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Jiahui Zhu

Publications and source records attributed to Jiahui Zhu.

28 records · Page 2Linked to original sources
arXiv2023

A Chain of AI-based Solutions for Resolving FQNs and Fixing Syntax Errors in Partial Code

API documentation, technical blogs and programming Q&A sites contain numerous partial code that can be reused in programming tasks, but often these code are uncompilable due to unresolved names and syntax errors. To facilitate partial code reuse, we propose the Partial Code Reuse Chain (PCR-Chain) for resolving fully-qualified names (FQNs) and fixing last-mile syntax errors in partial code based on a giant large language model (LLM) like ChatGPT. Methodologically, PCR-Chain is backed up by the underlying global-level prompt architecture (which combines three design ideas: hierarchical task breakdown, prompt composition, and a mix of prompt-based AI and non-AI units) and the local-level prompt design. Technically, we propose PCR-Chain, which employs in-context learning rather than symbolic, costly training methods. Experimental results demonstrate that in dynamically-typed languages (Python), PCR-Chain outperforms current state-of-the-art (SOTA) 5% accuracy like RING. For statically-type languages (Java), our approach achieves high accuracy of 80.5% in resolving both non-FQNs and last-mile syntax errors, surpassing SOTA methods (RING) that can only address last-mile syntax errors. The correct execution of the unit, module, and PCR-Chain demonstrates the effectiveness of the prompt design, composition, and architecture and opens up possibilities for building software engineering tools based on LLMs, replacing traditional program analysis methods.

cs.SE↗
arXiv2022

The stochastic nonlinear Schrödinger equations driven by pure jump noise

In this paper, we establish the existence and uniqueness of solutions of stochastic nonlinear Schrödinger equations with additive jump noise in $L^2(\mathbb{R}^d)$. Our results cover all either focusing or defocusing nonlinearity in the full subcritical range of exponents as in the deterministic case.

math.PR↗
arXiv2022

SVMAC: Unsupervised 3D Human Pose Estimation from a Single Image with Single-view-multi-angle Consistency

Recovering 3D human pose from 2D joints is still a challenging problem, especially without any 3D annotation, video information, or multi-view information. In this paper, we present an unsupervised GAN-based model consisting of multiple weight-sharing generators to estimate a 3D human pose from a single image without 3D annotations. In our model, we introduce single-view-multi-angle consistency (SVMAC) to significantly improve the estimation performance. With 2D joint locations as input, our model estimates a 3D pose and a camera simultaneously. During training, the estimated 3D pose is rotated by random angles and the estimated camera projects the rotated 3D poses back to 2D. The 2D reprojections will be fed into weight-sharing generators to estimate the corresponding 3D poses and cameras, which are then mixed to impose SVMAC constraints to self-supervise the training process. The experimental results show that our method outperforms the state-of-the-art unsupervised methods on Human 3.6M and MPI-INF-3DHP. Moreover, qualitative results on MPII and LSP show that our method can generalize well to unknown data.

cs.CV↗
arXiv2021

The Stochastic Strichartz estimates and stochastic nonlinear Schrödinger equations driven by Lévy noise

We establish a new version of the stochastic Strichartz estimate for the stochastic convolution driven by jump noise which we apply to the stochastic nonlinear Schrödinger equation with nonlinear multiplicative jump noise in the Marcus canonical form. With the help of the deterministic Strichartz estimates, we prove the existence and uniqueness of a global solution to stochastic nonlinear Schrödinger equation in $L^2(\RR^n)$ with either focusing or defocusing nonlinearity in the full subcritical range of exponents as in the deterministic case.

math.PR↗
arXiv2019

$\mathbb{L}^p$-solutions for stochastic Navier-Stokes equations with jump noise

We study the existence and uniqueness of solutions of 2D Stochastic Navier-Stokes equation with space irregular jump noise for initial data in certain Sobolev spaces of negative order. Comparing with the Galerkin approximation method, the main advantage of this work is to use an $\mathbb{L}^p$-setting to obtain the solution under much weaker assumptions on the noise and the initial condition.

math.PR↗
arXiv2019

Maximal inequalities and exponential estimates for stochastic convolutions driven by Lévy-type processes in Banach spaces with application to stochastic quasi-geostrophic equations

We present remarkably simple proofs of Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by Lévy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of Itô's formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with Lévy noise is established.

math.PR↗
arXiv2015

Maximal inequality of Stochastic convolution driven by compensated Poisson random measures in Banach spaces

Let $(E, \| \cdot\|)$ be a Banach space such that, for some $q\geq 2$, the function $x\mapsto \|x\|^q$ is of $C^2$ class and its first and second Fréchet derivatives are bounded by some constant multiples of $(q-1)$-th power of the norm and $(q-2)$-th power of the norm and let $S$ be a $C_0$-semigroup of contraction type on $(E, \| \cdot\|)$. We consider the following stochastic convolution process \begin{align*} u(t)=\int_0^t\int_ZS(t-s)ξ(s,z)\,\tilde{N}(\mathrm{d} s,\mathrm{d} z), \;\;\; t\geq 0, \end{align*} where $\tilde{N}$ is a compensated Poisson random measure on a measurable space $(Z,\mathcal{Z})$ and $ξ:[0,\infty)\timesΩ\times Z\rightarrow E$ is an $\mathbb{F}\otimes \mathcal{Z}$-predictable function. We prove that there exists a càdlàg modification a $\tilde{u}$ of the process $u$ which satisfies the following maximal inequality \begin{align*} \mathbb{E} \sup_{0\leq s\leq t} \|\tilde{u}(s)\|^{q^\prime}\leq C\ \mathbb{E} \left(\int_0^t\int_Z \|ξ(s,z) \|^{p}\,N(\mathrm{d} s,\mathrm{d} z)\right)^{\frac{q^\prime}{p}}, \end{align*} for all $ q^\prime \geq q$ and $1<p\leq 2$ with $C=C(q,p)$.

math.PR↗
arXiv2013

Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise

Motivated by applications to a manifold of semilinear and quasilinear stochastic partial differential equations (SPDEs) we establish the existence and uniqueness of strong solutions to coercive and locally monotone SPDEs driven by Lévy processes. We illustrate the main result of our paper by showing how it can be applied to various types of SPDEs such as stochastic reaction-diffusion equations, stochastic Burgers type equations, stochastic 2D hydrodynamical systems and stochastic equations of non-Newtonian fluids, which generalize many existing results in the literature.

math.AP↗
arXiv2011

A maximal inequality for stochastic convolutions in 2-smooth Banach spaces

Let (e^{tA})_{t \geq 0} be a C_0-contraction semigroup on a 2-smooth Banach space E, let (W_t)_{t \geq 0} be a cylindrical Brownian motion in a Hilbert space H, and let (g_t)_{t \geq 0} be a progressively measurable process with values in the space γ(H,E) of all γ-radonifying operators from H to E. We prove that for all 0<p<\infty there exists a constant C, depending only on p and E, such that for all T \geq 0 we have \E \sup_{0\le t\le T} || \int_0^t e^{(t-s)A} g_s dW_s \ ||^p \leq C \mathbb{E} (\int_0^T || g_t ||_{γ(H,E)}^2 dt)^\frac{p}{2}. For p \geq 2 the proof is based on the observation that ψ(x) = || x ||^p is Fréchet differentiable and its derivative satisfies the Lipschitz estimate || ψ'(x) - ψ'(y)|| \leq C(|| x || + || y ||)^{p-2} || x-y ||; the extension to 0<p<2 proceeds via Lenglart's inequality.

math.PR↗