In high-order and high-dimensional finite elements, ill-conditioned nodal distributions are often computationally cost-prohibitive. As a result, uniform distributions quickly fall apart. For tensor-product like elements, Gauss-Legendre-Lobatto (GLL) nodal distributions are often used as a substitute. Besides these, other efficient nodal distributions are difficult to create due to a desired symmetry within elements and conformity with neighboring elements. In this paper, we provide a general framework to construct symmetric, well-conditioned, cross-element compatible nodal distributions which can be used for high-order and high-dimensional finite elements. Starting from the inherent symmetries in any potential element, the framework is used to build up nodal groups in a structured and efficient manner utilizing the natural coordinates of each element, while ensuring nodes stay within the elements. By constructing constrained symmetry groups, the vertices, edges, and faces, of all elements are required to conform to their respective lower-dimensional distributions. Optimizing over these groups yields the desired optimized nodal distributions. We demonstrate the strength of this framework by creating and comparing optimized nodal distributions with GLL distributions (in elements such as the line, quadrilateral, and hexahedron), and its robustness by generating optimized nodal distributions for otherwise difficult elements (such as the triangle, tetrahedron, and triangular prism).
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The Scott-Vogelius element is a popular finite element for the discretization of the Stokes equations which enjoys inf-sup stability and gives divergence-free velocity approximation. However, it is well known that the convergence rates for the discrete pressure deteriorate in the presence of certain $critical$ $vertices$ in a triangulation of the domain. Modifications of the Scott-Vogelius element such as the recently introduced pressure-wired Stokes element also suffer from this effect. In this paper we introduce a simple modification strategy for these pressure spaces that preserves the inf-sup stability while the pressure converges at an optimal rate.
Simulation-based inference techniques are indispensable for parameter estimation of mechanistic and simulable models with intractable likelihoods. While traditional statistical approaches like approximate Bayesian computation and Bayesian synthetic likelihood have been studied under well-specified and misspecified settings, they often suffer from inefficiencies due to wasted model simulations. Neural approaches, such as sequential neural likelihood (SNL) avoid this wastage by utilising all model simulations to train a neural surrogate for the likelihood function. However, the performance of SNL under model misspecification is unreliable and can result in overconfident posteriors centred around an inaccurate parameter estimate. In this paper, we propose a novel SNL method, which through the incorporation of additional adjustment parameters, is robust to model misspecification and capable of identifying features of the data that the model is not able to recover. We demonstrate the efficacy of our approach through several illustrative examples, where our method gives more accurate point estimates and uncertainty quantification than SNL.
Mobile manipulators have been employed in many applications that are traditionally performed by either multiple fixed-base robots or a large robotic system. This capability is enabled by the mobility of the mobile base. However, the mobile base also brings redundancy to the system, which makes mobile manipulator motion planning more challenging. In this paper, we tackle the mobile manipulator motion planning problem under the end-effector trajectory continuity constraint in which the end-effector is required to traverse a continuous task-space trajectory (time-parametrized path), such as in mobile printing or spraying applications. Our method decouples the problem into: (1) planning an optimal base trajectory subject to geometric task constraints, end-effector trajectory continuity constraint, collision avoidance, and base velocity constraint; which ensures that (2) a manipulator trajectory is computed subsequently based on the obtained base trajectory. To validate our method, we propose a discrete optimal base trajectory planning algorithm to solve several mobile printing tasks in hardware experiment and simulations.
The permanent of a non-negative square matrix can be well approximated by finding the minimum of the Bethe free energy functions associated with some suitably defined factor graph; the resulting approximation to the permanent is called the Bethe permanent. Vontobel gave a combinatorial characterization of the Bethe permanent via degree-$M$ Bethe permanents, which is based on degree-$M$ covers of the underlying factor graph. In this paper, we prove a degree-$M$-Bethe-permanent-based lower bound on the permanent of a non-negative matrix, which solves a conjecture proposed by Vontobel in [IEEE Trans. Inf. Theory, Mar. 2013]. We also prove a degree-$M$-Bethe-permanent-based upper bound on the permanent of a non-negative matrix. In the limit $M \to \infty$, these lower and upper bounds yield known Bethe-permanent-based lower and upper bounds on the permanent of a non-negative matrix. Moreover, we prove similar results for an approximation to the permanent known as the (scaled) Sinkhorn permanent.
We introduce CrossNet, a complex spectral mapping approach to speaker separation and enhancement in reverberant and noisy conditions. The proposed architecture comprises an encoder layer, a global multi-head self-attention module, a cross-band module, a narrow-band module, and an output layer. CrossNet captures global, cross-band, and narrow-band correlations in the time-frequency domain. To address performance degradation in long utterances, we introduce a random chunk positional encoding. Experimental results on multiple datasets demonstrate the effectiveness and robustness of CrossNet, achieving state-of-the-art performance in tasks including reverberant and noisy-reverberant speaker separation. Furthermore, CrossNet exhibits faster and more stable training in comparison to recent baselines. Additionally, CrossNet's high performance extends to multi-microphone conditions, demonstrating its versatility in various acoustic scenarios.
For learning with noisy labels, the transition matrix, which explicitly models the relation between noisy label distribution and clean label distribution, has been utilized to achieve the statistical consistency of either the classifier or the risk. Previous researches have focused more on how to estimate this transition matrix well, rather than how to utilize it. We propose good utilization of the transition matrix is crucial and suggest a new utilization method based on resampling, coined RENT. Specifically, we first demonstrate current utilizations can have potential limitations for implementation. As an extension to Reweighting, we suggest the Dirichlet distribution-based per-sample Weight Sampling (DWS) framework, and compare reweighting and resampling under DWS framework. With the analyses from DWS, we propose RENT, a REsampling method with Noise Transition matrix. Empirically, RENT consistently outperforms existing transition matrix utilization methods, which includes reweighting, on various benchmark datasets. Our code is available at \url{//github.com/BaeHeeSun/RENT}.
Addressing the intricate challenge of modeling and re-rendering dynamic scenes, most recent approaches have sought to simplify these complexities using plane-based explicit representations, overcoming the slow training time issues associated with methods like Neural Radiance Fields (NeRF) and implicit representations. However, the straightforward decomposition of 4D dynamic scenes into multiple 2D plane-based representations proves insufficient for re-rendering high-fidelity scenes with complex motions. In response, we present a novel direction-aware representation (DaRe) approach that captures scene dynamics from six different directions. This learned representation undergoes an inverse dual-tree complex wavelet transformation (DTCWT) to recover plane-based information. DaReNeRF computes features for each space-time point by fusing vectors from these recovered planes. Combining DaReNeRF with a tiny MLP for color regression and leveraging volume rendering in training yield state-of-the-art performance in novel view synthesis for complex dynamic scenes. Notably, to address redundancy introduced by the six real and six imaginary direction-aware wavelet coefficients, we introduce a trainable masking approach, mitigating storage issues without significant performance decline. Moreover, DaReNeRF maintains a 2x reduction in training time compared to prior art while delivering superior performance.
Reservoir computing (RC) offers a neuromorphic framework that is particularly effective for processing spatiotemporal signals. Known for its temporal processing prowess, RC significantly lowers training costs compared to conventional recurrent neural networks. A key component in its hardware deployment is the ability to generate dynamic reservoir states. Our research introduces a novel dual-memory RC system, integrating a short-term memory via a WOx-based memristor, capable of achieving 16 distinct states encoded over 4 bits, and a long-term memory component using a TiOx-based memristor within the readout layer. We thoroughly examine both memristor types and leverage the RC system to process temporal data sets. The performance of the proposed RC system is validated through two benchmark tasks: isolated spoken digit recognition with incomplete inputs and Mackey-Glass time series prediction. The system delivered an impressive 98.84% accuracy in digit recognition and sustained a low normalized root mean square error (NRMSE) of 0.036 in the time series prediction task, underscoring its capability. This study illuminates the adeptness of memristor-based RC systems in managing intricate temporal challenges, laying the groundwork for further innovations in neuromorphic computing.
For the simulation-based design of fiber melt spinning processes, the accurate modeling of the processed polymer with regard to its material behavior is crucial. In this work, we develop a high-speed elongational rheometer for Carreau-type materials, making use of process simulations and fiber diameter measurements. The procedure is based on a unified formulation of the fiber spinning model for all material types (Newtonian and non-Newtonian), whose material laws are strictly monotone in the strain rate. The parametrically described material law for the elongational viscosity implies a nonlinear optimization problem for the parameter identification, for which we propose an efficient, robust gradient-based method. The work can be understood as a proof of concept, a generalization to other, more complex materials is possible.
Privacy protection methods, such as differentially private mechanisms, introduce noise into resulting statistics which often produces complex and intractable sampling distributions. In this paper, we propose a simulation-based "repro sample" approach to produce statistically valid confidence intervals and hypothesis tests, which builds on the work of Xie and Wang (2022). We show that this methodology is applicable to a wide variety of private inference problems, appropriately accounts for biases introduced by privacy mechanisms (such as by clamping), and improves over other state-of-the-art inference methods such as the parametric bootstrap in terms of the coverage and type I error of the private inference. We also develop significant improvements and extensions for the repro sample methodology for general models (not necessarily related to privacy), including 1) modifying the procedure to ensure guaranteed coverage and type I errors, even accounting for Monte Carlo error, and 2) proposing efficient numerical algorithms to implement the confidence intervals and $p$-values.
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