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Studies of the human brain during natural activities, such as locomotion, would benefit from the ability to image deep brain structures during these activities. While Positron Emission Tomography (PET) can image these structures, the bulk and weight of current scanners are not compatible with the desire for a wearable device. This has motivated the design of a robotic system to support a PET imaging system around the subject's head and to move the system to accommodate natural motion. We report here the design and experimental evaluation of a prototype robotic system that senses motion of a subject's head, using parallel string encoders connected between the robot-supported imaging ring and a helmet worn by the subject. This measurement is used to robotically move the imaging ring (coarse motion correction) and to compensate for residual motion during image reconstruction (fine motion correction). Minimization of latency and measurement error are the key design goals, respectively, for coarse and fine motion correction. The system is evaluated using recorded human head motions during locomotion, with a mock imaging system consisting of lasers and cameras, and is shown to provide an overall system latency of about 80 ms, which is sufficient for coarse motion correction and collision avoidance, as well as a measurement accuracy of about 0.5 mm for fine motion correction.

We study the numerical approximation of multidimensional stochastic differential equations (SDEs) with distributional drift, driven by a fractional Brownian motion. We work under the Catellier-Gubinelli condition for strong well-posedness, which assumes that the regularity of the drift is strictly greater than $1-1/(2H)$, where $H$ is the Hurst parameter of the noise. The focus here is on the case $H<1/2$, allowing the drift $b$ to be a distribution. We compare the solution $X$ of the SDE with drift $b$ and its tamed Euler scheme with mollified drift $b^n$, to obtain an explicit rate of convergence for the strong error. This extends previous results where $b$ was assumed to be a bounded measurable function. In addition, we investigate the limit case when the regularity of the drift is equal to $1-1/(2H)$, and obtain a non-explicit rate of convergence. As a byproduct of this convergence, there exists a strong solution that is pathwise unique in a class of H\"older continuous solutions. The proofs rely on stochastic sewing techniques, especially to deduce new regularising properties of the discrete-time fractional Brownian motion. In the limit case, we introduce a critical Gr\"onwall-type lemma to quantify the error. We also present several examples and numerical simulations that illustrate our results.

Numerical shock instability is a complexity which may occur in supersonic simulations. Riemann solver is usually the crucial factor that affects both the computation accuracy and numerical shock stability. In this paper, several classical Riemann solvers are discussed, and the intrinsic mechanism of shock instability is especially concerned. It can be found that the momentum perturbation traversing shock wave is a major reason that invokes instability. Furthermore, slope limiters used to depress oscillation across shock wave is also a key factor for computation stability. Several slope limiters can cause significant numerical errors near shock waves, and make the computation fail to converge. Extra dissipation of Riemann solvers and slope limiters can be helpful to eliminate instability, but reduces the computation accuracy. Therefore, to properly introduce numerical dissipation is critical for numerical computations. Here, pressure based shock indicator is used to show the position of shock wave and tunes the numerical dissipation. Overall, the presented methods are showing satisfactory results in both the accuracy and stability.

This study evaluates four fracture simulation methods, comparing their computational expenses and implementation complexities within the Finite Element (FE) framework when employed on multiphase materials. Fracture methods considered encompass the Cohesive Zone Model (CZM) using zero-thickness cohesive interface elements (CIEs), the Standard Phase-Field Fracture (SPFM) approach, the Cohesive Phase-Field fracture (CPFM) approach, and an innovative hybrid model. The hybrid approach combines the CPFM fracture method with the CZM, specifically applying the CZM within the interface zone. The finite element model studied is characterized by three specific phases: Inclusions, matrix, and interface zone. The thorough assessment of these modeling techniques indicates that the CPFM approach stands out as the most effective computational model provided that the thickness of the interface zone is not significantly smaller than that of the other phases. In materials like concrete the interface thickness is notably small when compared to other phases. This leads to the hybrid model standing as the most authentic finite element model, utilizing CIEs within the interface to simulate interface debonding. A significant finding from this investigation is that the CPFM method is in agreement with the hybrid model when the interface zone thickness is not excessively small. This implies that the CPFM fracture methodology may serve as a unified fracture approach for multiphase materials, provided the interface zone's thickness is comparable to that of the other phases. In addition, this research provides valuable insights that can advance efforts to fine-tune material microstructures. An investigation of the influence of the interface material properties, morphological features and spatial arrangement of inclusions showes a pronounced effect of these parameters on the fracture toughness of the material.

Likelihood-based inference in stochastic non-linear dynamical systems, such as those found in chemical reaction networks and biological clock systems, is inherently complex and has largely been limited to small and unrealistically simple systems. Recent advances in analytically tractable approximations to the underlying conditional probability distributions enable long-term dynamics to be accurately modelled, and make the large number of model evaluations required for exact Bayesian inference much more feasible. We propose a new methodology for inference in stochastic non-linear dynamical systems exhibiting oscillatory behaviour and show the parameters in these models can be realistically estimated from simulated data. Preliminary analyses based on the Fisher Information Matrix of the model can guide the implementation of Bayesian inference. We show that this parameter sensitivity analysis can predict which parameters are practically identifiable. Several Markov chain Monte Carlo algorithms are compared, with our results suggesting a parallel tempering algorithm consistently gives the best approach for these systems, which are shown to frequently exhibit multi-modal posterior distributions.

The ultimate goal of any numerical scheme for partial differential equations (PDEs) is to compute an approximation of user-prescribed accuracy at quasi-minimal computational time. To this end, algorithmically, the standard adaptive finite element method (AFEM) integrates an inexact solver and nested iterations with discerning stopping criteria balancing the different error components. The analysis ensuring optimal convergence order of AFEM with respect to the overall computational cost critically hinges on the concept of R-linear convergence of a suitable quasi-error quantity. This work tackles several shortcomings of previous approaches by introducing a new proof strategy. First, the algorithm requires several fine-tuned parameters in order to make the underlying analysis work. A redesign of the standard line of reasoning and the introduction of a summability criterion for R-linear convergence allows us to remove restrictions on those parameters. Second, the usual assumption of a (quasi-)Pythagorean identity is replaced by the generalized notion of quasi-orthogonality from [Feischl, Math. Comp., 91 (2022)]. Importantly, this paves the way towards extending the analysis to general inf-sup stable problems beyond the energy minimization setting. Numerical experiments investigate the choice of the adaptivity parameters.

We construct an efficient class of increasingly high-order (up to 17th-order) essentially non-oscillatory schemes with multi-resolution (ENO-MR) for solving hyperbolic conservation laws. The candidate stencils for constructing ENO-MR schemes range from first-order one-point stencil increasingly up to the designed very high-order stencil. The proposed ENO-MR schemes adopt a very simple and efficient strategy that only requires the computation of the highest-order derivatives of a part of candidate stencils. Besides simplicity and high efficiency, ENO-MR schemes are completely parameter-free and essentially scale-invariant. Theoretical analysis and numerical computations show that ENO-MR schemes achieve designed high-order convergence in smooth regions which may contain high-order critical points (local extrema) and retain ENO property for strong shocks. In addition, ENO-MR schemes could capture complex flow structures very well.

Video captioning models easily suffer from long-tail distribution of phrases, which makes captioning models prone to generate vague sentences instead of accurate ones. However, existing debiasing strategies tend to export external knowledge to build dependency trees of words or refine frequency distribution by complex losses and extra input features, which lack interpretability and are hard to train. To mitigate the impact of granularity bias on the model, we introduced a statistical-based bias extractor. This extractor quantifies the information content within sentences and videos, providing an estimate of the likelihood that a video-sentence pair is affected by granularity bias. Furthermore, with the growing trend of integrating contrastive learning methods into video captioning tasks, we use a bidirectional triplet loss to get more negative samples in a batch. Subsequently, we incorporate the margin score into the contrastive learning loss, establishing distinct training objectives for head and tail sentences. This approach facilitates the model's training effectiveness on tail samples. Our simple yet effective loss, incorporating Granularity bias, is referred to as the Margin-Contrastive Loss (GMC Loss). The proposed model demonstrates state-of-the-art performance on MSRVTT with a CIDEr of 57.17, and MSVD, where CIDEr reaches up to 138.68.

We consider the problem of aggregating the judgements of a group of experts to form a single prior distribution representing the judgements of the group. We develop a Bayesian hierarchical model to reconcile the judgements of the group of experts based on elicited quantiles for continuous quantities and probabilities for one-off events. Previous Bayesian reconciliation methods have not been used widely, if at all, in contrast to pooling methods and consensus-based approaches. To address this we embed Bayesian reconciliation within the probabilistic Delphi method. The result is to furnish the outcome of the probabilistic Delphi method with a direct probabilistic interpretation, with the resulting prior representing the judgements of the decision maker. We can use the rationales from the Delphi process to group the experts for the hierarchical modelling. We illustrate the approach with applications to studies evaluating erosion in embankment dams and pump failures in a water pumping station, and assess the properties of the approach using the TU Delft database of expert judgement studies. We see that, even using an off-the-shelf implementation of the approach, it out-performs individual experts, equal weighting of experts and the classical method based on the log score.

Common regularization algorithms for linear regression, such as LASSO and Ridge regression, rely on a regularization hyperparameter that balances the tradeoff between minimizing the fitting error and the norm of the learned model coefficients. As this hyperparameter is scalar, it can be easily selected via random or grid search optimizing a cross-validation criterion. However, using a scalar hyperparameter limits the algorithm's flexibility and potential for better generalization. In this paper, we address the problem of linear regression with l2-regularization, where a different regularization hyperparameter is associated with each input variable. We optimize these hyperparameters using a gradient-based approach, wherein the gradient of a cross-validation criterion with respect to the regularization hyperparameters is computed analytically through matrix differential calculus. Additionally, we introduce two strategies tailored for sparse model learning problems aiming at reducing the risk of overfitting to the validation data. Numerical examples demonstrate that our multi-hyperparameter regularization approach outperforms LASSO, Ridge, and Elastic Net regression. Moreover, the analytical computation of the gradient proves to be more efficient in terms of computational time compared to automatic differentiation, especially when handling a large number of input variables. Application to the identification of over-parameterized Linear Parameter-Varying models is also presented.