This paper presents a convergence analysis of a Krylov subspace spectral (KSS) method applied to a 1-D wave equation in an inhomogeneous medium. It will be shown that for sufficiently regular initial data, this KSS method yields unconditional stability, spectral accuracy in space, and second-order accuracy in time, in the case of constant wave speed and a bandlimited reaction term coefficient. Numerical experiments that corroborate the established theory are included, along with an investigation of generalizations, such as to higher space dimensions and nonlinear PDEs, that features performance comparisons with other Krylov subspace-based time-stepping methods. This paper also includes the first stability analysis of a KSS method that does not assume a bandlimited reaction term coefficient.
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This paper is concerned with inference on the conditional mean of a high-dimensional linear model when outcomes are missing at random. We propose an estimator which combines a Lasso pilot estimate of the regression function with a bias correction term based on the weighted residuals of the Lasso regression. The weights depend on estimates of the missingness probabilities (propensity scores) and solve a convex optimization program that trades off bias and variance optimally. Provided that the propensity scores can be consistently estimated, the proposed estimator is asymptotically normal and semi-parametrically efficient among all asymptotically linear estimators. The rate at which the propensity scores are consistent is essentially irrelevant, allowing us to estimate them via modern machine learning techniques. We validate the finite-sample performance of the proposed estimator through comparative simulation studies and the real-world problem of inferring the stellar masses of galaxies in the Sloan Digital Sky Survey.
The focus of this study is to investigate the impact of different initialization strategies for the weight matrix of Successor Features (SF) on learning efficiency and convergence in Reinforcement Learning (RL) agents. Using a grid-world paradigm, we compare the performance of RL agents, whose SF weight matrix is initialized with either an identity matrix, zero matrix, or a randomly generated matrix (using Xavier, He, or uniform distribution method). Our analysis revolves around evaluating metrics such as value error, step length, PCA of Successor Representation (SR) place field, and the distance of SR matrices between different agents. The results demonstrate that RL agents initialized with random matrices reach the optimal SR place field faster and showcase a quicker reduction in value error, pointing to more efficient learning. Furthermore, these random agents also exhibit a faster decrease in step length across larger grid-world environments. The study provides insights into the neurobiological interpretations of these results, their implications for understanding intelligence, and potential future research directions. These findings could have profound implications for the field of artificial intelligence, particularly in the design of learning algorithms.
In the context of the interaction between a moving plane shock wave and an inclined wall (wedge), it is possible to distinguish four distinct shock reflection configurations. These shock wave reflections, which depend on the characteristics of the incident shock wave and the geometry of the surface that it interacts with, are (i) regular reflection (RR), (ii) simple Mach reflection (SMR), (iii) transition Mach reflection (TMR), and (iv) double Mach reflection (DMR). The impact of these shock reflections on flow properties can be significant so understanding them is important when predicting the behavior of shock waves in more complex flow configurations. Previous research works have explored the referred shock reflections through both numerical and experimental approaches, employing various gases and different flow and geometrical configurations. The present study involves the use of a high-fidelity computational fluid dynamics (CFD) tool, known as PeleC, which is a compressible solver based on AMReX specifically designed to handle complex flow configurations. Accordingly, by solving the time-dependent Euler equations for various 2D flow configurations, this work studies shock wave reflections accounting for four different Mach-based operating conditions and compares and analyzes the resulting density profiles on the wedge wall with experimental data. To strike a balance between model accuracy and computational efficiency, adaptive mesh refinement (AMR) is incorporated, and a mesh independence study is performed by varying the number of AMR levels. The results of this study demonstrate the capabilities of the CFD tool employed as it accurately predicts the sensitivity of wave characteristics to different operating conditions.
The Mixture of Experts (MoE) is a widely known neural architecture where an ensemble of specialized sub-models optimizes overall performance with a constant computational cost. However, conventional MoEs pose challenges at scale due to the need to store all experts in memory. In this paper, we push MoE to the limit. We propose extremely parameter-efficient MoE by uniquely combining MoE architecture with lightweight experts.Our MoE architecture outperforms standard parameter-efficient fine-tuning (PEFT) methods and is on par with full fine-tuning by only updating the lightweight experts -- less than 1% of an 11B parameters model. Furthermore, our method generalizes to unseen tasks as it does not depend on any prior task knowledge. Our research underscores the versatility of the mixture of experts architecture, showcasing its ability to deliver robust performance even when subjected to rigorous parameter constraints. Our code used in all the experiments is publicly available here: //github.com/for-ai/parameter-efficient-moe.
Achieving real-time capability is an essential prerequisite for the industrial implementation of nonlinear model predictive control (NMPC). Data-driven model reduction offers a way to obtain low-order control models from complex digital twins. In particular, data-driven approaches require little expert knowledge of the particular process and its model, and provide reduced models of a well-defined generic structure. Herein, we apply our recently proposed data-driven reduction strategy based on Koopman theory [Schulze et al. (2022), Comput. Chem. Eng.] to generate a low-order control model of an air separation unit (ASU). The reduced Koopman model combines autoencoders and linear latent dynamics and is constructed using machine learning. Further, we present an NMPC implementation that uses derivative computation tailored to the fixed block structure of reduced Koopman models. Our reduction approach with tailored NMPC implementation enables real-time NMPC of an ASU at an average CPU time decrease by 98 %.
Since the introduction of DeepMimic [Peng et al. 2018], subsequent research has focused on expanding the repertoire of simulated motions across various scenarios. In this study, we propose an alternative approach for this goal, a deep reinforcement learning method based on the simulation of a single-rigid-body character. Using the centroidal dynamics model (CDM) to express the full-body character as a single rigid body (SRB) and training a policy to track a reference motion, we can obtain a policy that is capable of adapting to various unobserved environmental changes and controller transitions without requiring any additional learning. Due to the reduced dimension of state and action space, the learning process is sample-efficient. The final full-body motion is kinematically generated in a physically plausible way, based on the state of the simulated SRB character. The SRB simulation is formulated as a quadratic programming (QP) problem, and the policy outputs an action that allows the SRB character to follow the reference motion. We demonstrate that our policy, efficiently trained within 30 minutes on an ultraportable laptop, has the ability to cope with environments that have not been experienced during learning, such as running on uneven terrain or pushing a box, and transitions between learned policies, without any additional learning.
The purpose of this paper is to present the structure of the linear codes over a finite field with q elements that have a permutation automorphism of order m. These codes can be considered as generalized quasi-cyclic codes. Quasi-cyclic codes and almost quasi-cyclic codes are discussed in detail, presenting necessary and sufficient conditions for which linear codes with such an automorphism are self-orthogonal, self-dual, or linear complementary dual.
In this letter, the average mutual information (AMI) of generalized quadrature spatial modulation (GQSM) is first derived for continuous-input continuous-output channels. Our mathematical analysis shows that the calculation error induced by Monte Carlo integration increases exponentially with the signal-to-noise ratio. This nature of GQSM is resolved by deriving a closed-form expression. The derived AMI is compared with other related SM schemes and evaluated for different antenna activation patterns. Our results show that an equiprobable antenna selection method slightly decreases AMI of symbols, while the method significantly improves AMI in total.
In this paper we consider a mathematical model which describes the equilibrium of two elastic rods attached to a nonlinear spring. We derive the variational formulation of the model which is in the form of an elliptic quasivariational inequality for the displacement field. We prove the unique weak solvability of the problem, then we state and prove some convergence results, for which we provide the corresponding mechanical interpretation. Next, we turn to the numerical approximation of the problem based on a finite element scheme. We use a relaxation method to solve the discrete problems that we implement on the computer. Using this method, we provide numerical simulations which validate our convergence results.
Orthogonal time sequency multiplexing (OTSM) has been recently proposed as a single-carrier (SC) waveform offering similar bit error rate (BER) to multi-carrier orthogonal time frequency space (OTFS) modulation in doubly-spread channels under high mobilities; however, with much lower complexity making OTSM a promising candidate for low-power millimeter-wave (mmWave) vehicular communications in 6G wireless networks. In this paper, the performance of OTSM-based homodyne transceiver is explored under hardware impairments (HIs) including in-phase and quadrature imbalance (IQI), direct current offset (DCO), phase noise, power amplifier non-linearity, carrier frequency offset, and synchronization timing offset. First, the discrete-time baseband signal model is obtained in vector form under the mentioned HIs. Then, the system input-output relations are derived in time, delay-time, and delay-sequency (DS) domains in which the parameters of HIs are incorporated. Analytical studies demonstrate that noise stays white Gaussian and effective channel matrix is sparse in the DS domain under HIs. Also, DCO appears as a DC signal at receiver interfering with only the zero sequency over all delay taps in the DS domain; however, IQI redounds to self-conjugated fully-overlapping sequency interference. Simulation results reveal the fact that with no HI compensation (HIC), not only OTSM outperforms plain SC waveform but it performs close to uncompensated OTFS system; however, HIC is essentially needed for OTSM systems operating in mmWave and beyond frequency bands.
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