Temporal Error Research Articles

K-space dispersion error compensators for the fractional spatial derivatives based constant-Q viscoelastic wave equation modeling

Unlike the fractional temporal derivatives based constant-Q viscoelastic wave equation, the fractional spatial derivatives based one is preferred for the wavefield simulation due to its tractable memory consumption. This viscoelastic wave equation is often solved by the staggered-grid pseudospectral method, which has the spatial spectral accuracy but temporal low order accuracy. The k-space method is one of the potential remedies for the temporal discretization errors. However, it is difficult to fully remove the temporal dispersion errors by directly applying the elastic k-space error compensators to the viscoelastic wave equation, which contains additional temporal derivatives associated with the amplitude decay terms of the P- and S- waves. To tackle the issue, we have derived a new wavenumber-time domain viscoelastic k-space equation by the eigenvalue decomposition method and by the solution to the matrix differential equations. The error compensation terms inside our k-space equation can simultaneously correct the discretization errors of different types of temporal derivatives and are wave mode adaptive. The accuracy of the k-space wave equation is verified by the viscoelastic Green's function's solution. To further boost the simulation efficiency of the k-space method in heterogeneous media, we have proposed an improved k-space method, which decreases the number of the costly mixed-domain operators. The simulation results from the simple and complex models demonstrate that our two k-space methods are superior to the conventional staggered-grid pseudospectral method for the viscoelastic wave modeling.

Analysis of a Filtered Time-Stepping Finite Element Method for Natural Convection Problems

In this paper, we present, analyze, and test a novel low-complexity time-stepping finite element method for natural convection problems utilizing a time filter (TF). First, via a TF to postprocess the solutions of backward Euler (BE) schemes, we make a minimally intrusive modification to the existing codes to improve the time accuracy by one order. This also provides, at no extra complexity, an estimate of the temporal error,which is easy to construct a novel adaptive algorithm. Additionally, the TF can remove the overdamping of the BE scheme while remaining unconditionally energy stable. Hence, this paper addresses the question, how can one improve the time accuracy without increasing computational and cognitive complexity? Then long time stability and error estimates of BE plus time filter (BETF) with constant time stepsize are proved. Moreover, we construct adaptive algorithms by extending the approach to variable time stepsize, and we extend the methods to higher order algorithms. Finally, numerical tests confirm the convergence rates of our method and validate the theoretical results.

Training Spiking Neural Networks with Metaheuristic Algorithms

Taking inspiration from the brain, spiking neural networks (SNNs) have been proposed to understand and diminish the gap between machine learning and neuromorphic computing. Supervised learning is the most commonly used learning algorithm in traditional ANNs. However, directly training SNNs with backpropagation-based supervised learning methods is challenging due to the discontinuous and non-differentiable nature of the spiking neuron. To overcome these problems, this paper proposes a novel metaheuristic-based supervised learning method for SNNs by adapting the temporal error function. We investigated seven well-known metaheuristic algorithms called Harmony Search (HS), Cuckoo Search (CS), Differential Evolution (DE), Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Artificial Bee Colony (ABC), and Grammatical Evolution (GE) as search methods for carrying out network training. Relative target firing times were used instead of fixed and predetermined ones, making the computation of the error function simpler. The performance of our proposed approach was evaluated using five benchmark databases collected in the UCI Machine Learning Repository. The experimental results showed that the proposed algorithm had a competitive advantage in solving the four classification benchmark datasets compared to the other experimental algorithms, with accuracy levels of 0.9858, 0.9768, 0.7752, and 0.6871 for iris, cancer, diabetes, and liver datasets, respectively. Among the seven metaheuristic algorithms, CS reported the best performance.

A uniformly accurate method for the Klein-Gordon-Dirac system in the nonrelativistic regime

A multiscale time integrator Fourier pseudospectral (MTI-FP) method is presented for discretizing the massive Klein-Gordon-Dirac (KGD) system which involves a small dimensionless parameter 0<ε≤1. In the nonrelativistic limit regime, the KGD system admits rapid oscillations in time as ε→0+. In addition, the nonlinear Yukawa interaction and the indefinite Dirac operator bring other significant difficulties. The main idea of the MTI-FP method is to construct a precise multiscale decomposition by the frequency (MDF) to the solution of the KGD system at each time step and then employ the Fourier pseudospectral discretization for the spatial derivatives followed with the exponential wave integrator (EWI) for the time marching. This approach is explicit, easy to implement and preforms significantly better than the classical methods in the literature. More specifically, we rigorously establish the uniform error bounds at O(τ+hm0−1) for all ε∈(0,1] and optimal quadratic temporal error bounds at O(τ2) in the ε=O(1) regime, where τ is the time step size, h is mesh size and m0 depends on the regularity of the solution. Extensive numerical results demonstrate that our error bounds are optimal and sharp. Finally, we apply the MTI-FP method to numerically study the nonrelativistic limit behaviors of the KGD system when ε→0+.

A Space-Time Multiscale Mortar Mixed Finite Element Method for Parabolic Equations

We develop a space-time mortar mixed finite element method for parabolic problems. The domain is decomposed into a union of subdomains discretized with nonmatching spatial grids and asynchronous time steps. The method is based on a space-time variational formulation that couples mixed finite elements in space with discontinuous Galerkin in time. Continuity of flux (mass conservation) across space-time interfaces is imposed via a coarse-scale space-time mortar variable that approximates the primary variable. Uniqueness, existence, and stability as well as a priori error estimates for the spatial and temporal errors are established. A space-time nonoverlapping domain decomposition method is developed that reduces the global problem to a space-time coarse-scale mortar interface problem. Each interface iteration involves solving in parallel space-time subdomain problems. The spectral properties of the interface operator and the convergence of the interface iteration are analyzed. Numerical experiments are provided that illustrate the theoretical results and the flexibility of the method for modeling problems with features that are localized in space and time.

Small satellite formations and constellations for observing sub-daily mass changes in the Earth system

SUMMARY Satellite gravity missions so far are medium size satellites consisting of one or a pair of satellites flying in near-polar or sun-synchronous orbital planes. Due to the limited observation geometry and the related space–time sampling, high-frequency non-tidal mass variation signals from atmosphere and ocean cannot be observed and cause temporal aliasing. For current single-pair satellite gravimetry missions as GRACE and GRACE Follow-On (GRACE-FO) temporal aliasing is the limiting factor and represents the major error source in the gravity field time-series. Adding a second inclined satellite pair to a GRACE-like polar pair (Bender constellation) currently is the most promising solution to increase the spatio-temporal resolution and to significantly reduce the temporal aliasing error. This shall be implemented with the MAGIC mission in future. With the ongoing developments in miniaturization of satellites and gravity-relevant instruments (accelerometers and intersatellite ranging), in future constellations of multiple small satellite pairs may solve this problem even beyond the capabilities of a Bender constellation. Therefore, in this study the capabilities of such constellations flying in specific formations are investigated in order to enable a retrieval of the temporal gravity field on short time scales. We assess the performance of up to 18 satellite pairs. The satellite configurations cover satellite pairs in polar and inclined orbits flying in pair-wise or pearl-string formation with varying mean anomalies and right ascensions of the ascending node (RAAN). As future potential miniaturized instruments optomechanical accelerometers with similar performance as those flying on GRACE-FO are a candidate, while for the intersatellite ranging instrument still some technological development is required. Therefore, in this study a microwave ranging system equivalent to the GRACE and GRACE-FO instruments performance is taken as baseline assuming that such instruments can be miniaturized in future as well. In numerical closed-loop simulations, up to nine different satellite configurations with up to 18 satellite pairs are evaluated based on the retrieval of the non-tidal temporal gravity field on a monthly basis. From the simulation results it is concluded that the best-performing satellite constellation of 18 polar satellite pairs already is outperformed by a typical Bender-like constellation of 1 polar and 1 inclined pair. In general, we identify that increasing the number of satellite pairs leads to an improved gravity field retrieval, either at low spherical harmonic degree and order (d/o) by the shift in RAAN or at high d/o by the shift in mean anomaly. By a two-step simulation approach, co-estimating also (sub-)daily gravity fields for selected configurations with a large number of satellite pairs distributed equally over the globe, it is possible to retrieve stand-alone gravity fields at 24, 12 and 6 hr temporal resolution. Ultimately it is concluded that a network of miniaturized satellites with instrument performances similar to GRACE-FO and flying in a well-defined constellation has the potential to observe (sub-)daily mass variations and therefore could drastically reduce the problem of temporal aliasing due to high frequency mass variations in the Earth system.

Efficient temporal high-order staggered-grid scheme with a dispersion-relation-preserving method for the scalar wave modeling

Staggered-grid finite-difference (FD) method is widely used to solve the wave equation for the numerical seismic modeling, and it is a key step of the advanced seismic imaging and inversion problem. However, the conventional FD method is prone to instability and dispersion error due to the insufficient approximation accuracy. In this work, we propose an efficient temporal high-order finite-difference (FD) scheme with the cross-rhombus stencil. Compared with the standard cross-rhombus method, the new method has less computational cost due to we simplify the FD scheme. Moreover, the dispersion relation of the new method is easy to obtain the dispersion-relation-preserving (DRP) FD coefficients, which can significantly alleviate the spatial and temporal dispersion errors. Dispersion and stability analyses indicate that the new scheme has better performance in seismic modeling than the conventional method, and numerical experiments also indicate that the new scheme can effectively mitigate dispersion error and improve the numerical accuracy.

Slow-rising and fast-falling dopaminergic dynamics jointly adjust negative prediction error in the ventral striatum.

The greater the reward expectations are, the more different the brain's physiological response will be. Although it is well-documented that better-than-expected outcomes are encoded quantitatively via midbrain dopaminergic (DA) activity, it has been less addressed experimentally whether worse-than-expected outcomes are expressed quantitatively as well. We show that larger reward expectations upon unexpected reward omissions are associated with the preceding slower rise and following larger decrease (DA dip) in the DA concentration at the ventral striatum of mice. We set up a lever press task on a fixed ratio (FR) schedule requiring five lever presses as an effort for a food reward (FR5). The mice occasionally checked the food magazine without a reward before completing the task. The percentage of this premature magazine entry (PME) increased as the number of lever presses approached five, showing rising expectations with increasing proximity to task completion, and hence greater reward expectations. Fibre photometry of extracellular DA dynamics in the ventral striatum using a fluorescent protein (genetically encoded GPCR activation-based DA sensor: GRABDA2m ) revealed that the slow increase and fast decrease in DA levels around PMEs were correlated with the PME percentage, demonstrating a monotonic relationship between the DA dip amplitude and degree of expectations. Computational modelling of the lever press task implementing temporal difference errors and state transitions replicated the observed correlation between the PME frequency and DA dip amplitude in the FR5 task. Taken together, these findings indicate that the DA dip amplitude represents the degree of reward expectations monotonically, which may guide behavioural adjustment.

Cognitive and plastic recurrent neural network clock model for the judgment of time and its variations

The aim of this study in the field of computational neurosciences was to simulate and predict inter-individual variability in time judgements with different neuropsychological properties. We propose and test a Simple Recurrent Neural Network-based clock model that is able to account for inter-individual variability in time judgment by adding four new components into the clock system: the first relates to the plasticity of the neural system, the second to the attention allocated to time, the third to the memory of duration, and the fourth to the learning of duration by iteration. A simulation with this model explored its fit with participants’ time estimates in a temporal reproduction task undertaken by both children and adults, whose varied cognitive abilities were assessed with neuropsychological tests. The simulation successfully predicted 90% of temporal errors. Our Cognitive and Plastic RNN-Clock model (CP-RNN-Clock), that takes into account the interference arising from a clock system grounded in cognition, was thus validated.

New machine learning application platform for spatial–temporal thermal error prediction and control with STFGCN for ball screw system

The big data platform, which has a high control accuracy and efficiency, is expected to realize the high-accuracy prediction and real-time control of the thermal error (TE) for the ball screw system. But the TE of a ball screw system has significant dynamic spatial–temporal behaviors. The TE model, which fully captures its dynamic spatial–temporal behaviors, has not been reported so far. Moreover, the embedding of the TE model into the big data platform is extremely difficult, and the deployment of the big data platform is time-consuming and relies on the expert knowledge, and the cloud computing used in the big data platform also faces a limited bandwidth pressure of the industrial Internet, leading to the failure of the real-time control. To overcome the above limitations, a novel machine learning application platform (MLAP) is designed based on the cloud-terminal architecture. Moreover, the spatial–temporal fusion graph convolutional network (STFGCN) is proposed for the first time to fully capture of the dynamic spatial–temporal behaviors, and the proposed STFGCN is embedded into the designed MLAP to expedite the training process of the STFGCN model and reduce the bandwidth pressure on the industrial Internet. The prediction performance and robustness of the proposed STFGCN model are compared with that of the traditional multiple linear regression (MLR), long short-term memory (LSTM), gated recurrent unit (GRU), convolutional neural network-long short-term memory (CNN-LSTM), temporal-graph convolutional network (T-GCN), hypergraph neural network (HGNN), spatial–temporal graph convolutional network (STGCN), and attention-based spatial–temporal graph convolutional network (ASTGCN). The results suggest that the evaluation metrics R2 of the MLR, LSTM, GRU, CNN-LSTM, T-GCN, HGNN, STGCN, ASTGCN, and STFGCN are 0.8039, 0.9500, 0.9441, 0.9414, 0.9571, 0.9512, 0.9689, 0.9712, and 0.9812 to show the prediction performance, respectively. Moreover, the training rate is improved by adding the number of the virtual machine nodes on the MLAP. The reduction rate of the machining error is the ratio of the machining error reduction to the original machining error. The positioning error and its fluctuation range of the ball screw system are reduced effectively, and the machining errors with the implementation of the MLAP are decreased by more than 33% and 72% compared with that with the pitch error control and without the TE control, respectively.

Optimal error estimates of a linearized second-order BDF scheme for a nonlocal parabolic problem

PurposeThis paper focuses on the unconditionally optimal error estimates of a linearized second-order scheme for a nonlocal nonlinear parabolic problem. The first step of the scheme is based on Crank–Nicholson method while the second step is the second-order BDF method.Design/methodology/approachA rigorous error analysis is done, and optimal L2 error estimates are derived using the error splitting technique. Some numerical simulations are presented to confirm the study’s theoretical analysis.FindingsOptimal L2 error estimates and energy norm.Originality/valueThe goal of this research article is to present and establish the unconditionally optimal error estimates of a linearized second-order BDF finite element scheme for the reaction-diffusion problem. An optimal error estimate for the proposed methods is derived by using the temporal-spatial error splitting techniques, which split the error between the exact solution and the numerical solution into two parts, that is, the temporal error and the spatial error. Since the spatial error is not dependent on the time step, the boundedness of the numerical solution in L∞-norm follows an inverse inequality immediately without any restriction on the grid mesh.

Optimal a priori error estimate of relaxation-type linear finite element method for nonlinear Schrödinger equation

In this paper, we study, analyze and numerically validate a conservative relaxation-type linear finite element method (FEM) for the nonlinear Schrödinger equation. The method avoids solving the complex nonlinear system and preserves the discrete mass and energy. A key to our analysis is that the errors are split into the temporal error and the spatial error by introducing the corresponding time-discrete system. Therefore, we derive the optimal L2 and semi-H1 error estimates without any coupling condition between time step τ and space size h. Some numerical experiments not only demonstrate the method’s optimal convergence rates but also confirm the conservation laws during long time simulations.

A new portable open-path instrument for ambient NH3 and on-road emission measurements

Increased attentions to vehicle emission of NH3 have been paid since it is generally regarded as an important source in urban areas. Here, we developed a movable instrument based on Differential Optical Absorption Spectroscopy (DOAS) principle for detecting on-road NH3, which can avoid the losses in the sampling process attributed to the non-sampling methods. For this mobile DOAS, the temporal resolution, detection limit and relative error for NH3 were 1 min, 2.29 ppbv and 4.57% ± 2.44%, respectively. By employed to the on-road measurements along the arterial highway in Shanghai, the spatial distributions of NH3 and NO were obtained, and their dependence of traffic and road conditions were studied. The slopes of linear regression between NH3 and NO were 0.40, 0.02 and 0.07 on the Middle Ring Road, Outer Ring Road and Chongming Island Ring Road. It indicates that light gasoline vehicles (LGVs) were found to be the main contributor to NH3 emissions, while heavy-duty diesel vehicles (HDVs) mainly emitted NO. Based on the measured NH3 in the tunnel, the mileage-based NH3 emission factor per vehicle was estimated to be 17.9 ± 6.3 mg/km. The reported open-path instrument can be broadly used in on-road pollutant monitoring or vehicle emissions, and the measurements can reveal the real situation of emission characteristics, even find the abnormal operations of vehicle catalyst system.

Low regularity integrators for semilinear parabolic equations with maximum bound principles

This paper is concerned with conditionally structure-preserving, low regularity time integration methods for a class of semilinear parabolic equations of Allen–Cahn type. Important properties of such equations include maximum bound principle (MBP) and energy dissipation law; for the former, that means the absolute value of the solution is pointwisely bounded for all the time by some constant imposed by appropriate initial and boundary conditions. The model equation is first discretized in space by the central finite difference, then by iteratively using Duhamel’s formula, first- and second-order low regularity integrators (LRIs) are constructed for time discretization of the semi-discrete system. The proposed LRI schemes are proved to preserve the MBP and the energy stability in the discrete sense. Furthermore, their temporal error estimates are also successfully derived under a low regularity requirement that the exact solution of the semi-discrete problem is only assumed to be continuous in time. Numerical results show that the proposed LRI schemes are more accurate and have better convergence rates than classic exponential time differencing schemes, especially when the interfacial parameter approaches zero.

TRACK—a new algorithm and open-source tool for the analysis of pursuit-tracking sensorimotor integration processes

In daily life, sensorimotor integration processes are fundamental for many cognitive operations. The pursuit-tracking paradigm is an ecological and valid paradigm to examine sensorimotor integration processes in a more complex environment than many established tasks that assess simple motor responses. However, the analysis of pursuit-tracking performance is complicated, and parameters quantified to examine performance are sometimes ambiguous regarding their interpretation. We introduce an open-source algorithm (TRACK) to calculate a new tracking error metric, the spatial error, based on the identification of the intended target position for the respective cursor position. The identification is based on assigning cursor and target direction changes to each other as key events, based on the assumptions of similarity and proximity. By applying our algorithm to pursuit-tracking data, beyond replication of known effects such as learning or practice effects, we show a higher precision of the spatial tracking error, i.e., it fits our behavioral data better than the temporal tracking error and thus provides new insights and parameters for the investigation of pursuit-tracking behavior. Our work provides an important step towards fully utilizing the potential of pursuit-tracking tasks for research on sensorimotor integration processes.

What is a static head measurement? Evaluating structural error and implications for regional groundwater modeling

AbstractMost groundwater modelers avoid using static heads measured from active production wells because they can introduce a bias into model calibration. However, in the deep confined Cambrian‐Ordovician Sandstone Aquifer System in the Central Midcontinent of North America, dedicated observation wells are sparse and remote from areas of most concentrated pumping. As a result, in areas where drawdown is the greatest and modeling is most needed, only static heads from production wells are available for calibration. This paper evaluates two leading sources of discrepancies in using production well data, spatial and temporal structural error (S.E.). A simple Theis solution is used to evaluate the potential magnitude of spatial S.E. when calibrating a regional MODFLOW model with coarse cell resolution. Despite theoretical analyses indicating that spatial S.E. could be significant, statistical analysis of the model results suggests that temporal S.E. is dominant. Long (ranging over decades) or frequent (monthly) head datasets are key in understanding temporal S.E., to better capture water‐level variability. In this study, the range in static head observations impacted estimates of the remaining time a well could extract water from the aquifer by 0.1 to 16.0 years. This uncertainty in future water supply is highly relevant to stakeholders and must be assessed in hydrographs depicting risk.

Temporal prediction error triggers amygdala-dependent memory updating in appetitive operant conditioning in rats.

Reinforcement learning theories postulate that prediction error, i.e., a discrepancy between the actual and expected outcomes, drives reconsolidation and new learning, inducing an updating of the initial memory. Pavlovian studies have shown that prediction error detection is a fundamental mechanism in triggering amygdala-dependent memory updating, where the temporal relationship between stimuli plays a critical role. However, in contrast to the well-established findings in aversive situations (e.g., fear conditioning), only few studies exist on prediction error in appetitive operant conditioning, and even less with regard to the role of temporal parameters. To explore if temporal prediction error in an appetitive operant paradigm could generate an updating and consequent reconsolidation and/or new learning of temporal association, we ran four experiments in adult male rats. Experiment 1 verified whether an unexpected delay in the time of reward's availability (i.e., a negative temporal prediction error) in a single session produces an updating in long-term memory of temporal expectancy in an appetitive operant conditioning. Experiment 2 showed that negative prediction errors, either due to the temporal change or through reward omission, increased in the basolateral amygdala nucleus (BLA) the activation of a protein that is critical for memory formation. Experiment 3 revealed that the presence of a protein synthesis inhibitor (anisomycin) in the BLA during the session when the reward was delayed (Error session) affected the temporal updating. Finally, Experiment 4 showed that anisomycin, when infused immediately after the Error session, interfered with the long-term memory of the temporal updating. Together, our study demonstrated an involvement of BLA after a change in temporal and reward contingencies, and in the resulting updating in long-term memory in appetitive operant conditioning.

Degradation modeling based on the gamma process with random initial degradation level and random threshold

ABSTRACT The stochastic modeling of performance characteristics is an important approach for the quality assessment of products. As part of the stochastic modeling, it is possible to consider different variation sources such as temporal, unit-to-unit heterogeneity and measurement error. These sources have been incorporated into different stochastic processes in the literature. Considering the unit-to-unit heterogeneity, two sources of variation can be found for certain products, particularly when the initial level and the critical threshold vary from unit-to-unit. Although, these sources have been studied separately, they have not been studied simultaneously. In this paper, a modeling approach is proposed which considers the gamma process to take into account the temporal uncertainty and the initial degradation level and the critical threshold level as two additional sources of variation. The modeling consists in first obtaining the distribution of the subtraction of the critical threshold and the initial level via a deconvolution operation. Then, the cumulative distribution function of the first passage time distribution is obtained through numerical integration. A simulation study is conducted to evaluate the performance of the proposed method, where different scenarios are analyzed. Furthermore, this approach is applied to a fatigue-crack propagation case study, from which a reliability assessment is performed.

Superconvergence analysis of a conservative mixed finite element method for the nonlinear Klein–Gordon–Schrödinger equations

AbstractIn this paper, a linearized mass and energy conservative mixed finite element method (MFEM) is proposed for solving the nonlinear Klein–Gordon–Schrödinger equations. Optimal error estimates without grid‐ratio condition are derived by some rigorous analysis and an error splitting technique, that is, one is the temporal error which is only τ‐dependent and the other is the spatial error which is only h‐dependent. Furthermore, the superclose results are obtained by using the idea of combination of interpolation and projection. Besides, a so‐called “lifting” approach also play an important role to obtain the superclose results. With the above achievements, the global superconvergent properties are deduced through the interpolated post processing operators. Finally, three numerical examples are given to validate the convergence order, unconditional stability, mass, and energy conservation.

Binary Change Guided Hyperspectral Multiclass Change Detection.

Characterized by tremendous spectral information, hyperspectral image is able to detect subtle changes and discriminate various change classes for change detection. The recent research works dominated by hyperspectral binary change detection, however, cannot provide fine change classes information. And most methods incorporating spectral unmixing for hyperspectral multiclass change detection (HMCD), yet suffer from the neglection of temporal correlation and error accumulation. In this study, we proposed an unsupervised Binary Change Guided hyperspectral multiclass change detection Network (BCG-Net) for HMCD, which aims at boosting the multiclass change detection result and unmixing result with the mature binary change detection approaches. In BCG-Net, a novel partial-siamese united-unmixing module is designed for multi-temporal spectral unmixing, and a groundbreaking temporal correlation constraint directed by the pseudo-labels of binary change detection result is developed to guide the unmixing process from the perspective of change detection, encouraging the abundance of the unchanged pixels more coherent and that of the changed pixels more accurate. Moreover, an innovative binary change detection rule is put forward to deal with the problem that traditional rule is susceptible to numerical values. The iterative optimization of the spectral unmixing process and the change detection process is proposed to eliminate the accumulated errors and bias from unmixing result to change detection result. The experimental results demonstrate that our proposed BCG-Net could achieve comparative or even outstanding performance of multiclass change detection among the state-of-the-art approaches and gain better spectral unmixing results at the same time.