arXiv subjects

Xiaobing Feng

Publications and source records attributed to Xiaobing Feng.

At least 37 records · Page 2Linked to original sources

arXiv2022

Fully discrete finite element methods for nonlinear stochastic elastic wave equations with multiplicative noise

This paper is concerned with fully discrete finite element methods for approximating variational solutions of nonlinear stochastic elastic wave equations with multiplicative noise. A detailed analysis of the properties of the weak solution is carried out and a fully discrete finite element method is proposed. Strong convergence in the energy norm with rate $\cal{O}(k+h^r)$ is proved, where $k$ and $h$ denote respectively the temporal and spatial mesh sizes, and $r(\geq 1)$ is the order of the finite element. Numerical experiments are provided to test the efficiency of proposed numerical methods and to validate the theoretical error estimate results.

math.NA↗

arXiv2022

Higher order time discretization for the stochastic semilinear wave equation with multiplicative noise

In this paper, a higher-order time-discretization scheme is proposed, where the iterates approximate the solution of the stochastic semilinear wave equation driven by multiplicative noise with general drift and diffusion. We employ a variational method for its error analysis and prove an improved convergence order of 3/2 for the approximates of the solution. The core of the analysis is Holder continuity in time and moment bounds for the solutions of the continuous and the discrete problem. Computational experiments are also presented.

math.NA↗

arXiv2022

An efficient multi-modes Monte Carlo homogenization method for random materials

In this paper, we propose and analyze a new stochastic homogenization method for diffusion equations with random and fast oscillatory coefficients. In the proposed method, the homogenized solutions are sought through a two-stage procedure. In the first stage, the original oscillatory diffusion equation is approximated, for each fixed random sample w, by a spatially homogenized diffusion equation with piecewise constant coefficients, resulting a random diffusion equation. In the second stage, the resulted random diffusion equation is approximated and computed by using an efficient multi-modes Monte Carlo method which only requires to solve a diffusion equation with a constant diffusion coefficient and a random right-hand side. The main advantage of the proposed method is that it separates the computational difficulty caused by the spatial fast oscillation of the solution and that caused by the randomness of the solution, so they can be overcome separately using different strategies. The convergence of the solution of the spatially homogenized equation (from the first stage) to the solution of the original random diffusion equation is established and the optimal rate of convergence is also obtained for the proposed multi-modes Monte Carlo method. Numerical experiments on some benchmark test problems for random composite materials are also presented to gauge the efficiency and accuracy of the proposed two-stage stochastic homogenization method.

math.NA↗

arXiv2022

A practical framework for multi-domain speech recognition and an instance sampling method to neural language modeling

Automatic speech recognition (ASR) systems used on smart phones or vehicles are usually required to process speech queries from very different domains. In such situations, a vanilla ASR system usually fails to perform well on every domain. This paper proposes a multi-domain ASR framework for Tencent Map, a navigation app used on smart phones and in-vehicle infotainment systems. The proposed framework consists of three core parts: a basic ASR module to generate n-best lists of a speech query, a text classification module to determine which domain the speech query belongs to, and a reranking module to rescore n-best lists using domain-specific language models. In addition, an instance sampling based method to training neural network language models (NNLMs) is proposed to address the data imbalance problem in multi-domain ASR. In experiments, the proposed framework was evaluated on navigation domain and music domain, since navigating and playing music are two main features of Tencent Map. Compared to a general ASR system, the proposed framework achieves a relative 13% $\sim$ 22% character error rate reduction on several test sets collected from Tencent Map and our in-car voice assistant.

eess.AS↗

arXiv2022

A narrow-stencil framework for convergent numerical approximations of fully nonlinear second order PDEs

This paper develops a unified general framework for designing convergent finite difference and discontinuous Galerkin methods for approximating viscosity and regular solutions of fully nonlinear second order PDEs. Unlike the well-known monotone (finite difference) framework, the proposed new framework allows for the use of narrow stencils and unstructured grids which makes it possible to construct high order methods. The general framework is based on the concepts of consistency and g-monotonicity which are both defined in terms of various numerical derivative operators. Specific methods that satisfy the framework are constructed using numerical moments. Admissibility, stability, and convergence properties are proved, and numerical experiments are provided along with some computer implementation details.

math.NA↗

arXiv2021

An efficient iterative method for solving parameter-dependent and random convention-diffusion problems

This paper develops and analyzes a general iterative framework for solving parameter-dependent and random convection-diffusion problems. It is inspired by the multi-modes method of [7,8] and the ensemble method of [20] and extends those methods into a more general and unified framework. The main idea of the framework is to reformulate the underlying problem into another problem with parameter-independent convection and diffusion coefficients and a parameter-dependent (and solution-dependent) right-hand side, a fixed-point iteration is then employed to compute the solution of the reformulated problem. The main benefit of the proposed approach is that an efficient direct solver and a block Krylov subspace iterative solver can be used at each iteration, allowing to reuse the $LU$ matrix factorization or to do an efficient matrix-matrix multiplication for all parameters, which in turn results in significant computation saving. Convergence and rates of convergence are established for the iterative method both at the variational continuous level and at the finite element discrete level under some structure conditions. Several strategies for establishing reformulations of parameter-dependent and random diffusion and convection-diffusion problems are proposed and their computational complexity is analyzed. Several 1-D and 2-D numerical experiments are also provided to demonstrate the efficiency of the proposed iterative method and to validate the theoretical convergence results.

math.NA↗

arXiv2021

Analysis of Chorin-Type Projection Methods for the Stochastic Stokes Equations with General Multiplicative Noises

This paper is concerned with numerical analysis of two fully discrete Chorin-type projection methods for the stochastic Stokes equations with general non-solenoidal multiplicative noise. The first scheme is the standard Chorin scheme and the second one is a modified Chorin scheme which is designed by employing the Helmholtz decomposition on the noise function at each time step to produce a projected divergence-free noise and a "pseudo pressure" after combining the original pressure and the curl-free part of the decomposition. Optimal order rates of the convergence are proved for both velocity and pressure approximations of these two (semi-discrete) Chorin schemes. It is crucial to measure the errors in appropriate norms. The fully discrete finite element methods are formulated by discretizing both semi-discrete Chorin schemes in space by the standard finite element method. Suboptimal order error estimates are derived for both fully discrete methods. It is proved that all spatial error constants contain a growth factor $k^{-1/2}$, where $k$ denotes the time step size, which explains the deteriorating performance of the standard Chorin scheme when $k\to 0$ and the space mesh size is fixed as observed earlier in the numerical tests of [9]. Numerical results are also provided to guage the performance of the proposed numerical methods and to validate the sharpness of the theoretical error estimates.

math.NA↗

arXiv2021

On a New Class of Fractional Calculus of Variations and Related Fractional Differential Equations

This paper is concerned with analyzing a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on the classical notion of fractional derivatives, the fractional calculus of variations considered in this paper is based on a newly developed notion of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new types of calculus of variations and fractional differential equations as well as nonstandard Neumann boundary operators. The primary objective of this paper is to establish the well-posedness and regularities for a class of fractional calculus of variations problems and their Euler-Lagrange (fractional differential) equations. This is achieved first for one-sided Dirichlet energy functionals which lead to one-sided fractional Laplace equations, then for more general energy functionals which give rise to more general fractional differential equations.

math.AP↗

arXiv2021

Improving Speech Recognition Accuracy of Local POI Using Geographical Models

Nowadays voice search for points of interest (POI) is becoming increasingly popular. However, speech recognition for local POI has remained to be a challenge due to multi-dialect and massive POI. This paper improves speech recognition accuracy for local POI from two aspects. Firstly, a geographic acoustic model (Geo-AM) is proposed. The Geo-AM deals with multi-dialect problem using dialect-specific input feature and dialect-specific top layer. Secondly, a group of geo-specific language models (Geo-LMs) are integrated into our speech recognition system to improve recognition accuracy of long tail and homophone POI. During decoding, specific language models are selected on demand according to users' geographic location. Experiments show that the proposed Geo-AM achieves 6.5%$\sim$10.1% relative character error rate (CER) reduction on an accent testset and the proposed Geo-AM and Geo-LM totally achieve over 18.7% relative CER reduction on Tencent Map task.

eess.AS↗

arXiv2021

Pinpointing the Memory Behaviors of DNN Training

The training of deep neural networks (DNNs) is usually memory-hungry due to the limited device memory capacity of DNN accelerators. Characterizing the memory behaviors of DNN training is critical to optimize the device memory pressures. In this work, we pinpoint the memory behaviors of each device memory block of GPU during training by instrumenting the memory allocators of the runtime system. Our results show that the memory access patterns of device memory blocks are stable and follow an iterative fashion. These observations are useful for the future optimization of memory-efficient training from the perspective of raw memory access patterns.

cs.PF↗

arXiv2021

High-order mass- and energy-conserving SAV-Gauss collocation finite element methods for the nonlinear Schrödinger equation

A family of arbitrarily high-order fully discrete space-time finite element methods are proposed for the nonlinear Schrödinger equation based on the scalar auxiliary variable formulation, which consists of a Gauss collocation temporal discretization and the finite element spatial discretization. The proposed methods are proved to be well-posed and conserving both mass and energy at the discrete level. An error bound of the form $O(h^p+τ^{k+1})$ in the $L^\infty(0,T;H^1)$-norm is established, where $h$ and $τ$ denote the spatial and temporal mesh sizes, respectively, and $(p,k)$ is the degree of the space-time finite elements. Numerical experiments are provided to validate the theoretical results on the convergence rates and conservation properties. The effectiveness of the proposed methods in preserving the shape of a soliton wave is also demonstrated by numerical results.

math.NA↗

arXiv2020

Fusion-Catalyzed Pruning for Optimizing Deep Learning on Intelligent Edge Devices

The increasing computational cost of deep neural network models limits the applicability of intelligent applications on resource-constrained edge devices. While a number of neural network pruning methods have been proposed to compress the models, prevailing approaches focus only on parametric operators (e.g., convolution), which may miss optimization opportunities. In this paper, we present a novel fusion-catalyzed pruning approach, called FuPruner, which simultaneously optimizes the parametric and non-parametric operators for accelerating neural networks. We introduce an aggressive fusion method to equivalently transform a model, which extends the optimization space of pruning and enables non-parametric operators to be pruned in a similar manner as parametric operators, and a dynamic filter pruning method is applied to decrease the computational cost of models while retaining the accuracy requirement. Moreover, FuPruner provides configurable optimization options for controlling fusion and pruning, allowing much more flexible performance-accuracy trade-offs to be made. Evaluation with state-of-the-art residual neural networks on five representative intelligent edge platforms, Jetson TX2, Jetson Nano, Edge TPU, NCS, and NCS2, demonstrates the effectiveness of our approach, which can accelerate the inference of models on CIFAR-10 and ImageNet datasets.

cs.NE↗

arXiv2020

Accelerating Deep Learning Inference with Cross-Layer Data Reuse on GPUs

Accelerating the deep learning inference is very important for real-time applications. In this paper, we propose a novel method to fuse the layers of convolutional neural networks (CNNs) on Graphics Processing Units (GPUs), which applies data reuse analysis and access optimization in different levels of the memory hierarchy. To achieve the balance between computation and memory access, we explore the fusion opportunities in the CNN computation graph and propose three fusion modes of convolutional neural networks: straight, merge and split. Then, an approach for generating efficient fused code is designed, which goes deeper in multi-level memory usage for cross-layer data reuse. The effectiveness of our method is evaluated with the network layers from state-of-the-art CNNs on two different GPU platforms, NVIDIA TITAN Xp and Tesla P4. The experiments show that the average speedup is 2.02x on representative structures of CNNs, and 1.57x on end-to-end inference of SqueezeNet.

cs.DC↗

arXiv2020

LANCE: Efficient Low-Precision Quantized Winograd Convolution for Neural Networks Based on Graphics Processing Units

Accelerating deep convolutional neural networks has become an active topic and sparked an interest in academia and industry. In this paper, we propose an efficient low-precision quantized Winograd convolution algorithm, called LANCE, which combines the advantages of fast convolution and quantization techniques. By embedding linear quantization operations into the Winograd-domain, the fast convolution can be performed efficiently under low-precision computation on graphics processing units. We test neural network models with LANCE on representative image classification datasets, including SVHN, CIFAR, and ImageNet. The experimental results show that our 8-bit quantized Winograd convolution improves the performance by up to 2.40x over the full-precision convolution with trivial accuracy loss.

cs.CV↗

arXiv2020

A new theory of fractional differential calculus

This paper presents a self-contained new theory of weak fractional differential calculus in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions and characterize what functions are fractionally differentiable. Various calculus rules including a fundamental theorem calculus, product and chain rules, and integration by parts formulas are established for weak fractional derivatives. Additionally, relationships with classical fractional derivatives and detailed characterizations of weakly fractional differentiable functions are also established. Furthermore, the notion of weak fractional derivatives is also systematically extended to general distributions instead of only to some special distributions. This new theory lays down a solid theoretical foundation for systematically and rigorously developing new theories of fractional Sobolev spaces, fractional calculus of variations, and fractional PDEs as well as their numerical solutions in subsequent works. This paper is a concise presentation of the materials of Sections 1-4 and 6 of reference [9].

math.FA↗

arXiv2020

On New Families of Fractional Sobolev Spaces

This paper presents three new families of fractional Sobolev spaces and their accompanying theory in one-dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural generalizations of the well-established integer order Sobolev spaces and theory. In particular, two new families of one-sided fractional Sobolev spaces are introduced and analyzed, they reveal more insights about another family of so-called symmetric fractional Sobolev spaces. Many key theorems/properties, such as density/approximation theorem, extension theorems, one-sided trace theorem, and various embedding theorems and Sobolev inequalities in those Sobolev spaces are established. Moreover, a few relationships with existing fractional Sobolev spaces are also discovered. The results of this paper lay down a solid theoretical foundation for systematically developing a fractional calculus of variations theory and a fractional PDE theory as well as their numerical solutions in subsequent works. This paper is a concise presentation of the materials of Sections 1, 4 and 5 of reference [7].

math.FA↗

arXiv2020

Optimally Convergent Mixed Finite Element Methods for the Stochastic Stokes Equations

We propose some new mixed finite element methods for the time dependent stochastic Stokes equations with multiplicative noise, which use the Helmholtz decomposition of the driving multiplicative noise. It is known [16] that the pressure solution has a low regularity, which manifests in sub-optimal convergence rates for well-known inf-sup stable mixed finite element methods in numerical simulations, see [10]. We show that eliminating this gradient part from the noise in the numerical scheme leads to optimally convergent mixed finite element methods, and that this conceptual idea may be used to retool numerical methods that are well-known in the deterministic setting, including pressure stabilization methods, so that their optimal convergence properties can still be maintained in the stochastic setting. Computational experiments are also provided to validate the theoretical results and to illustrate the conceptional usefulness of the proposed numerical approach.

math.NA↗

arXiv2020

A new theory of fractional differential calculus and fractional Sobolev spaces: One-dimensional case

This paper presents a self-contained new theory of weak fractional differential calculus and fractional Sobolev spaces in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions and characterize what functions are fractionally differentiable. Various calculus rules including a fundamental theorem of calculus, product and chain rules, and integration by parts formulas are established for weak fractional derivatives and relationships with classical derivatives are also obtained. Based on the weak fractional derivative notion, new fractional order Sobolev spaces are introduced and many important theorems and properties, such as density/approximation theorem, extension theorems, trace theorem, and various embedding theorems in these Sobolev spaces are established. Moreover, a few relationships with existing fractional Sobolev spaces are also established. Furthermore, the notion of weak fractional derivatives is also systematically extended to general distributions instead of only to some special distributions. The new theory lays down a solid theoretical foundation for systematically and rigorously developing a fractional calculus of variations theory and a fractional PDE theory as well as their numerical solutions in subsequent works.

math.CA↗