Inverse Uncertainty Research Articles

Nonlinear Inversion of Ultrasonic Dispersion Curves for Cortical Bone Thickness and Elastic Velocities.

In this study, a nonlinear grid-search inversion has been developed to estimate the thickness and elastic velocities of long cortical bones, which are important determinants of bone strength, from axially-transmitted ultrasonic data. The inversion scheme is formulated in the dispersive frequency-phase velocity domain to recover bone properties. The method uses ultrasonic guided waves to retrieve overlying soft tissue thickness, cortical thickness, compressional, and shear-wave velocities of the cortex. The inversion strategy requires systematic examination of a large set of trial dispersion-curve solutions within a pre-defined model space to match the data with minimum cost in a least-squares sense. The theoretical dispersion curves required to solve the inverse problem are computed for bilayered bone models using a semi-analytical finite-element method. The feasibility of the proposed approach was demonstrated by the numerically simulated data for a 1 mm soft tissue-5 mm bone bilayer and ex-vivo data from a bovine femur plate with an overlying 2 mm-thick soft-tissue mimic. The bootstrap method was employed to evaluate the inversion uncertainty and stability. Our results have shown that the cortical thickness and wave speeds could be recovered with fair accuracy.

Sensorimotor beta power reflects the precision-weighting afforded to sensory prediction errors

It has been proposed that accurate motor control relies on Bayesian inference that integrates sensory input with prior contextual knowledge (Bays and Wolpert, 2007; Körding and Wolpert, 2004; Wolpert et al., 1995). Recent evidence has suggested that modulations in beta power (∼12–30 Hz) measured over sensorimotor cortices using electroencephalography (EEG) may represent parameters of Bayesian inference. While the well characterised post-movement beta synchronisation has been shown to correlate with prediction error (H. Tan, Jenkinson, & Brown, 2014; Huiling Tan, Wade, & Brown, 2016), recent evidence suggests that beta power may also represent uncertainty measures (Tan et al., 2016; Tzagarakis et al., 2015). The current study aimed to measure the neurophysiological correlates of uncertainty mediating Bayesian updating during a visuomotor adaptation paradigm in healthy human participants. In particular, sensory uncertainty was directly modulated to measure its effect on sensorimotor beta power. Participant's behaviour was modelled using the Hierarchical Gaussian Filter (HGF) in order to extract the latent variables involved in learning actions required by the task and correlate these with the measured EEG. We found that sensorimotor beta power correlated with inverse uncertainty afforded to sensory prediction errors both prior to and following a movement. This suggests that sensorimotor beta oscillations may more readily represent relative uncertainty within the sensorimotor system rather than error. Neurophysiological models describing the generation of beta oscillations offer a potential mechanism by which this neural signature may encode latent uncertainty parameters. This is essential for understanding how the brain controls behaviour.

An uncertain two-echelon fixed charge transportation problem

In the present study, a two-echelon fixed charge transportation problem is investigated under uncertainty. Due to the existence of considerable amount of uncertainties, the demands, supplies, availabilities, fixed charges and transported quantities in this problem are assumed as uncertain variables. The aim is to maximize the total profit under uncertain environments. The expected value model, chance-constrained model and measure chance model are developed, and the deterministic equivalent forms of these models are obtained by inverse uncertainty distribution. Genetic algorithm and particle swarm optimization are proposed to solve the equivalent forms of the models based on the structure of the problem. To verify the effectiveness of these proposed approaches, numerical experiments are performed.

Feedforward Motion Control With a Variable Stiffness Actuator Inspired by Muscle Cross-Bridge Kinematics

High mechanical impedance, sensor resolution, and computing bandwidth are desirable for achieving stability in feedback control. In contrast, although the human body is physically inferior to man-made systems in terms of feedback control due to signal transmission delay, humans can achieve not only stable but also robust and adaptive control ability. For these reasons, the significance of the feedforward control of the human has been emphasized in neuroscience. In previous studies, virtual trajectory control and internal model hypotheses were used to explain the principle of human feedforward control in view of the mechanical stiffness of joints and the internal model of the brain, respectively. Inspired by these insights, in this paper, we focus on the relationship between the joint stiffness and inverse model accuracy and attempted to apply it to the field of robotics. We present a variable stiffness actuator developed using variable radius gear transmission inspired by muscle cross-bridge kinematics. The developed mechanism allows the joint stiffness to be accurately controlled without any sensor-based feedback control. Using the developed actuator, we conduct a feedforward motion generating experiment with respect to variations in the inverse dynamics model uncertainty and verified that joint stiffness can compensate for inverse model uncertainty and external disturbances. These results indicate that the developed variable stiffness actuator can be applied to the robotics field for feedforward applications and support the hypothesis that humans may utilize joint stiffness to compensate for the inverse dynamics model uncertainty.

Characterizing the Variability of the Structure Parameter in the PROSPECT Leaf Optical Properties Model

Radiative transfer model (RTM) inversion allows for the quantitative estimation of vegetation biochemical composition from satellite sensor data, but large uncertainties associated with inversion make accurate estimation difficult. The leaf structure parameter (Ns) is one of the largest sources of uncertainty in inversion of the widely used leaf-level PROSPECT model, since it is the only parameter that cannot be directly measured. In this study, we characterize Ns as a function of phenology by collecting an extensive dataset of leaf measurements from samples of three dicotyledon species (hard red wheat, soft white wheat, and upland rice) and one monocotyledon (soy), grown under controlled conditions over two full growth seasons. A total of 230 samples were collected: measured leaf reflectance and transmittance were used to estimate Ns from each sample. These experimental data were used to investigate whether Ns depends on phenological stages (early/mid/late), and/or irrigation regime (irrigation at 85%, 75%, 60% of the initial saturated tray weight, and pre-/post-irrigation). The results, supported by the extensive experimental data set, indicate a significant difference between Ns estimated on monocotyledon and dicotyledon plants, and a significant difference between Ns estimated at different phenological stages. Different irrigation regimes did not result in significant Ns differences for either monocotyledon or dicotyledon plant types. To our knowledge, this study provides the first systematic record of Ns as a function of phenology for common crop species.

An e-commerce facility location problem under uncertainty

Facility location problem is a branch of operational research and computational geometry. It involves the best allocation of facilities to minimize transportation costs, while considering factors such as avoiding placing dangerous materials near the premises and the facilities of competitors. According to B2C e-commerce unique customer characteristics and fierce market competition, two facility location models in e-commerce under uncertainty are proposed, i.e., expected value model and pessimistic value model. It is proved these models can be converted into equivalent models based on inverse uncertainty distribution method. A hybrid algorithm is proposed to solve these models. Some numerical experiments are used to demonstrate the effectiveness of the proposed models and approach.

Covariance of uncertain random variables and its application to portfolio optimization

Covariance is a device to measure the joint variability of two uncertain random variables. If the greater values of one uncertain random variable associated with the greater values of the other uncertain random variable, and the same state holds for the lesser values, i.e., the uncertain random variables have similar behavior, the covariance is positive. On the other hand, when the greater values of one uncertain random variable associate with the lesser values of the other, i.e., the variables tend to show opposite behavior, the covariance is negative. Since, interpretation of covariance is not easy, we consider the concept of correlation coefficient for two uncertain random variables as a normalized version of covariance. For calculating the covariance of uncertain random variables, some formulas are provided through the inverse uncertainty distribution. As an application of variance-covariance, portfolio selection problem is optimized by mean-variance model. The main results are explained by using several examples.

Estimation of probability Density Functions for model input parameters using inverse uncertainty quantification with bias terms

The documentation of most nuclear thermal-hydraulics codes does not provide sufficient information on uncertainty of physical models (e.g. interfacial heat transfer coefficients). These models were derived based on experimental data and implemented as empirical correlations in the computational code. The uncertainty quantification for the relevant output quantity (e.g. Peak Cladding Temperature) requires estimation of the Probability Density Functions (PDFs) of the code inputs, such as physical models. In this paper, we investigate the effect of boundary conditions (outlet pressure, inlet liquid temperature, and inlet flow rate) on the uncertainty of two physical models (the interfacial friction coefficient and the wall to liquid friction coefficient). The boundary conditions effect was accounted for by adding a bias term to the mathematical framework of two existing methods for Inverse Uncertainty Quantification (IUQ): the Maximum Likelihood Estimation (MLE) method and the Maximum A Posterior (MAP) method. The two methods were demonstrated using the BFBT benchmark, experimental data was compared to code predictions of the RSTART thermal-hydraulics code for two different cases: without and with bias term. The results show an evident improvement in code prediction when the bias term is used. Finally, a validation set of experimental data was used to investigate the possibility of data overfitting, and the proposed methodology showed absence of overfitting when bias terms are used.

Effect of mesh refinement on the estimation of model input parameters using Inverse Uncertainty Quantification

Proper quantification of the uncertainties in the input parameters of a thermal-hydraulics code (e.g. physical models) is an essential part of the forward Uncertainty Quantification (UQ) problem. Such quantification can be systematically achieved by solving the Inverse Uncertainty Quantification (IUQ) problem using data from code predictions and experimental measurements. The IUQ problem is highly dependent on several factors such as geometry and discretization. In this paper, we study the effect of mesh refinement on the statistical parameters of three uncertain physical models (interfacial friction coefficient, wall to liquid friction coefficient and critical heat flux). A mathematical framework based on MLE and MAP algorithms is implemented to perform IUQ for the physical models of the thermal-hydraulics code RSTART based on experimental data from the OECD/NEA BWR Full-size Fine-mesh Bundle Test (BFBT) benchmark. Sensitivity analysis required for the IUQ problem was achieved by implementing discrete adjoint methods in the thermal-hydraulics code. The statistical parameters of the three physical models were obtained by implementing the MLE and MAP methods using code predictions that correspond to three different mesh sizes. Results from MLE and MAP showed a noticeable effect of mesh refinement on the estimates of the mean and a negligible effect on the estimates of the variance of the physical models. The compensation of error due to mesh refinement was studied by running the code with perturbation values calculated using the three different mesh sizes. It was shown that the results based on the most refined mesh led to best agreement between experimental data and code prediction.

Evaluation of ice-stream model sensitivities for parameter estimation

Large-ensemble perturbed-parameter forward ice-flow modeling can provide useful insights to uncertainties in inversions for basal drag or other ice-flow parameters. Inversion and data assimilation provide estimates of poorly known parameters that are essential for accurate prognostic modeling. Because ice flow depends on many such parameters with their associated uncertainties, which may interact in nonlinear ways, full uncertainty assessment for parameter estimates is challenging. With rising computational power, it is increasingly practicable to explore co-dependencies and sensitivities. Here, we use a well-characterized higher-order flowline model configured for a well-lubricated (“shelfy”) ice stream to run large ensembles, perturbing the magnitude and spatial pattern of basal drag, basal topography, and input flux. We find for steady state that ice-stream velocity and thickness along the entire domain are especially correlated to drag at the downstream end, but with greater local correlation during transients. The modeled effects of basal topographic perturbations on velocity and ice thickness are primarily local. Perturbations of input ice fluxes interact with the others in interesting ways. These insights point to the value of inversions informed by concentrated observations during forced transients such as lake-drainage events, accumulation-rate fluctuations or ice-shelf losses, and to the care needed when interpreting local results of some inversions for basal-drag parameters.

The role of geological uncertainty in developing combined geological and potential field inversions

Recently, implicit model building techniques have been developed which use, for example, geo-statistical methods to interpolate boundary orientations as a scalar field. Boundaries are implicitly formulated as iso-values of that field. Using more than one potential allows modelling for intrusion and unconformities. This technique is attractive because it makes 3D geological modeling a repeatable task and model uncertainty can be estimated from the geostatistical estimators. However, this uncertainty does not take into account the error on field measurements, nor estimates of how relevant are the measurements to the modelled structures. Lastly, this uncertainty does not take into account possible variation in the knowledge based information which is very often interpretative, such as structural evolution history, fault network, and overprinting relationships with themselves and the different formations. We present an innovative method that will simulate numerous (millions of) geological models from a single initial structural dataset, taking into account these variables. These models are used to estimate geological uncertainty, highlighting future areas of research or data collection. A series of best models are then assessed against potential field inversions and modified to better fit potential field data. In the end, the models that both better fit geological input data and potential field data will be retained. A best probable model will be proposed that will satisfy geophysical data as well as geological data. To assess the models against the initial geological data input, we develop geological objective functions based on (e.g.) locations and gradients of boundaries. It is our intent to combine these objectives functions with classical geophysical objective functions to provide a new method for combined geological and potential field inversions.

Crustal thickness and Poisson's ratios of South China revealed from joint inversion of receiver function and gravity data

Crustal thickness and Poisson's ratio are two important parameters for understanding the crustal structure, composition, and deformation of the South China block. Due to the complicated crustal structure and poor signal-to-noise ratio of teleseismic data at dozens of stations in South China, the receiver-function H–κ stacking produces large uncertainty and deviation when estimating these two crustal parameters. In this paper, we improve the technique of the joint inversion of receiver function and gravity data and utilize it to improve the estimates of crustal thickness and Poisson's ratio in South China. We calculate the receiver functions of 245 permanent seismic stations from the teleseismic data recorded during 2013–2015 and then carry out the joint inversion on the receiver functions and the complete Bouguer gravity anomalies data. The joint inversion reduces the uncertainty of each single-method inversion and enhances the accuracy of estimation. Our results demonstrate that the crustal thickness, ranging 26.1–46.5 km, is large in the NW (38–46.5 km) and small in the SE (26–34 km) with a NNE-trending gradient zone along the line of Yichang–Jishou–Baise in central South China. The crustal Poisson's ratio, ranging 0.20–0.31, is high (0.28–0.31) in most of the northern South China, intermediate (0.26–0.29) around the eastern coastal area, very low (0.20–0.24) within the Jiangnan orogenic belt and around the southern coastal area, and slightly low (0.22–0.26) in other places. With the constraints of previous geological and geophysical studies, we suggested that the notable low-value belt of crustal Poisson's ratio (between the lines of Shitai–Jiujiang–Yiyang–Jishou–Baise and Shaoxing–Jiangshan–Pingxiang–Yongzhou–Guigang–Beihai) represents the possible suture zone between Yangtze and Cathaysia sub-blocks in South China during the Neoproterozoic.

Uncertainty metric in model-based eddy current inversion using the adaptive Monte Carlo method

The accuracy of model-based inversion largely depends on the gap between simulated and observed results. However, the gap does not disappear even if a wide range of calibrations or compensations are made. In essence, the gap varies statistically due to random noises and poorly known parameters, causing predicted results to change in a random manner. In this work, an adaptive Monte Carlo method is proposed to evaluate the uncertainty of model-based inversion for eddy current nondestructive evaluation. Using the presented Monte Carlo method, the influences of excitation frequencies on uncertainty metric were identified, and uncertainty metric was calculated when the conductivity, thickness and liftoff distance were inferred for characterization of a plate. The presented Monte Carlo based method allows efficient and cost-effective assessment of uncertainty and could enable us to identify the factors that play a dominant role in the performances of model-based inversion.

An uncertain parallel machine problem with deterioration and learning effect

An uncertain uniform parallel machine scheduling problem with job deterioration and a learning effect is considered. Job processing times, due dates, deterioration rates and learning rates are assumed to be uncertain variables. The objective functions are the total weight earliness, tardiness and makespan. Three mathematical programming models are presented, i.e., expected value model, pessimistic value model and measure chance model. These models can be converted into equivalent crisp models by the inverse uncertainty distribution method. A hybrid algorithm mixed with dispatching rules based on structural features is employed to solve the problem. Finally, computational experiments are presented to illustrate the effectiveness of proposed algorithms.

大口径压电快摆镜机构迟滞非线性补偿与控制

In order to improve the control accuracy of large-aperture piezoelectric fast steering mirror(FSM) in precision image stabilization system in space telescope, the compound control strategy combining hysteresis feed forward compensation with the optimal PID control algorithm was adopted. According to the problems that the reversibility of Prandtl-Ishlinskii (PI) model based on Play operator was limited by the constraint condition and the estimation error accumulation of model parameters in the inverse process, the PI inverse model based on the generalized Stop operator was proposed to compensate the PZT hysteresis. The optimal PID closed loop controller was added in the control system for the problems of the inverse hysteresis model uncertainty and the poor anti-interference ability of direct feed forward control. The adaptive differential evolution(ADE) was used to optimize the hysteresis inverse model and PID controller parameters and the chaos search mechanism was introduced to improve the performance of ADE. The experimental results show that, compared with the traditional analytic method to obtain the inverse model, the hysteresis inverse model based on the Stop operator can better describe the inverse hysteresis curve, and the fitting precision increases by 78.04% when fitting the hysteresis curve with 1 Hz frequency; the tracking accuracy of compound control algorithm increases by 38.56%,22.92% and 13.5% respectively compared with the direct feedforward control, in the real-time tracking for target swinging displacements of large-aperture piezoelectric fast steering mirror with 1 Hz,10 Hz and 20 Hz frequencies.

WASSERSTEIN METRIC-DRIVEN BAYESIAN INVERSION WITH APPLICATIONS TO SIGNAL PROCESSING

We present a Bayesian framework based on a new exponential likelihood function driven by the quadratic Wasserstien metric. Compared to conventional Bayesian models based on Gaussian likelihood functions driven by the least-squares norm ($L_2$ norm), the new framework features several advantages. First, the new framework does not rely on the likelihood of the measurement noise and hence can treat complicated noise structures such as combined additive and multiplicative noise. Secondly, unlike the normal likelihood function, the Wasserstein-based exponential likelihood function does not usually generate multiple local extrema. As a result, the new framework features better convergence to correct posteriors when a Markov Chain Monte Carlo sampling algorithm is employed. Thirdly, in the particular case of signal processing problems, while a normal likelihood function measures only the amplitude differences between the observed and simulated signals, the new likelihood function can capture both the amplitude and the phase differences. We apply the new framework to a class of signal processing problems, that is, the inverse uncertainty quantification of waveforms, and demonstrate its advantages compared to Bayesian models with normal likelihood functions.

Novel Lidar algorithm for horizontal visibility measurement and sea fog monitoring.

Traditionally, Klett and Fernald inversion estimates an initial value using the slope method for horizontal visibility, which causes inversion uncertainty. We proposed an algorithm to retrieve the extinction coefficient and visibility distribution information from scanning Lidar to overcome instability due to initial atmospheric extinction coefficient choice and assuming the Lidar ratio. Numerical simulations showed that extinction coefficient maximum relative was much larger for inhomogeneous atmosphere using the Klett method, reaching 0.31. In contrast, it is only 0.049 using the proposed algorithm. Experimental showed that the proposed algorithm and scanning Lidar system provide very high stability and accuracy, can work in different weather conditions and monitor sea fog evolution over real time, and is suitable for various situations with different visibility.

Uncertainty analysis for infrasound waveform inversion: Application to explosion yield estimation.

While the acoustic waveform inversion method is increasingly used in geophysical acoustics to constrain source parameters, the inversion results are often provided without any uncertainty analysis. This study presents a probabilistic representation for acoustic waveform inversion and method to evaluate the inversion uncertainty using ground-truth data. A posteriori probability distribution of source estimate is described by a priori waveform misfit covariance and the variance of acoustic source model. The probabilistic framework is applied to local explosion infrasound to estimate the yields of explosions and uncertainty. Estimated yields showed overall good agreement with the true yields (less than 25% errors). The uncertainty of the estimated yield is represented by the sum of the waveform inversion uncertainty and source model uncertainty. It is shown that the yield uncertainty attributed to local infrasound inversion (within 10 km) is as small as the uncertainty caused by 10% prediction errors in the acoustic source model. These results indicate that the acoustic source model uncertainty should also be considered for accurate yield estimation and that local infrasound can be a valuable tool to understand the magnitude of the source uncertainty.

Inversion of the density structure of the lithosphere in the North China Craton from GOCE satellite gravity gradient data

Lithospheric thinning in the North China Craton (NCC), one of the oldest cratons on Earth, has been a focus of geoscientists across the globe for a long time. In this study, seismic tomographic P-wave velocity variations are converted to density perturbations as the initial constraint values, and the four components of the GOCE Level-2 gravity gradient product are subjected to topographic effect correction, relief correction on the Moho and sedimentary layer boundary, and long-wavelength correction. In addition, the anomalous gravity gradient effect resulting from the uncertainty of the Moho and sedimentary layer depth is considered. Considering the effects of temperature variations in the lithosphere, which leads to the non-uniform density distribution, the anomalous gravity gradient effect resulting from the temperature variations is corrected at the first time in this area. Inverse computation is performed based on the preconditioned conjugate gradient algorithm. Because the Lagrange empirical parameter in the algorithm exhibits uncertainty in data inversion, the regularization parameter, which is typically an empirical parameter, is replaced with the value of the inflection point of the L-curve. The results show the followings. The non-uniform density distribution in the lithosphere is related not only to the composition but also to the internal temperature variations in the lithosphere. The density of the lithosphere in the NCC is significantly non-uniform in both the horizontal and vertical directions and exhibits a notable segment-wise spatial distribution pattern. Two tectonic units, namely the Taihangshan tectonic zone and the Linfen–Weihe graben, constitute a central gravity gradient transition zone. There is a significant difference in the density distribution on the two sides of this central transition zone.

Development of a Pragmatic Approach to Model Input Uncertainty Quantification for BEPU Applications

In the framework of the Organisation for Economic Co-operation and Development/Nuclear Energy Agency PREMIUM [Post-BEMUSE (Best-Estimate Methods Uncertainty and Sensitivity Evaluation) REflood Model Input Uncertainty Methods] benchmark (2012–2015), Tractebel has contributed to the development and the proof-of-concept application of a sampling-based inverse uncertainty quantification (IUQ) approach with the DAKOTA statistical uncertainty and sensitivity analysis tool. This IUQ approach has been applied to quantify the RELAP5/MOD3.3 reflood-related model input uncertainties, based on selected reflood tests [FEBA (Flooding Experiments with Blocked Arrays) and PERICLES]. This paper presents the Tractebel IUQ approach as well as the results of applications to the PREMIUM benchmark. Lessons learned and perspectives for future development are also discussed.