field models.

On Discrete Subproblems in Integer Optimal Control with Total Variation Regularization in Two Dimensions

We analyze integer linear programs that we obtain after discretizing two-dimensional subproblems arising from a trust-region algorithm for mixed integer optimal control problems with total variation regularization. We discuss NP-hardness of the discretized problems and the connection to graph-based problems. We show that the underlying polyhedron exhibits structural restrictions in its vertices with regard to which variables can attain fractional values at the same time. Based on this property, we derive cutting planes by employing a relation to shortest-path and minimum bisection problems. We propose a branching rule and a primal heuristic which improves previously found feasible points. We validate the proposed tools with a numerical benchmark in a standard integer programming solver. We observe a significant speedup for medium-sized problems. Our results give hints for scaling toward larger instances in the future. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the Deutsche Forschungsgemeinschaft [Grant MA 10080/2-1]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0680 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0680 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Grassmann time-evolving matrix product operators: An efficient numerical approach for fermionic path integral simulations.

Developing numerical exact solvers for open quantum systems is a challenging task due to the non-perturbative and non-Markovian nature when coupling to structured environments. The Feynman-Vernon influence functional approach is a powerful analytical tool to study the dynamics of open quantum systems. Numerical treatments of the influence functional including the quasi-adiabatic propagator technique and the tensor-network-based time-evolving matrix product operator method have proven to be efficient in studying open quantum systems with bosonic environments. However, the numerical implementation of the fermionic path integral suffers from the Grassmann algebra involved. In this work, we present a detailed introduction to the Grassmann time-evolving matrix product operator method for fermionic open quantum systems. In particular, we introduce the concepts of Grassmann tensor, signed matrix product operator, and Grassmann matrix product state to handle the Grassmann path integral. Using the single-orbital Anderson impurity model as an example, we review the numerical benchmarks for structured fermionic environments for real-time nonequilibrium dynamics, real-time and imaginary-time equilibration dynamics, and its application as an impurity solver. These benchmarks show that our method is a robust and promising numerical approach to study strong coupling physics and non-Markovian dynamics. It can also serve as an alternative impurity solver to study strongly correlated quantum matter with dynamical mean-field theory.

IMST-based identification of time-varying cable forces in cable-supported bridges: Numerical, experimental, and field test validation

Accurately assessing the structural integrity of cable-supported bridges hinges upon the precise determination of time-varying cable forces. In this study, a novel method is proposed for identifying these forces, even with the challenge of having only a single accelerometer available. Leveraging an improved Multisynchrosqueezing Transform (IMST) with an efficient ridge extraction algorithm, this method offers a robust solution. The IMST is employed to construct a clearer time-frequency spectrum based on the monitored acceleration response of a cable. Subsequently, the time-frequency ridge extraction algorithm enables the derivation of the cable's instantaneous frequency. Ultimately, the time-varying cable forces are computed by using the axially loaded beam theory and considering the influence of bending stiffness. Validations through a numerical benchmark model and a scaled cable testing confirm the efficacy of this approach. Result indicates that the average error in identifying cable forces is less than 2.50% in the numerical simulation. Moreover, the method is successfully applied to a real-world bridge scenario, demonstrating its ability to accurately evaluate cable forces despite having access to only a single-point noisy acceleration response.

Investigating the Performance of the Informer Model for Streamflow Forecasting

Recent studies have shown the potential of transformer-based neural networks in increasing prediction capacity. However, classical transformers present several problems such as computational time complexity and high memory requirements, which make Long Sequence Time-Series Forecasting (LSTF) challenging. The contribution to the prediction of time series of flood events using deep learning techniques is examined, with a particular focus on evaluating the performance of the Informer model (a particular implementation of transformer architecture), which attempts to address the previous issues. The predictive capabilities of the Informer model are explored and compared to statistical methods, stochastic models and traditional deep neural networks. The accuracy, efficiency as well as the limits of the approaches are demonstrated via numerical benchmarks relating to real river streamflow applications. Using daily flow data from the River Test in England as the main case study, we conduct a rigorous evaluation of the Informer efficacy in capturing the complex temporal dependencies inherent in streamflow time series. The analysis is extended to encompass diverse time series datasets from various locations (>100) in the United Kingdom, providing insights into the generalizability of the Informer. The results highlight the superiority of the Informer model over established forecasting methods, especially regarding the LSTF problem. For a forecast horizon of 168 days, the Informer model achieves an NSE of 0.8 and maintains a MAPE below 10%, while the second-best model (LSTM) only achieves −0.63 and 25%, respectively. Furthermore, it is observed that the dependence structure of time series, as expressed by the climacogram, affects the performance of the Informer network.

Soil sampling design matters - Enhancing the efficiency of digital soil mapping at the field scale

Optimisation of sampling design (methods chosen to select the samples) and sample size (number of samples) remains a key challenge in digital soil mapping, especially in the area of precision farming with the expected economic benefits from the introduction of new technologies. As the existing information is available in the form of relevant environmental covariates, its combination with non-parametric machine learning techniques requires careful planning from the initial field sampling to the final production of digital soil maps. The aim of this study is to compare widely used covariate-wise sampling designs combined with variable sample sizes for supervised prediction of common soil drivers of agricultural productivity (pH, soil organic carbon, soil macronutrients) in a real case study of a field (35 ha) with heterogeneous soil properties. From a total of 200 samples, we evaluated different sample sets where 10, 30 and 60 field samples were selected by conditioned Latin Hypercube Sampling (cLHS) and Feature Space Coverage Sampling (FSCS) to calibrate random forest (RF) models. The evaluation was performed on independently in-situ sampled test points. In addition to these datasets, we also compared the investigated methods with Simple Random Sampling (SRS) in a numerical benchmark experiment with increasing sample size, comparing the global accuracies of the predicted maps on the test points, but using interpolated maps as the artificial true population for each soil characteristic. The results of the study in both the field experiment and the numerical experiment showed slightly better results for the FSCS method, especially when the number of samples was small. At smaller training sample sizes, the risk of insufficiently accurate prediction models was slightly lower for FSCS and the difference decreased as the sample size increased. Nevertheless, sample size proved to be the most important factor in the accuracy of RF models, regardless of the sampling technique. The results suggest that a sample size between 18 and 30 training samples (0.6 to 1 sample ha−1) seems plausible for covariate-wise predictions using RF at field scale in our case study. The relative importance of each auxiliary variable for each RF calibration was also assessed for the field experiment. The results showed that the additional introduction of spatial proxies overshadowed the importance of other covariates, but only significantly improved the model calibration at larger sample sizes. The calibrated models without spatial proxies showed the strongest effect of remotely sensed surface characteristics.

Meta Invariance Defense Towards Generalizable Robustness to Unknown Adversarial Attacks.

Despite providing high-performance solutions for computer vision tasks, the deep neural network (DNN) model has been proved to be extremely vulnerable to adversarial attacks. Current defense mainly focuses on the known attacks, but the adversarial robustness to the unknown attacks is seriously overlooked. Besides, commonly used adaptive learning and fine-tuning technique is unsuitable for adversarial defense since it is essentially a zero-shot problem when deployed. Thus, to tackle this challenge, we propose an attack-agnostic defense method named Meta Invariance Defense (MID). Specifically, various combinations of adversarial attacks are randomly sampled from a manually constructed Attacker Pool to constitute different defense tasks against unknown attacks, in which a student encoder is supervised by multi-consistency distillation to learn the attack-invariant features via a meta principle. The proposed MID has two merits: 1) Full distillation from pixel-, feature- and prediction-level between benign and adversarial samples facilitates the discovery of attack-invariance. 2) The model simultaneously achieves robustness to the imperceptible adversarial perturbations in high-level image classification and attack-suppression in low-level robust image regeneration. Theoretical and empirical studies on numerous benchmarks such as ImageNet verify the generalizable robustness and superiority of MID under various attacks.

Research on Optimization of Visual Object Tracking Algorithm Based on Deep Learning

The appearance of deep gaining knowledge has extensively advanced the sphere of visible item tracking, permitting greater robust and accurate monitoring of items throughout complex scenes. This study optimises a visual item tracking set of rules based on the Siamese region inspiration network (Siamese RPN) monitoring algorithm, aiming to beautify its efficiency and effectiveness in actual-time packages. The Siamese RPN algorithm, acknowledged for its stability among accuracy and velocity because of its architecture that mixes the Siamese network for characteristic extraction with a vicinity suggestion community for item localisation, provides a promising basis for improvement. This examination introduces numerous optimisations to the authentic Siamese RPN framework. First, we endorse an improved feature extraction model that leverages a more efficient deep neural community structure, lowering computational load while preserving excessive accuracy. 2nd, we optimise the place concept mechanism by incorporating an adaptive anchor scaling method that dynamically adjusts the scale and ratio of anchors based on the object’s scale variations, enhancing the tracking accuracy across distinctive object sizes and aspect ratios. Moreover, we introduce a unique training method that employs an aggregate of actual global and synthetically generated statistics to beautify the robustness of the monitoring algorithm towards various demanding situations, including occlusions, speedy moves, and illumination adjustments. The effectiveness of the proposed optimizations is evaluated through complete experiments on numerous benchmark datasets, consisting of OTB, VOT, and LaSOT, demonstrating extensive upgrades in tracking accuracy and speed as compared to the authentic Siamese RPN algorithm and different modern-day tracking techniques. The outcomes of this study no longer underscore the potential of optimised Siamese RPN algorithms in visible item monitoring but additionally lay the basis for future explorations into actual-time, green, and strong tracking systems. Those improvements keep great promise for a wide variety of packages, from surveillance and protection to autonomous cars and augmented truth structures, where particular and dependable item monitoring is paramount.

Spin Exchange-Enabled Quantum Simulator for Large-Scale Non-Abelian Gauge Theories

A central requirement for the faithful implementation of large-scale lattice gauge theories (LGTs) on quantum simulators is the protection of the underlying gauge symmetry. Recent advancements in the experimental realizations of large-scale LGTs have been impressive, albeit mostly restricted to Abelian gauge groups. Guided by this requirement for gauge protection, we propose an experimentally feasible approach to implement large-scale non-Abelian SU(N) and U(N) LGTs with dynamical matter in d+1D, enabled by two-body spin-exchange interactions realizing local emergent gauge-symmetry stabilizer terms. We present two concrete proposals for 2+1DSU(2) and U(2) LGTs, including dynamical bosonic matter and induced plaquette terms, that can be readily implemented in current ultracold-molecule and next-generation ultracold-atom platforms. We provide numerical benchmarks showcasing experimentally accessible dynamics, and demonstrate the stability of the underlying non-Abelian gauge invariance. We develop a method to obtain the effective gauge-invariant model featuring the relevant magnetic plaquette and minimal gauge-matter coupling terms. Our approach paves the way towards near-term realizations of large-scale non-Abelian quantum link models in analog quantum simulators. Published by the American Physical Society 2024

Numerical investigation of pile foundation systems employing an enhanced embedded finite element

The design of large-scale pile foundation systems is routinely assisted by finite element simulations. To a large extent, both the modeling and the computational effort of such analyses are governed by the adopted pile modeling technique. The traditional approach to this problem fully resolves the pile and soil domain employing solid elements, resulting in considerable meshing constraints and high simulation runtimes that may be regarded as unbearable for many practical purposes. As an attractive alternative to circumvent these obstacles, embedded FE models have become increasingly popular in solving this modeling task, mainly due to their flexible meshing procedure and significantly enhanced runtime efficiency. In a preceding contribution, the authors have proposed an extended formulation that provides a rigorous framework to capture soil-structure interaction effects at the physical soil-pile contacts. As a key feature, the implemented combined soil-pile coupling scheme explicitly accounts for endpoint interaction. However, validation studies have been constrained to single pile analyses to date. The present work expands this validation scope to large-scale boundary value problems involving multiple piles and investigates the model performance based on three different case studies. The results are compared to both, measurements and numerical benchmark solutions and provide exclusive insight into the numerical fidelity of the developed embedded FE model, with a view to increasing its potential for take-up in engineering practice.

On Complexity Bounds and Confluence of Parallel Term Rewriting*

We revisit parallel-innermost term rewriting as a model of parallel computation on inductive data structures and provide a corresponding notion of runtime complexity parametric in the size of the start term. We propose automatic techniques to derive both upper and lower bounds on parallel complexity of rewriting that enable a direct reuse of existing techniques for sequential complexity. Our approach to find lower bounds requires confluence of the parallel-innermost rewrite relation, thus we also provide effective sufficient criteria for proving confluence. The applicability and the precision of the method are demonstrated by the relatively light effort in extending the program analysis tool APROVE and by experiments on numerous benchmarks from the literature.

Thermally radiative chemically reactive flow of Reiner-Rivlin nanofluid through porous medium with Newtonian conditions

The nanofluids have well-known implications in pharmaceutical industries, microelectronics, heat exchangers, engine cooling, hybrid-powered engines, thermal management of vehicles, refrigerator, machining, chillers, grinding, and in boiler fuel to reduce the gas temperature. Due to such dominant implications of nanofluids, the boundary layer Reiner-Rivlin nanomaterial flow over chemically reactive stretched sheet is considered. Radiation term is incorporated in the energy equation for heat transportation analysis. Newtonian thermal and solutal conditions are implemented at the boundaries of the surface. Similarity variables are utilized for the conversion of nonlinear partial differential equations into the system of one independent variable equations. The resulted system of equations is calculated analytically with the help of homotopy analysis method. Convergence of the calculated results is verified through plots and numeric benchmark. The results of various pertinent parameters on quantities of physical importance are drawn and discussed in detail. Results revealed that the incremented cross viscous constraint resulted higher velocity and lower temperature profiles. An augmentation in radiative constraint gives rise to temperature field. Concentration and temperature have reverse trends against the rising Brownian motion constraint values.

Fast empirical scenarios

We seek to extract a small number of representative scenarios from large panel data that are consistent with sample moments. Among two novel algorithms, the first identifies scenarios that have not been observed before, and comes with a scenario-based representation of covariance matrices. The second proposal selects important data points from states of the world that have already realized, and are consistent with higher-order sample moment information. Both algorithms are efficient to compute and lend themselves to consistent scenario-based modeling and multi-dimensional numerical integration that can be used for interpretable decision-making under uncertainty. Extensive numerical benchmarking studies and an application in portfolio optimization favor the proposed algorithms.

Learning integral operators via neural integral equations

Nonlinear operators with long-distance spatiotemporal dependencies are fundamental in modelling complex systems across sciences; yet, learning these non-local operators remains challenging in machine learning. Integral equations, which model such non-local systems, have wide-ranging applications in physics, chemistry, biology and engineering. We introduce the neural integral equation, a method for learning unknown integral operators from data using an integral equation solver. To improve scalability and model capacity, we also present the attentional neural integral equation, which replaces the integral with self-attention. Both models are grounded in the theory of second-kind integral equations, where the indeterminate appears both inside and outside the integral operator. We provide a theoretical analysis showing how self-attention can approximate integral operators under mild regularity assumptions, further deepening previously reported connections between transformers and integration, as well as deriving corresponding approximation results for integral operators. Through numerical benchmarks on synthetic and real-world data, including Lotka–Volterra, Navier–Stokes and Burgers’ equations, as well as brain dynamics and integral equations, we showcase the models’ capabilities and their ability to derive interpretable dynamics embeddings. Our experiments demonstrate that attentional neural integral equations outperform existing methods, especially for longer time intervals and higher-dimensional problems. Our work addresses a critical gap in machine learning for non-local operators and offers a powerful tool for studying unknown complex systems with long-range dependencies.

The value of simplified models of radionuclide transport for the safety assessment of nuclear waste repositories: A benchmark study

In order to assess sites for a deep geological repository for storing high-level nuclear waste safely in Germany, various numerical models and tools will be in use. For their interaction within one workflow, their reproducibility, and reliability version-controlled open-source solutions and careful documentation of model setups, results and verifications are of special value. However, spatially fully resolved models including all relevant physical and chemical processes are neither computationally feasible for large domains nor is the data typically available to parameterize such models. Thus, simplified models are crucial for the pre-assessment of possible sites to narrow down the list of suitable candidates for which detailed site investigations and fully resolved models will be done at a later stage. Still, the accuracy of these simplified models is of importance as the pre-assessment of suitable sites will be based on them. In this study, we compare the modelling capabilities of TransPyREnd, a one-dimensional transport code based on finite differences, specifically developed for the fast estimation of radionuclide transport by the German federal company for radioactive waste disposal (BGE), with OpenGeoSys, which is a modelling platform based on finite elements in up to three spatial dimensions. Both codes are used in the site selection procedure for the German nuclear waste repository. The comparison of the model results of TransPyREnd and OpenGeoSys is augmented by comparisons with an analytical solution for a homogeneous material. For the purpose of numerical benchmarking, we consider a geological profile located in southern Germany as an example where the hypothetical repository is located in a clay-stone formation. TransPyREnd and OpenGeoSys yield overall similar results. However, both codes use different discretizations which impact is the highest for strongly sorbing compounds, while the difference gets negligible for less sorbing and more diffusive compounds as higher diffusion tends to blur the initial conditions. Overall, the OpenGeoSys model is more exact whereas the TransPyREnd model has considerable faster run times. We found in our example, that significant substance amounts are only leaving the host rock formation, if apparent diffusion is high, for which case both codes give similar results, while relative differences are considerable for strongly sorbing compounds. However, in the latter case no significant substance amount of radionuclides leaves the host-rock formation, thus deeming the differences in the model results minor for the overall safety assessments of sites.

GX: a GPU-native gyrokinetic turbulence code for tokamak and stellarator design

GX is a code designed to solve the nonlinear gyrokinetic system for low-frequency turbulence in magnetized plasmas, particularly tokamaks and stellarators. In GX, our primary motivation and target is a fast gyrokinetic solver that can be used for fusion reactor design and optimization along with wide-ranging physics exploration. This has led to several code and algorithm design decisions, specifically chosen to prioritize time to solution. First, we have used a discretization algorithm that is pseudospectral in the entire phase space, including a Laguerre–Hermite pseudospectral formulation of velocity space, which allows for smooth interpolation between coarse gyrofluid-like resolutions and finer conventional gyrokinetic resolutions and efficient evaluation of a model collision operator. Additionally, we have built GX to natively target graphics processors (GPUs), which are among the fastest computational platforms available today. Finally, we have taken advantage of the reactor-relevant limit of small $\rho _*$ by using the radially local flux-tube approach. In this paper we present details about the gyrokinetic system and the numerical algorithms used in GX to solve the system. We then present several numerical benchmarks against established gyrokinetic codes in both tokamak and stellarator magnetic geometries to verify that GX correctly simulates gyrokinetic turbulence in the small $\rho _*$ limit. Moreover, we show that the convergence properties of the Laguerre–Hermite spectral velocity formulation are quite favourable for nonlinear problems of interest. Coupled with GPU acceleration, which we also investigate with scaling studies, this enables GX to be able to produce useful turbulence simulations in minutes on one (or a few) GPUs and higher fidelity results in a few hours using several GPUs. GX is open-source software that is ready for fusion reactor design studies.

Toxicity of rare earth elements (REEs) to marine organisms: Using species sensitivity distributions to establish water quality guidelines for protecting marine life

A lack of chronic rare earth element (REE) toxicity data for marine organisms has impeded the establishment of numerical REE water quality benchmarks (e.g., guidelines) to protect marine life and assess ecological risk. This study determined the chronic no (significant) effect concentrations (N(S)ECs) and median-effect concentrations (EC50s) of eight key REEs (yttrium (Y), lanthanum (La), cerium (Ce), praseodymium (Pr), neodymium (Nd), gadolinium (Gd), dysprosium (Dy) and lutetium (Lu)) for 30 coastal marine organisms (encompassing 22 phyla and five trophic levels from temperate and tropical habitats). Organisms with calcifying life stages were most vulnerable to REEs, which competitively inhibit calcium uptake. The most sensitive organism was a sea urchin, with N(S)ECs ranging from 0.64 μg/L for Y to 1.9 μg/L for La and Pr, and EC50s ranging from 4.3 μg/L for Y to 14.4 μg/L for Pr. Conversely, the least sensitive organism was a cyanobacterium, with N(S)ECs ranging from 121 μg/L for Y to 469 μg/L for Pr, and EC50s ranging from 889 μg/L for Y to 3000 μg/L for Pr. Median sensitivity varied 215-fold across all organisms. The two-fold difference in median toxicity (μmol/L EC50) among REEs (Y ∼ Gd > Lu ∼ Nd ∼ Dy ∼ Ce > La ∼ Pr) was attributed to offset differences in binding affinity (log K) to cell surface receptors and the percentage of free metal ion (REE3+) in the test waters. The toxicity (EC50) of the remaining REEs (samarium, europium, terbium, holmium, thulium and ytterbium) was predicted using a combination of physicochemical data and measured EC50s for the eight tested REEs, with good agreement between predicted and measured EC50s for selected organisms. Numerical REE water quality guidelines to protect marine life were established using species sensitivity distributions (e.g., for 95 % species protection, values ranged from 1.1 μg/L for Y to 3.0 μg/L for La, Pr or Lu).

The enriched multinode Shepard collocation method for solving elliptic problems with singularities

In this paper, the multinode Shepard method is adopted for the first time to numerically solve a differential problem with a discontinuity in the boundary. Starting from previous studies on elliptic boundary value problems, here the Shepard method is employed to catch the singularity on the boundary. Enrichments of the functional space spanned by the multinode cardinal Shepard basis functions are proposed to overcome the difficulties encountered. The Motz's problem is considered as numerical benchmark to assess the method. Numerical results are presented to show the effectiveness of the proposed approach.