The bubble transform is a procedure to decompose differential forms, which are piecewise smooth with respect to a given triangulation of the domain, into a sum of local bubbles. In this paper, an improved version of a construction in the setting of the de Rham complex previously proposed by the authors is presented. The major improvement in the decomposition is that unlike the previous results, in which the individual bubbles were rational functions with the property that groups of local bubbles summed up to preserve piecewise smoothness, the new decomposition is strictly space-preserving in the sense that each local bubble preserves piecewise smoothness. An important property of the transform is that the construction only depends on the given triangulation of the domain and is independent of any finite element space. On the other hand, all the standard piecewise polynomial spaces are invariant under the transform. Other key properties of the transform are that it commutes with the exterior derivative, is bounded in L^2, and satisfies the stable decomposition property.
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We propose a novel neural network architecture based on conformer transducer that adds contextual information flow to the ASR systems. Our method improves the accuracy of recognizing uncommon words while not harming the word error rate of regular words. We explore the uncommon words accuracy improvement when we use the new model and/or shallow fusion with context language model. We found that combination of both provides cumulative gain in uncommon words recognition accuracy.
We provide a new theoretical framework for the variable-step deferred correction (DC) methods based on the well-known BDF2 formula. By using the discrete orthogonal convolution kernels, some high-order BDF2-DC methods are proven to be stable on arbitrary time grids according to the recent definition of stability (SINUM, 60: 2253-2272). It significantly relaxes the existing step-ratio restrictions for the BDF2-DC methods (BIT, 62: 1789-1822). The associated sharp error estimates are established by taking the numerical effects of the starting approximations into account, and they suggest that the BDF2-DC methods have no aftereffect, that is, the lower-order starting scheme for the BDF2 scheme will not cause a loss in the accuracy of the high-order BDF2-DC methods. Extensive tests on the graded and random time meshes are presented to support the new theory.
Gaussian binomial coefficients are q-analogues of the binomial coefficients of integers. On the other hand, binomial coefficients have been extended to finite words, i.e., elements of the finitely generated free monoids. In this paper we bring together these two notions by introducing q-analogues of binomial coefficients of words. We study their basic properties, e.g., by extending classical formulas such as the q-Vandermonde and Manvel's et al. identities to our setting. As a consequence, we get information about the structure of the considered words: these q-deformations of binomial coefficients of words contain much richer information than the original coefficients. From an algebraic perspective, we introduce a q-shuffle and a family q-infiltration products for non-commutative formal power series. Finally, we apply our results to generalize a theorem of Eilenberg characterizing so-called p-group languages. We show that a language is of this type if and only if it is a Boolean combination of specific languages defined through q-binomial coefficients seen as polynomials over $\mathbb{F}_p$.
The design of particle simulation methods for collisional plasma physics has always represented a challenge due to the unbounded total collisional cross section, which prevents a natural extension of the classical Direct Simulation Monte Carlo (DSMC) method devised for the Boltzmann equation. One way to overcome this problem is to consider the design of Monte Carlo algorithms that are robust in the so-called grazing collision limit. In the first part of this manuscript, we will focus on the construction of collision algorithms for the Landau-Fokker-Planck equation based on the grazing collision asymptotics and which avoids the use of iterative solvers. Subsequently, we discuss problems involving uncertainties and show how to develop a stochastic Galerkin projection of the particle dynamics which permits to recover spectral accuracy for smooth solutions in the random space. Several classical numerical tests are reported to validate the present approach.
Deep denoisers have shown excellent performance in solving inverse problems in signal and image processing. In order to guarantee the convergence, the denoiser needs to satisfy some Lipschitz conditions like non-expansiveness. However, enforcing such constraints inevitably compromises recovery performance. This paper introduces a novel training strategy that enforces a weaker constraint on the deep denoiser called pseudo-contractiveness. By studying the spectrum of the Jacobian matrix, relationships between different denoiser assumptions are revealed. Effective algorithms based on gradient descent and Ishikawa process are derived, and further assumptions of strict pseudo-contractiveness yield efficient algorithms using half-quadratic splitting and forward-backward splitting. The proposed algorithms theoretically converge strongly to a fixed point. A training strategy based on holomorphic transformation and functional calculi is proposed to enforce the pseudo-contractive denoiser assumption. Extensive experiments demonstrate superior performance of the pseudo-contractive denoiser compared to related denoisers. The proposed methods are competitive in terms of visual effects and quantitative values.
The elasticity of the DTW metric provides a more flexible comparison between time series and is used in numerous machine learning domains such as classification or clustering. However, it does not align the measurements at the beginning and end of time series if they have a shift occurring right at the start of one series, with the omitted part appearing at the end of that series. Due to the cyclicity of such series - which lack a definite beginning or end - we rely on the Cyclic DTW approach to propose a less computationally expensive approximation of this calculation method. This approximation will then be employed in conjunction with the K-Means clustering method.
We formulate a uniform tail bound for empirical processes indexed by a class of functions, in terms of the individual deviations of the functions rather than the worst-case deviation in the considered class. The tail bound is established by introducing an initial "deflation" step to the standard generic chaining argument. The resulting tail bound is the sum of the complexity of the "deflated function class" in terms of a generalization of Talagrand's $\gamma$ functional, and the deviation of the function instance, both of which are formulated based on the natural seminorm induced by the corresponding Cram\'{e}r functions. We also provide certain approximations for the mentioned seminorm when the function class lies in a given (exponential type) Orlicz space, that can be used to make the complexity term and the deviation term more explicit.
Acoustic wave equation is a partial differential equation (PDE) which describes propagation of acoustic waves through a material. In general, the solution to this PDE is nonunique. Therefore, it is necessary to impose initial conditions in the form of Cauchy conditions for obtaining a unique solution. Theoretically, solving the wave equation is equivalent to representing the wavefield in terms of a radiation source which possesses finite energy over space and time.The radiation source is represented by a forcing term in the right-hand-side of the wave equation. In practice, the source may be represented in terms of normal derivative of pressure or normal velocity over a surface. The pressure wavefield is then calculated by solving an associated boundary-value problem via imposing conditions on the boundary of a chosen solution space. From ananalytic point of view, this manuscript aims to review typical approaches for obtaining unique solution to the acoustic wave equation in terms of either a volumetric radiation source, or a surface source in terms of normal derivative of pressure or normal velocity. A numerical approximation of the derived formulae will then be explained. The key step for numerically approximating the derived analytic formulae is inclusion of source, and will be studied carefully in this manuscript.
We propose a new concept of lifts of reversible diffusion processes and show that various well-known non-reversible Markov processes arising in applications are lifts in this sense of simple reversible diffusions. Furthermore, we introduce a concept of non-asymptotic relaxation times and show that these can at most be reduced by a square root through lifting, generalising a related result in discrete time. Finally, we demonstrate how the recently developed approach to quantitative hypocoercivity based on space-time Poincar\'e inequalities can be rephrased and simplified in the language of lifts and how it can be applied to find optimal lifts.
We consider nonparametric Bayesian inference in a multidimensional diffusion model with reflecting boundary conditions based on discrete high-frequency observations. We prove a general posterior contraction rate theorem in $L^2$-loss, which is applied to Gaussian priors. The resulting posteriors, as well as their posterior means, are shown to converge to the ground truth at the minimax optimal rate over H\"older smoothness classes in any dimension. Of independent interest and as part of our proofs, we show that certain frequentist penalized least squares estimators are also minimax optimal.
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