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A recently developed upscaling technique, the multicontinuum homogenization method, has gained significant attention for its effectiveness in modeling complex multiscale systems. This method defines multiple continua based on distinct physical properties and solves a series of constrained cell problems to capture localized information for each continuum. However, solving all these cell problems on very fine grids at every macroscopic point is computationally expensive, which is a common limitation of most homogenization approaches for non-periodic problems. To address this challenge, we propose a hierarchical multicontinuum homogenization framework. The core idea is to define hierarchical macroscopic points and solve the constrained problems on grids of varying resolutions. We assume that the local solutions can be represented as a combination of a linear interpolation of local solutions from preceding levels and an additional correction term. This combination is substituted into the original constrained problems, and the correction term is resolved using finite element (FE) grids of varying sizes, depending on the level of the macropoint. By normalizing the computational cost of fully resolving the local problem to $\mathcal{O}(1)$, we establish that our approach incurs a cost of $\mathcal{O}(L \eta^{(1-L)d})$, highlighting substantial computational savings across hierarchical layers $L$, coarsening factor $\eta$, and spatial dimension $d$. Numerical experiments validate the effectiveness of the proposed method in media with slowly varying properties, underscoring its potential for efficient multiscale modeling.

This paper concerns the numerical approximation for the invariant distribution of Markovian switching L\'evy-driven stochastic differential equations. By combining the tamed-adaptive Euler-Maruyama scheme with the Multi-level Monte Carlo method, we propose an approximation scheme that can be applied to stochastic differential equations with super-linear growth drift and diffusion coefficients.

Designing neural network architectures that can handle data symmetry is crucial. This is especially important for geometric graphs whose properties are equivariance under Euclidean transformations. Current equivariant graph neural networks (EGNNs), particularly those using message passing, have a limitation in expressive power. Recent high-order graph neural networks can overcome this limitation, yet they lack equivariance properties, representing a notable drawback in certain applications in chemistry and physical sciences. In this paper, we introduce the Clifford Group Equivariant Graph Neural Networks (CG-EGNNs), a novel EGNN that enhances high-order message passing by integrating high-order local structures in the context of Clifford algebras. As a key benefit of using Clifford algebras, CG-EGNN can learn functions that capture equivariance from positional features. By adopting the high-order message passing mechanism, CG-EGNN gains richer information from neighbors, thus improving model performance. Furthermore, we establish the universality property of the $k$-hop message passing framework, showcasing greater expressive power of CG-EGNNs with additional $k$-hop message passing mechanism. We empirically validate that CG-EGNNs outperform previous methods on various benchmarks including n-body, CMU motion capture, and MD17, highlighting their effectiveness in geometric deep learning.

We completely determine the depth and regularity of symbolic powers of the fiber product of two homogeneous ideals in disjoint sets of variables, given knowledge of the symbolic powers of each factor. Generalizing previous joint work with Vu, we provide exact, characteristic-independent formulas for the depth and regularity of ordinary powers of such fiber products.

Ego-velocity estimation from point cloud measurements of a millimeter-wave frequency-modulated continuous wave (mmWave FMCW) radar has become a crucial component of radar-inertial odometry (RIO) systems. Conventional approaches often perform poorly when the number of point cloud outliers exceeds that of inliers. In this paper, we propose CREVE, an acceleration-based inequality constraints filter that leverages additional measurements from an inertial measurement unit (IMU) to achieve robust ego-velocity estimations. To further enhance accuracy and robustness against sensor errors, we introduce a practical accelerometer bias estimation method and a parameter adaptation rule. The effectiveness of the proposed method is evaluated using five open-source drone datasets. Experimental results demonstrate that our algorithm significantly outperforms three existing state-of-the-art methods, achieving reductions in absolute trajectory error of approximately 53%, 84%, and 35% compared to them., Comment: 7 pages, conference

We present a fully autonomous self-recharging drone system capable of long-duration sustained operations near powerlines. The drone is equipped with a robust onboard perception and navigation system that enables it to locate powerlines and approach them for landing. A passively actuated gripping mechanism grasps the powerline cable during landing after which a control circuit regulates the magnetic field inside a split-core current transformer to provide sufficient holding force as well as battery recharging. The system is evaluated in an active outdoor three-phase powerline environment. We demonstrate multiple contiguous hours of fully autonomous uninterrupted drone operations composed of several cycles of flying, landing, recharging, and takeoff, validating the capability of extended, essentially unlimited, operational endurance., Comment: Accepted for publication at the 2024 IEEE International Conference on Robotics and Automation (ICRA). Video at https://www.youtube.com/watch?v=C-uekD6VTIQ

In this paper, we propose a radar odometry structure that directly utilizes radar velocity measurements for dead reckoning while maintaining its ability to update estimations within the Kalman filter framework. Specifically, we employ the Doppler velocity obtained by a 4D Frequency Modulated Continuous Wave (FMCW) radar in conjunction with gyroscope data to calculate poses. This approach helps mitigate high drift resulting from accelerometer biases and double integration. Instead, tilt angles measured by gravitational force are utilized alongside relative distance measurements from radar scan matching for the filter's measurement update. Additionally, to further enhance the system's accuracy, we estimate and compensate for the radar velocity scale factor. The performance of the proposed method is verified through five real-world open-source datasets. The results demonstrate that our approach reduces position error by 62% and rotation error by 66% on average compared to the state-of-the-art radar-inertial fusion method in terms of absolute trajectory error., Comment: 9 pages, 5 figures, 1 table, IROS 2024

This paper describes the NOWJ1 Team's approach for the Automated Legal Question Answering Competition (ALQAC) 2023, which focuses on enhancing legal task performance by integrating classical statistical models and Pre-trained Language Models (PLMs). For the document retrieval task, we implement a pre-processing step to overcome input limitations and apply learning-to-rank methods to consolidate features from various models. The question-answering task is split into two sub-tasks: sentence classification and answer extraction. We incorporate state-of-the-art models to develop distinct systems for each sub-task, utilizing both classic statistical models and pre-trained Language Models. Experimental results demonstrate the promising potential of our proposed methodology in the competition., Comment: ISAILD@KSE 2023

Innovative enhancement in embedded system platforms, specifically hardware accelerations, significantly influence the application of deep learning in real-world scenarios. These innovations translate human labor efforts into automated intelligent systems employed in various areas such as autonomous driving, robotics, Internet-of-Things (IoT), and numerous other impactful applications. NVIDIA's Jetson platform is one of the pioneers in offering optimal performance regarding energy efficiency and throughput in the execution of deep learning algorithms. Previously, most benchmarking analysis was based on 2D images with a single deep learning model for each comparison result. In this paper, we implement an end-to-end video-based crime-scene anomaly detection system inputting from surveillance videos and the system is deployed and completely operates on multiple Jetson edge devices (Nano, AGX Xavier, Orin Nano). The comparison analysis includes the integration of Torch-TensorRT as a software developer kit from NVIDIA for the model performance optimisation. The system is built based on the PySlowfast open-source project from Facebook as the coding template. The end-to-end system process comprises the videos from camera, data preprocessing pipeline, feature extractor and the anomaly detection. We provide the experience of an AI-based system deployment on various Jetson Edge devices with Docker technology. Regarding anomaly detectors, a weakly supervised video-based deep learning model called Robust Temporal Feature Magnitude Learning (RTFM) is applied in the system. The approach system reaches 47.56 frames per second (FPS) inference speed on a Jetson edge device with only 3.11 GB RAM usage total. We also discover the promising Jetson device that the AI system achieves 15% better performance than the previous version of Jetson devices while consuming 50% less energy power., Comment: Accepted in Future of Information and Communication Conference (FICC) 2024

We study the Electrical Impedance Tomography Bayesian inverse problem for recovering the conductivity given noisy measurements of the voltage on some boundary surface electrodes. The uncertain conductivity depends linearly on a countable number of uniformly distributed random parameters in a compact interval, with the coefficient functions in the linear expansion decaying at an algebraic rate. We analyze the surrogate Markov Chain Monte Carlo (MCMC) approach for sampling the posterior probability measure, where the multivariate sparse adaptive interpolation, with interpolating points chosen according to a lower index set, is used for approximating the forward map. The forward equation is approximated once before running the MCMC for all the realizations, using interpolation on the finite element (FE) approximation at the parametric interpolating points. When evaluation of the solution is needed for a realization, we only need to compute a polynomial, thus cutting drastically the computation time. We contribute a rigorous error estimate for the MCMC convergence. In particular, we show that there is a nested sequence of interpolating lower index sets for which we can derive an interpolation error estimate in terms of the cardinality of these sets, uniformly for all the parameter realizations. An explicit convergence rate for the MCMC sampling of the posterior expectation of the conductivity is rigorously derived, in terms of the interpolating point number, the accuracy of the FE approximation of the forward equation, and the MCMC sample number. We perform numerical experiments using an adaptive greedy approach to construct the sets of interpolation points. We show the benefits of this approach over the simple MCMC where the forward equation is repeatedly solved for all the samples and the non-adaptive surrogate MCMC with an isotropic index set treating all the random parameters equally.

We propose a multilevel Monte Carlo-FEM algorithm to solve elliptic Bayesian inverse problems with "Besov random tree prior". These priors are given by a wavelet series with stochastic coefficients, and certain terms in the expansion vanishing at random, according to the law of so-called Galton-Watson trees. This allows to incorporate random fractal structures and large deviations in the log-diffusion, which occur naturally in many applications from geophysics or medical imaging. This framework entails two main difficulties: First, the associated diffusion coefficient does not satisfy a uniform ellipticity condition, which leads to non-integrable terms and thus divergence of standard multilevel estimators. Secondly, the associated space of parameters is Polish, but not a normed linear space. We address the first point by introducing cut-off functions in the estimator to compensate for the non-integrable terms, while the second issue is resolved by employing an independence Metropolis-Hastings sampler. The resulting algorithm converges in the mean-square sense with essentially optimal asymptotic complexity, and dimension-independent acceptance probabilities., Comment: 31 pages. arXiv admin note: text overlap with arXiv:2302.00522

Sheaf theory, which is a complex but powerful tool supported by topological theory, offers more flexibility and precision than traditional graph theory when it comes to modeling relationships between multiple features. In the realm of air quality monitoring, this can be incredibly useful in detecting sudden changes in local dust particle density, which can be difficult to accurately measure using commercial instruments. Traditional methods for air quality measurement often rely on calibrating the measurement with public standard instruments or calculating the measurements moving average over a constant period. However, this can lead to an incorrect index at the measurement location, as well as an oversmoothing effect on the signal. In this study, we propose a compact device that uses sheaf theory to detect and count vehicles as a local air quality change-causing factor. By inferring the number of vehicles into the PM2.5 index and propagating it into the recorded PM2.5 index from low-cost air monitoring sensors such as PMS7003 and BME280, we can achieve self-correction in real-time. Plus, the sheaf-theoretic method allows for easy scaling to multiple nodes for further filtering effects. By implementing sheaf theory in air quality monitoring, we can overcome the limitations of traditional methods and provide more accurate and reliable results.

Vietnam requires a sustainable urbanization, for which city sensing is used in planning and de-cision-making. Large cities need portable, scalable, and inexpensive digital technology for this purpose. End-to-end air quality monitoring companies such as AirVisual and Plume Air have shown their reliability with portable devices outfitted with superior air sensors. They are pricey, yet homeowners use them to get local air data without evaluating the causal effect. Our air quality inspection system is scalable, reasonably priced, and flexible. Minicomputer of the sys-tem remotely monitors PMS7003 and BME280 sensor data through a microcontroller processor. The 5-megapixel camera module enables researchers to infer the causal relationship between traffic intensity and dust concentration. The design enables inexpensive, commercial-grade hardware, with Azure Blob storing air pollution data and surrounding-area imagery and pre-venting the system from physically expanding. In addition, by including an air channel that re-plenishes and distributes temperature, the design improves ventilation and safeguards electrical components. The gadget allows for the analysis of the correlation between traffic and air quali-ty data, which might aid in the establishment of sustainable urban development plans and poli-cies.

We consider discrete (time and space) random walks confined to the quarter plane, with jumps only in directions $(i,j)$ with $i+j \geq 0$ and small negative jumps, i.e., $i,j \geq -1$. These walks are called singular, and were recently intensively studied from a combinatorial point of view. In this paper, we show how the compensation approach introduced in the 90ies by Adan, Wessels and Zijm may be applied to compute positive harmonic functions with Dirichlet boundary conditions. In particular, in case the random walks have a drift with positive coordinates, we derive an explicit formula for the escape probability, which is the probability to tend to infinity without reaching the boundary axes. These formulas typically involve famous recurrent sequences, such as the Fibonacci numbers. As a second step, we propose a probabilistic interpretation of the previously constructed harmonic functions and prove that they allow to compute all positive harmonic functions of these singular walks. To that purpose, we derive the asymptotics of the Green functions in all directions of the quarter plane and use Martin boundary theory., Comment: 38 pages, 4 figures

We construct harmonic functions in the quarter plane for discrete Laplace operators. In particular, the functions are conditioned to vanish on the boundary and the Laplacians admit coefficients associated with transition probabilities of non-symmetric random walks. By solving a boundary value problem for generating functions of harmonic functions, we deduce explicit expressions for the generating functions in terms of conformal mappings. These mappings are yielded from a conformal welding problem with quasisymmetric shift and contain information about the growth of harmonic functions. Further, we describe the set of harmonic functions as a vector space isomorphic to the space of formal power series., Comment: 39 pages, 9 figures

We develop an essentially optimal finite element approach for solving ergodic stochastic two-scale elliptic equations whose two-scale coefficient may depend also on the slow variable. We solve the limiting stochastic two-scale homogenized equation obtained from the stochastic two-scale convergence in the mean (A. Bourgeat, A. Mikelic and S. Wright, J. reine angew. Math, Vol. 456, 1994), whose solution comprises of the solution to the homogenized equation and the corrector, by truncating the infinite domain of the fast variable and using the sparse tensor product finite elements. We show that the convergence rate in terms of the truncation level is equivalent to that for solving the cell problems in the same truncated domain. Solving this equation, we obtain the solution to the homogenized equation and the corrector at the same time, using only a number of degrees of freedom that is essentially equivalent to that required for solving one cell problem. Optimal complexity is obtained when the corrector possesses sufficient regularity with respect to both the fast and the slow variables. Although the regularity norm of the corrector depends on the size of the truncated domain, we show that the convergence rate of the approximation for the solution to the homogenized equation is independent of the size of the truncated domain. With the availability of an analytic corrector, we construct a numerical corrector for the solution of the original stochastic two-scale equation from the finite element solution to the truncated stochastic two-scale homogenized equation. Numerical examples of quasi-periodic two-scale equations, and a stochastic two-scale equation of the checker board type, whose coefficient is discontinuous, confirm the theoretical results.