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Zhi Zhou

Publications and source records attributed to Zhi Zhou.

At least 127 records · Page 7Linked to original sources

Robust Deep Semi-Supervised Learning: A Brief Introduction

Semi-supervised learning (SSL) is the branch of machine learning that aims to improve learning performance by leveraging unlabeled data when labels are insufficient. Recently, SSL with deep models has proven to be successful on standard benchmark tasks. However, they are still vulnerable to various robustness threats in real-world applications as these benchmarks provide perfect unlabeled data, while in realistic scenarios, unlabeled data could be corrupted. Many researchers have pointed out that after exploiting corrupted unlabeled data, SSL suffers severe performance degradation problems. Thus, there is an urgent need to develop SSL algorithms that could work robustly with corrupted unlabeled data. To fully understand robust SSL, we conduct a survey study. We first clarify a formal definition of robust SSL from the perspective of machine learning. Then, we classify the robustness threats into three categories: i) distribution corruption, i.e., unlabeled data distribution is mismatched with labeled data; ii) feature corruption, i.e., the features of unlabeled examples are adversarially attacked; and iii) label corruption, i.e., the label distribution of unlabeled data is imbalanced. Under this unified taxonomy, we provide a thorough review and discussion of recent works that focus on these issues. Finally, we propose possible promising directions within robust SSL to provide insights for future research.

cs.LG↗

iGniter: Interference-Aware GPU Resource Provisioning for Predictable DNN Inference in the Cloud

GPUs are essential to accelerating the latency-sensitive deep neural network (DNN) inference workloads in cloud datacenters. To fully utilize GPU resources, spatial sharing of GPUs among co-located DNN inference workloads becomes increasingly compelling. However, GPU sharing inevitably brings severe performance interference among co-located inference workloads, as motivated by an empirical measurement study of DNN inference on EC2 GPU instances. While existing works on guaranteeing inference performance service level objectives (SLOs) focus on either temporal sharing of GPUs or reactive GPU resource scaling and inference migration techniques, how to proactively mitigate such severe performance interference has received comparatively little attention. In this paper, we propose iGniter, an interference-aware GPU resource provisioning framework for cost-efficiently achieving predictable DNN inference in the cloud. iGniter is comprised of two key components: (1) a lightweight DNN inference performance model, which leverages the system and workload metrics that are practically accessible to capture the performance interference; (2) A cost-efficient GPU resource provisioning strategy that jointly optimizes the GPU resource allocation and adaptive batching based on our inference performance model, with the aim of achieving predictable performance of DNN inference workloads. We implement a prototype of iGniter based on the NVIDIA Triton inference server hosted on EC2 GPU instances. Extensive prototype experiments on four representative DNN models and datasets demonstrate that iGniter can guarantee the performance SLOs of DNN inference workloads with practically acceptable runtime overhead, while saving the monetary cost by up to 25% in comparison to the state-of-the-art GPU resource provisioning strategies.

cs.DC↗

GNN at the Edge: Cost-Efficient Graph Neural Network Processing over Distributed Edge Servers

Edge intelligence has arisen as a promising computing paradigm for supporting miscellaneous smart applications that rely on machine learning techniques. While the community has extensively investigated multi-tier edge deployment for traditional deep learning models (e.g. CNNs, RNNs), the emerging Graph Neural Networks (GNNs) are still under exploration, presenting a stark disparity to its broad edge adoptions such as traffic flow forecasting and location-based social recommendation. To bridge this gap, this paper formally studies the cost optimization for distributed GNN processing over a multi-tier heterogeneous edge network. We build a comprehensive modeling framework that can capture a variety of different cost factors, based on which we formulate a cost-efficient graph layout optimization problem that is proved to be NP-hard. Instead of trivially applying traditional data placement wisdom, we theoretically reveal the structural property of quadratic submodularity implicated in GNN's unique computing pattern, which motivates our design of an efficient iterative solution exploiting graph cuts. Rigorous analysis shows that it provides parameterized constant approximation ratio, guaranteed convergence, and exact feasibility. To tackle potential graph topological evolution in GNN processing, we further devise an incremental update strategy and an adaptive scheduling algorithm for lightweight dynamic layout optimization. Evaluations with real-world datasets and various GNN benchmarks demonstrate that our approach achieves superior performance over de facto baselines with more than 95.8% cost eduction in a fast convergence speed.

cs.DC↗

USB: A Unified Semi-supervised Learning Benchmark for Classification

Semi-supervised learning (SSL) improves model generalization by leveraging massive unlabeled data to augment limited labeled samples. However, currently, popular SSL evaluation protocols are often constrained to computer vision (CV) tasks. In addition, previous work typically trains deep neural networks from scratch, which is time-consuming and environmentally unfriendly. To address the above issues, we construct a Unified SSL Benchmark (USB) for classification by selecting 15 diverse, challenging, and comprehensive tasks from CV, natural language processing (NLP), and audio processing (Audio), on which we systematically evaluate the dominant SSL methods, and also open-source a modular and extensible codebase for fair evaluation of these SSL methods. We further provide the pre-trained versions of the state-of-the-art neural models for CV tasks to make the cost affordable for further tuning. USB enables the evaluation of a single SSL algorithm on more tasks from multiple domains but with less cost. Specifically, on a single NVIDIA V100, only 39 GPU days are required to evaluate FixMatch on 15 tasks in USB while 335 GPU days (279 GPU days on 4 CV datasets except for ImageNet) are needed on 5 CV tasks with TorchSSL.

cs.LG↗

Inverse Problems for Subdiffusion from Observation at an Unknown Terminal Time

Inverse problems of recovering space-dependent parameters, e.g., initial condition, space-dependent source or potential coefficient, in a subdiffusion model from the terminal observation have been extensively studied in recent years. However, all existing studies have assumed that the terminal time at which one takes the observation is exactly known. In this work, we present uniqueness and stability results for three canonical inverse problems, e.g., backward problem, inverse source and inverse potential problems, from the terminal observation at an unknown time. The subdiffusive nature of the problem indicates that one can simultaneously determine the terminal time and space-dependent parameter. The analysis is based on explicit solution representations, asymptotic behavior of the Mittag-Leffler function, and mild regularity conditions on the problem data. Further, we present several one- and two-dimensional numerical experiments to illustrate the feasibility of the approach.

math.NA↗

Recovery of a Space-Time Dependent Diffusion Coefficient in Subdiffusion: Stability, Approximation and Error Analysis

In this work, we study an inverse problem of recovering a space-time dependent diffusion coefficient in the subdiffusion model from the distributed observation, where the mathematical model involves a Djrbashian-Caputo fractional derivative of order $α\in(0,1)$ in time. The main technical challenges of both theoretical and numerical analysis lie in the limited smoothing properties due to the fractional differential operator and high degree of nonlinearity of the forward map from the unknown diffusion coefficient to the distributed observation. We establish two conditional stability results using a novel test function, which leads to a stability bound in $L^2(0,T;L^2(Ω))$ under a suitable positivity condition. The positivity condition is verified for a large class of problem data. Numerically, we develop a rigorous procedure for recovering the diffusion coefficient based on a regularized least-squares formulation, which is then discretized by the standard Galerkin method with continuous piecewise linear elements in space and backward Euler convolution quadrature in time. We provide a complete error analysis of the fully discrete formulation, by combining several new error estimates for the direct problem (optimal in terms of data regularity), a discrete version of fractional maximal $L^p$ regularity, and a nonstandard energy argument. Under the positivity condition, we obtain a standard $\ell^2(L^2(Ω))$ error estimate consistent with the conditional stability. Further, we illustrate the analysis with some numerical examples.

math.NA↗

Stability and numerical analysis of backward problem for subdiffusion with time-dependent coefficients

Our aim is to study the backward problem, i.e. recover the initial data from the terminal observation, of the subdiffusion with time dependent coefficients. First of all, by using the smoothing property of solution operators and a perturbation argument of freezing the diffusion coefficients, we show a stability estimate in Sobolev spaces, under some smallness/largeness condition on the terminal time. Moreover, in case of noisy observation, we apply a quasi-boundary value method to regularize the problem and then show the convergence of the regularization scheme. Finally, to numerically reconstruct the initial data, we propose a completely discrete scheme by applying the finite element method in space and backward Euler convolution quadrature in time. An \textsl{a priori} error estimate is then established. The proof is heavily built on a perturbation argument dealing with time dependent coefficients and some nonstandard error estimates for the direct problem. The error estimate gives a useful guide for balancing discretization parameters, regularization parameter and noise level. Some numerical experiments are presented to illustrate our theoretical results.

math.NA↗

Modified BDF2 schemes for subdiffusion models with a singular source term

The aim of this paper is to study the time stepping scheme for approximately solving the subdiffusion equation with a weakly singular source term. In this case, many popular time stepping schemes, including the correction of high-order BDF methods, may lose their high-order accuracy. To fill in this gap, in this paper, we develop a novel time stepping scheme, where the source term is regularized by using a $k$-fold integral-derivative and the equation is discretized by using a modified BDF2 convolution quadrature. We prove that the proposed time stepping scheme is second-order, even if the source term is nonsmooth in time and incompatible with the initial data. Numerical results are presented to support the theoretical results.

math.NA↗

Drag force on spherical particles trapped at a liquid interface

The dynamics of particles attached to an interface separating two immiscible fluids are encountered in a wide variety of applications. Here we present a combined asymptotic and numerical investigation of the fluid motion past spherical particles attached to a deformable interface undergoing uniform creeping flows in the limit of small Capillary number and small deviation of the contact angle from 90 degrees. Under the assumption of a constant three-phase contact angle, we calculate the interfacial deformation around an isolated particle and a particle pair. Applying the Lorentz reciprocal theorem to the zeroth-order approximation corresponding to spherical particles at a flat interface and the first correction in Capillary number and correction contact angle allows us to obtain explicit analytical expressions for the hydrodynamic drag in terms of the zeroth-order approximations and the correction deformations. The drag coefficients are computed as a function of the three-phase contact angle, the viscosity ratio of the two fluids, the Bond number, and the separation distance between the particles. In addition, the capillary force acting on the particles due to the interfacial deformation is calculated.

physics.flu-dyn↗

Edge Robotics: Edge-Computing-Accelerated Multi-Robot Simultaneous Localization and Mapping

With the wide penetration of smart robots in multifarious fields, Simultaneous Localization and Mapping (SLAM) technique in robotics has attracted growing attention in the community. Yet collaborating SLAM over multiple robots still remains challenging due to performance contradiction between the intensive graphics computation of SLAM and the limited computing capability of robots. While traditional solutions resort to the powerful cloud servers acting as an external computation provider, we show by real-world measurements that the significant communication overhead in data offloading prevents its practicability to real deployment. To tackle these challenges, this paper promotes the emerging edge computing paradigm into multi-robot SLAM and proposes RecSLAM, a multi-robot laser SLAM system that focuses on accelerating map construction process under the robot-edge-cloud architecture. In contrast to conventional multi-robot SLAM that generates graphic maps on robots and completely merges them on the cloud, RecSLAM develops a hierarchical map fusion technique that directs robots' raw data to edge servers for real-time fusion and then sends to the cloud for global merging. To optimize the overall pipeline, an efficient multi-robot SLAM collaborative processing framework is introduced to adaptively optimize robot-to-edge offloading tailored to heterogeneous edge resource conditions, meanwhile ensuring the workload balancing among the edge servers. Extensive evaluations show RecSLAM can achieve up to 39% processing latency reduction over the state-of-the-art. Besides, a proof-of-concept prototype is developed and deployed in real scenes to demonstrate its effectiveness.

cs.RO↗

Identification of potential in diffusion equations from terminal observation: analysis and discrete approximation

The aim of this paper is to study the recovery of a spatially dependent potential in a (sub)diffusion equation from overposed final time data. We construct a monotone operator one of whose fixed points is the unknown potential. The uniqueness of the identification is theoretically verified by using the monotonicity of the operator and a fixed point argument. Moreover, we show a conditional stability in Hilbert spaces under some suitable conditions on the problem data. Next, a completely discrete scheme is developed, by using Galerkin finite element method in space and finite difference method in time, and then a fixed point iteration is applied to reconstruct the potential. We prove the linear convergence of the iterative algorithm by the contraction mapping theorem, and present a thorough error analysis for the reconstructed potential. Our derived \textsl{a priori} error estimate provides a guideline to choose discretization parameters according to the noise level. The analysis relies heavily on some suitable nonstandard error estimates for the direct problem as well as the aforementioned conditional stability. Numerical experiments are provided to illustrate and complement our theoretical analysis.

math.NA↗

Backward diffusion-wave problem: stability, regularization and approximation

We aim at the development and analysis of the numerical schemes for approximately solving the backward diffusion-wave problem, which involves a fractional derivative in time with order $α\in(1,2)$. From terminal observations at two time levels, i.e., $u(T_1)$ and $u(T_2)$, we simultaneously recover two initial data $u(0)$ and $u_t(0)$ and hence the solution $u(t)$ for all $t > 0$. First of all, existence, uniqueness and Lipschitz stability of the backward diffusion-wave problem were established under some conditions about $T_1$ and $T_2$. Moreover, for noisy data, we propose a quasi-boundary value scheme to regularize the "mildly" ill-posed problem, and show the convergence of the regularized solution. Next, to numerically solve the regularized problem, a fully discrete scheme is proposed by applying finite element method in space and convolution quadrature in time. We establish error bounds of the discrete solution in both cases of smooth and nonsmooth data. The error estimate is very useful in practice since it indicates the way to choose discretization parameters and regularization parameter, according to the noise level. The theoretical results are supported by numerical experiments.

math.NA↗

Robust Convergence of Parareal Algorithms with Arbitrarily High-order Fine Propagators

The aim of this paper is to analyze the robust convergence of a class of parareal algorithms for solving parabolic problems. The coarse propagator is fixed to the backward Euler method and the fine propagator is a high-order single step integrator. Under some conditions on the fine propagator, we show that there exists some critical $J_*$ such that the parareal solver converges linearly with a convergence rate near $0.3$, provided that the ratio between the coarse time step and fine time step named $J$ satisfies $J \ge J_*$. The convergence is robust even if the problem data is nonsmooth and incompatible with boundary conditions. The qualified methods include all absolutely stable single step methods, whose stability function satisfies $|r(-\infty)|<1$, and hence the fine propagator could be arbitrarily high-order. Moreover, we examine some popular high-order single step methods, e.g., two-, three- and four-stage Lobatto IIIC methods, and verify that the corresponding parareal algorithms converge linearly with a factor $0.31$ and the threshold for these cases is $J_* = 2$. Intensive numerical examples are presented to support and complete our theoretical predictions.

math.NA↗

Recovering the Potential and Order in One-Dimensional Time-Fractional Diffusion with Unknown Initial Condition and Source

This paper is concerned with an inverse problem of recovering a potential term and fractional order in a one-dimensional subdiffusion problem, which involves a Djrbashian-Caputo fractional derivative of order $α\in(0,1)$ in time, from the lateral Cauchy data. In the model, we do not assume a full knowledge of the initial data and the source term, since they might be unavailable in some practical applications. We prove the unique recovery of the spatially-dependent potential coefficient and the order $α$ of the derivation simultaneously from the measured trace data at one end point, when the model is equipped with a boundary excitation with a compact support away from $t=0$. One of the initial data and the source can also be uniquely determined, provided that the other is known. The analysis employs a representation of the solution and the time analyticity of the associated function. Further, we discuss a two-stage procedure, directly inspired by the analysis, for the numerical identification of the order and potential coefficient, and illustrate the feasibility of the recovery with several numerical experiments.

math.AP↗

A parallel-in-time algorithm for high-order BDF methods for diffusion and subdiffusion equations

In this paper, we propose a parallel-in-time algorithm for approximately solving parabolic equations. In particular, we apply the $k$-step backward differentiation formula, and then develop an iterative solver by using the waveform relaxation technique. Each resulting iteration represents a periodic-like system, which could be further solved in parallel by using the diagonalization technique. The convergence of the waveform relaxation iteration is theoretically examined by using the generating function method. The approach we established in this paper extends the existing argument of single-step methods in Gander and Wu [Numer. Math., 143 (2019), pp. 489--527] to general BDF methods up to order six. The argument could be further applied to the time-fractional subdiffusion equation, whose discretization shares common properties of the standard BDF methods, because of the nonlocality of the fractional differential operator. Illustrative numerical results are presented to complement the theoretical analysis.

math.NA↗

Reconstruction of a Space-Time Dependent Source in Subdiffusion Models via a Perturbation Approach

In this article we study inverse problems of recovering a space-time dependent source component from the lateral boundary observation in a subidffusion model. The mathematical model involves a Djrbashian-Caputo fractional derivative of order $α\in(0,1)$ in time, and a second-order elliptic operator with time-dependent coefficients. We establish a well-posedness and a conditional stability result for the inverse problems using a novel perturbation argument and refined regularity estimates of the associated direct problem. Further, we present an algorithm for efficiently and accurately reconstructing the source component, and provide several two-dimensional numerical results showing the feasibility of the recovery.

math.NA↗

Stability implies robust convergence of a class of preconditioned parallel-in-time iterative algorithms

Solving evolutionary equations in a parallel-in-time manner is an attractive topic and many algorithms are proposed in recent two decades. The algorithm based on the block $α$-circulant preconditioning technique has shown promising advantages, especially for wave propagation problems. By fast Fourier transform for factorizing the involved circulant matrices, the preconditioned iteration can be computed efficiently via the so-called diagonalization technique, which yields a direct parallel implementation across all time levels. In recent years, considerable efforts have been devoted to exploring the convergence of the preconditioned iteration by studying the spectral radius of the iteration matrix, and this leads to many case-by-case studies depending on the used time-integrator. In this paper, we propose a unified convergence analysis for the algorithm applied to $u'+Au=f$, where $σ(A)\subset\mathbb{C}^+$ with $σ(A)$ being the spectrum of $A\in\mathbb{C}^{m\times m}$. For any one-step method (such as the Runge-Kutta methods) with stability function $\mathcal{R}(z)$, we prove that the decay rate of the global error is bounded by $α/(1-α)$, provided the method is stable, i.e., $\max_{λ\inσ(A)}|\mathcal{R}(Δtλ)|\leq1$. For any linear multistep method, such a bound becomes $cα/(1-cα)$, where $c\geq1$ is a constant specified by the multistep method itself. Our proof only relies on the stability of the time-integrator and the estimate is independent of the step size $Δt$ and the spectrum $σ(A)$.

math.NA↗

Arbitrarily High-order Maximum Bound Preserving Schemes with Cut-off Postprocessing for Allen-Cahn Equations

We develop and analyze a class of maximum bound preserving schemes for approximately solving Allen--Cahn equations. We apply a $k$th-order single-step scheme in time (where the nonlinear term is linearized by multi-step extrapolation), and a lumped mass finite element method in space with piecewise $r$th-order polynomials and Gauss--Lobatto quadrature. At each time level, a cut-off post-processing is proposed to eliminate extra values violating the maximum bound principle at the finite element nodal points. As a result, the numerical solution satisfies the maximum bound principle (at all nodal points), and the optimal error bound $O(τ^k+h^{r+1})$ is theoretically proved for a certain class of schemes. These time stepping schemes under consideration includes algebraically stable collocation-type methods, which could be arbitrarily high-order in both space and time. Moreover, combining the cut-off strategy with the scalar auxiliary value (SAV) technique, we develop a class of energy-stable and maximum bound preserving schemes, which is arbitrarily high-order in time. Numerical results are provided to illustrate the accuracy of the proposed method.

math.NA↗