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Jiakun Liu

Publications and source records attributed to Jiakun Liu.

At least 55 records · Page 3Linked to original sources
arXiv2023

$C^{2,α}$ regularity of free boundaries in optimal transportation

The regularity of the free boundary in optimal transportation is equivalent to that of the potential function along the free boundary. By establishing new geometric estimates of the free boundary and studying the second boundary value problem of the Monge-Ampère equation, we obtain the $C^{2,α}$ regularity of the potential function as well as that of the free boundary, thereby resolve an open problem raised by Caffarelli and McCann in \cite{CM}.

math.AP↗
arXiv2022

An Indirect Rate-Distortion Characterization for Semantic Sources: General Model and the Case of Gaussian Observation

A new source model, which consists of an intrinsic state part and an extrinsic observation part, is proposed and its information-theoretic characterization, namely its rate-distortion function, is defined and analyzed. Such a source model is motivated by the recent surge of interest in the semantic aspect of information: the intrinsic state corresponds to the semantic feature of the source, which in general is not observable but can only be inferred from the extrinsic observation. There are two distortion measures, one between the intrinsic state and its reproduction, and the other between the extrinsic observation and its reproduction. Under a given code rate, the tradeoff between these two distortion measures is characterized by the rate-distortion function, which is solved via the indirect rate-distortion theory and is termed as the semantic rate-distortion function of the source. As an application of the general model and its analysis, the case of Gaussian extrinsic observation is studied, assuming a linear relationship between the intrinsic state and the extrinsic observation, under a quadratic distortion structure. The semantic rate-distortion function is shown to be the solution of a convex programming programming problem with respect to an error covariance matrix, and a reverse water-filling type of solution is provided when the model further satisfies a diagonalizability condition.

cs.IT↗
arXiv2021

Global regularity for the Monge-Ampère equation with natural boundary condition

In this paper, we establish the global $C^{2,α}$ and $W^{2,p}$ regularity for the Monge-Ampère equation $\det\,D^2u = f$ subject to boundary condition $Du(Ω) = Ω^*$, where $Ω$ and $Ω^*$ are bounded convex domains in the Euclidean space $\mathbb{R}^n$ with $C^{1,1}$ boundaries, and $f$ is a Hölder continuous function. This boundary value problem arises naturally in optimal transportation and many other applications.

math.AP↗
arXiv2021

A Rate-Distortion Framework for Characterizing Semantic Information

A rate-distortion problem motivated by the consideration of semantic information is formulated and solved. The starting point is to model an information source as a pair consisting of an intrinsic state which is not observable, corresponding to the semantic aspect of the source, and an extrinsic observation which is subject to lossy source coding. The proposed rate-distortion problem seeks a description of the information source, via encoding the extrinsic observation, under two distortion constraints, one for the intrinsic state and the other for the extrinsic observation. The corresponding state-observation rate-distortion function is obtained, and a few case studies of Gaussian intrinsic state estimation and binary intrinsic state classification are studied.

cs.IT↗
arXiv2021

An Exploratory Study on the Repeatedly Shared External Links on Stack Overflow

On Stack Overflow, users reuse 11,926,354 external links to share the resources hosted outside the Stack Overflow website. The external links connect to the existing programming-related knowledge and extend the crowdsourced knowledge on Stack Overflow. Some of the external links, so-called as repeated external links, can be shared for multiple times. We observe that 82.5% of the link sharing activities (i.e., sharing links in any question, answer, or comment) on Stack Overflow share external resources, and 57.0% of the occurrences of the external links are sharing the repeated external links. However, it is still unclear what types of external resources are repeatedly shared. To help users manage their knowledge, we wish to investigate the characteristics of the repeated external links in knowledge sharing on Stack Overflow. In this paper, we analyze the repeated external links on Stack Overflow. We observe that external links that point to the text resources (hosted in documentation websites, tutorial websites, etc.) are repeatedly shared the most. We observe that: 1) different users repeatedly share the same knowledge in the form of repeated external links, thus increasing the maintenance effort of knowledge (e.g., update invalid links in multiple posts), 2) the same users can repeatedly share the external links for the purpose of promotion, and 3) external links can point to webpages with an overload of information that is difficult for users to retrieve relevant information. Our findings provide insights to Stack Overflow moderators and researchers. For example, we encourage Stack Overflow to centrally manage the commonly occurring knowledge in the form of repeated external links in order to better maintain the crowdsourced knowledge on Stack Overflow.

cs.SE↗
arXiv2021

An Exploratory Study on the Introduction and Removal of Different Types of Technical Debt

To complete tasks faster, developers often have to sacrifice the quality of the software. Such compromised practice results in the increasing burden to developers in future development. The metaphor, technical debt, describes such practice. Prior research has illustrated the negative impact of technical debt, and many researchers investigated how developers deal with a certain type of technical debt. However, few studies focused on the removal of different types of technical debt in practice. To fill this gap, we use the introduction and removal of different types of self-admitted technical debt (i.e., SATD) in 7 deep learning frameworks as an example. This is because deep learning frameworks are some of the most important software systems today due to their prevalent use in life-impacting deep learning applications. Moreover, the field of the development of different deep learning frameworks is the same, which enables us to find common behaviors on the removal of different types of technical debt across projects. By mining the file history of these frameworks, we find that design debt is introduced the most along the development process. As for the removal of technical debt, we find that requirement debt is removed the most, and design debt is removed the fastest. Most of test debt, design debt, and requirement debt are removed by the developers who introduced them. Based on the introduction and removal of different types of technical debt, we discuss the evolution of the frequencies of different types of technical debt to depict the unresolved sub-optimal trade-offs or decisions that are confronted by developers along the development process. We also discuss the removal patterns of different types of technical debt, highlight future research directions, and provide recommendations for practitioners.

cs.SE↗
arXiv2020

An Auto-Generated Geometry-Based Discrete Finite Element Model for Damage Evolution in Composite Laminates with Arbitrary Stacking Sequence

Stiffness degradation and progressive failure of composite laminates are complex processes involving evolution and multi-mode interactions among fiber fractures, intra-ply matrix cracks and inter-ply delaminations. This paper presents a novel finite element model capable of explicitly treating such discrete failures in laminates of random layup. Matching of nodes is guaranteed at potential crack bifurcations to ensure correct displacement jumps near crack tips and explicit load transfer among cracks. The model is entirely geometry-based (no mesh prerequisite) with distinct segments assembled together using surface-based tie constraints, and thus requires no element partitioning or enrichment. Several numerical examples are included to demonstrate the model's ability to generate results that are in qualitative and quantitative agreement with experimental observations on both damage evolution and tensile strength of specimens. The present model is believed unique in realizing simultaneous and accurate coupling of all three types of failures in laminates having arbitrary ply angles and layup.

math.NA↗
arXiv2020

Broken External Links on Stack Overflow

Stack Overflow hosts valuable programming-related knowledge with 11,926,354 links that reference to the third-party websites. The links that reference to the resources hosted outside the Stack Overflow websites extend the Stack Overflow knowledge base substantially. However, with the rapid development of programming-related knowledge, many resources hosted on the Internet are not available anymore. Based on our analysis of the Stack Overflow data that was released on Jun. 2, 2019, 14.2% of the links on Stack Overflow are broken links. The broken links on Stack Overflow can obstruct viewers from obtaining desired programming-related knowledge, and potentially damage the reputation of the Stack Overflow as viewers might regard the posts with broken links as obsolete. In this paper, we characterize the broken links on Stack Overflow. 65% of the broken links in our sampled questions are used to show examples, e.g., code examples. 70% of the broken links in our sampled answers are used to provide supporting information, e.g., explaining a certain concept and describing a step to solve a problem. Only 1.67% of the posts with broken links are highlighted as such by viewers in the posts' comments. Only 5.8% of the posts with broken links removed the broken links. Viewers cannot fully rely on the vote scores to detect broken links, as broken links are common across posts with different vote scores. The websites that host resources that can be maintained by their users are referenced by broken links the most on Stack Overflow -- a prominent example of such websites is GitHub. The posts and comments related to the web technologies, i.e., JavaScript, HTML, CSS, and jQuery, are associated with more broken links. Based on our findings, we shed lights for future directions and provide recommendations for practitioners and researchers.

cs.SE↗
arXiv2020

Regularity of free boundaries in optimal transportation

In this paper, we obtain some regularities of the free boundary in optimal transportation with the quadratic cost. Our first result is about the $C^{1,α}$ regularity of the free boundary for optimal partial transport between convex domains for densities $f, g$ bounded from below and above. When $f, g \in C^α$, and $\partialΩ, \partialΩ^*\in C^{1,1}$ are far apart, by adopting our recent results on boundary regularity of Monge-Ampère equations \cite{CLW1}, our second result shows that the free boundaries are $C^{2,α}$. As an application, in the last we also obtain these regularities of the free boundary in an optimal transport problem with two separate targets.

math.AP↗
arXiv2019

A class of nonlinear optimisation and applications

In this paper, we introduce a class of nonlinear optimisation problems. Under mild assumptions, we obtain the existence of potential functions and show that the potential function is a generalised solution of a Monge-Ampère type equation. We also present some interesting applications in optimal transportation and geometric optics problems.

math.AP↗
arXiv2019

Global regularity of optimal mappings in non-convex domains

In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument, using a recent global regularity of optimal transportation in convex domains by the authors.

math.AP↗
arXiv2018

Noncompact $L_p$-Minkowski problems

In this paper we prove the existence of complete, noncompact convex hypersurfaces whose $p$-curvature function is prescribed on a domain in the unit sphere. This problem is related to the solvability of Monge-Ampère type equations subject to certain boundary conditions depending on the value of $p$. The special case of $p=1$ was previously studied by Pogorelov and Chou-Wang. Here, we give some sufficient conditions for the solvability for general $p\neq1$.

math.AP↗
arXiv2018

$L_p$-Brunn-Minkowski inequality for $p\in (1-\frac{c}{n^{\frac{3}{2}}}, 1)$

Kolesnikov-Milman [9] established a local $L_p$-Brunn-Minkowski inequality for $p\in(1-c/n^{\frac{3}{2}},1).$ Based on their local uniqueness results for the $L_p$-Minkowski problem, we prove in this paper the (global) $L_p$-Brunn-Minkowski inequality. Two uniqueness results are also obtained: the first one is for the $L_p$-Minkowski problem when $p\in (1-c/n^{\frac{3}{2}}, 1)$ for general measure with even positive $C^α$ density, and the second one is for the Logarithmic Minkowski problem when the density of measure is a small $C^α$ even perturbation of the uniform density.

math.AP↗
arXiv2018

Optimal transport with discrete long range mean field interactions

We study an optimal transport problem where, at some intermediate time, the mass is accelerated by either an external force field, or self-interacting. We obtain regularity of the velocity potential, intermediate density, and optimal transport map, under conditions on the interaction potential that are related to the so-called Ma-Trudinger-Wang condition from optimal transport.

math.AP↗
arXiv2018

Boundary regularity for the second boundary-value problem of Monge-Ampère equations in dimension two

In this paper, we introduce an iteration argument to prove that a convex solution to the Monge-Ampère equation $\mbox{det } D^2 u =f $ in dimension two subject to the natural boundary condition $Du(Ω) = Ω^*$ is $C^{2,α}$ smooth up to the boundary. We establish the estimate under the sharp conditions that the inhomogeneous term $f\in C^α$ and the domains are convex and $C^{1,α}$ smooth. When $f\in C^0$ (resp. $1/C<f<C$ for some positive constant $C$), we also obtain the global $W^{2,p}$ (resp. $W^{2,1+ε}$) regularity.

math.AP↗
arXiv2018

Stochastic continuity of random fields governed by a system of stochastic PDEs

This paper constructs a solvability theory for a system of stochastic partial differential equations. On account of the Kolmogorov continuity theorem, solutions are looked for in certain Hölder-type classes in which a random field is treated as a space-time function taking values in $L^{p}$-space of random variables. A modified stochastic parabolicity condition involving $p$ is proposed to ensure the finiteness of the associated norm of the solution, which is showed to be sharp by examples. The Schauder-type estimates and the solvability theorem are proved.

math.PR↗