arXiv subjects

Qinglan Xia

Publications and source records attributed to Qinglan Xia.

15 recordsLinked to original sources

arXiv2025

Optimal transport paths with capacity induced cost function

This article generalizes the study of ramified optimal transport with capacity constraint in transport multi-paths by generalizing the $\mathbf{M}_{\alpha}$ cost to $\mathbf{M}_{\alpha,c}$, which incorporates capacity constraints into the cost function. Equipped with $\mathbf{M}_{\alpha,c}$ cost, we prove the existence of optimal transport path, $\mathbf{M}_{\alpha,c}$ related inequalities, decomposition of any general transport paths, and occurrence of direct line segments in an optimal transport path.

math.OC↗

arXiv2024

General fractals represented by $\mathcal{F}$-limit sets of compression maps

In this article, we provide a simple and systematic way to represent general (inhomogeneous) fractals that may look different at different scales and places. By using set-valued compression maps, we express these general fractals as $\mathcal{F}$-limit sets, which are represented as sequences of points in a fixed parameterization space $M$. By choosing different types of sequences in $M$, we get various types of fractals: from self-simlilar to non self-similar, and from deterministic to random. The computational complexity of producing a general fractal is independent of the sequence in $M$, and as a result, is the same as that of an iterated function system obtained from a constant sequence. In the metric space setting, we also estimate the Hausdorff dimension of limit sets for collections of sets that do not necessarily satisfy the Moran structure conditions. In particular, we introduce the concept ``uniform covering condition" for the study of the lower bound of the Hausdorff dimension of the limit set, and provide sufficient conditions for this condition. Specific examples (Cantor-like sets, Sierpiński-like Triangles, etc.) with the calculations of their corresponding Hausdorff dimensions are also studied.

math.CA↗

arXiv2024

Transport multi-paths with capacity constraints

This article generalizes the study of branched/ramified optimal transportation to those with capacity constraints. Each admissible transport network studied here is represented by a transport multi-path between measures, with a capacity constraint on each of its components. The associated transport cost is given by the sum of the $\textbf{M}_α$-cost of each component. Using this new formulation, we prove the existence of an optimal solution and provide an upper bound on the number of components for the solution. Additionally, we conduct analytical examinations of the properties (e.g. ``map-compatibility", and ``simple common-source property") of each solution component and explore the interplay among components, particularly in the discrete case.

math.OC↗

arXiv2023

Map-compatible decomposition of transport paths

In the Monge-Kantorovich transport problem, the transport cost is expressed in terms of transport maps or transport plans, which play crucial roles there. A variant of the Monge-Kantorovich problem is the ramified (branching) transport problem that models branching transport systems via transport paths. In this article, we showed that any cycle-free transport path between two atomic measures can be decomposed into the sum of a map-compatible path and a plan-compatible path. Moreover, we showed that each stair-shaped transport path can be decomposed into the difference of two map-compatible transport paths.

math.AP↗

arXiv2023

Partial Plateau's Problem with $H$-mass

Classically, Plateau's problem asks to find a surface of the least area with a given boundary $B$. In this article, we investigate a version of Plateau's problem, where the boundary of an admissible surface is only required to partially span $B$. Our boundary data is given by a flat $(m-1)$-chain $B$ and a smooth compactly supported differential $(m-1)$-form $Φ$. We are interested in minimizing $ \mathbf{M}(T) - \int_{\partial T} Φ$ over all $m$-dimensional rectifiable currents $T$ in $\mathbb{R}^n$ such that $\partial T$ is a subcurrent of the given boundary $B$. The existence of a rectifiable minimizer is proven with Federer and Fleming's compactness theorem. We generalize this problem by replacing the mass $\mathbf{M}$ with the $H$-mass of rectifiable currents. By minimizing over a larger class of objects, called scans with boundary, and by defining their $H$-mass as a type of lower-semicontinuous envelope over the $H$-mass of rectifiable currents, we prove an existence result for this problem by using Hardt and De Pauw's BV compactness theorem.

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arXiv2021

On the Curvature of Metric Triples

In this article, we introduce a notion of curvature, denoted by $ k_X(T)$, for a metric triple $T$ inside a (possibly discrete) metric space $X$. Such a notion enables us to consider curvature information of any metric space, including discrete metric spaces such as those generated by scientific data. To define the notion, we employ the information consisting of side lengths of the triple as well as the minimum total distance from vertices of the triple to points of the metric space. Those information provides us a unique number $k_X(T)$ such that the triple $T$ can be isometrically embedded into the model space $M_k^2$ up to $k\le k_X(T)$. The value $k_X(T)$ agrees with the usual curvature when $X$ is a convex subset of a model space. We also show that the curvature $k_X(T)$ of any metric triple $T$ inside a $CAT(k)$ space is bounded above by $k$.

math.MG↗

arXiv2021

Ramified optimal transportation with payoff on the boundary

This paper studies a variant of ramified/branched optimal transportation problems. Given the distributions of production capacities and market sizes, a firm looks for an allocation of productions over factories, a distribution of sales across markets, and a transport path that delivers the product to maximize its profit. Mathematically, given any two measures $μ$ and $ν$ on $X$, and a payoff function $h$, the planner wants to minimize $\mathbf{M}_{α}(T)-\int_{X}hd(\partial T)$ among all transport paths $T$ from $\tildeμ$ to $\tildeν$ with $\tildeμ\leq μ$ and $\tildeν\leq ν$, where $\mathbf{M}_{α}$ is the standard cost functional used in ramified transportation. After proving the existence result, we provide a characterization of the boundary measures of the optimal solution. They turn out to be the original measures restricted on some Borel subsets up to a Delta mass on each connected component. Our analysis further finds that as the boundary payoff increases, the corresponding solution of the current problem converges to an optimal transport path, which is the solution of the standard ramified transportation.

math.OC↗

arXiv2020

The existence of minimizers for an isoperimetric problem with Wasserstein penalty term in unbounded domains

In this article, we consider the (double) minimization problem $$\min\left\{P(E;Ω)+λW_p(E,F):~E\subseteqΩ,~F\subseteq \mathbb{R}^d,~\lvert E\cap F\rvert=0,~ \lvert E\rvert=\lvert F\rvert=1\right\},$$ where $p\geqslant 1$, $Ω$ is a (possibly unbounded) domain in $\mathbb{R}^d$, $P(E;Ω)$ denotes the relative perimeter of $E$ in $Ω$ and $W_p$ denotes the $p$-Wasserstein distance. When $Ω$ is unbounded and $d\geqslant 3$, it is an open problem proposed by Buttazzo, Carlier and Laborde in the paper ON THE WASSERSTEIN DISTANCE BETWEEN MUTUALLY SINGULAR MEASURES. We prove the existence of minimizers to this problem when $\frac{1}{p}+\frac{2}{d}>1$, $Ω=\mathbb{R}^d$ and $λ$ is sufficiently small.

math.CA↗

arXiv2017

A fractal shape optimization problem in branched transport

We investigate the following question: what is the set of unit volume which can be best irrigated starting from a single source at the origin, in the sense of branched transport? We may formulate this question as a shape optimization problem and prove existence of solutions, which can be considered as a sort of "unit ball" for branched transport. We establish some elementary properties of optimizers and describe these optimal sets A as sublevel sets of a so-called landscape function which is now classical in branched transport. We prove $β$-H{ö}lder regularity of the landscape function, allowing us to get an upper bound on the Minkowski dimension of the boundary: dim $\partial$A $\le$ d -- $β$ (where $β$ := d($α$ -- (1 -- 1/d)) $\in$ (0, 1) is a relevant exponent in branched transport, associated with the exponent $α$ > 1 -- 1/d appearing in the cost). We are not able to prove the upper bound, but we conjecture that $\partial$A is of non-integer dimension d -- $β$. Finally, we make an attempt to compute numerically an optimal shape, using an adaptation of the phase-field approximation of branched transport introduced some years ago by Oudet and the second author.

math.OC↗

arXiv2011

On the Ramified Optimal Allocation Problem

This paper proposes an optimal allocation problem with ramified transport technology in a spatial economy. Ramified transportation is used to model the transport economy of scale in group transportation observed widely in both nature and efficiently designed transport systems of branching structures. The ramified allocation problem aims at finding an optimal allocation plan as well as an associated optimal allocation path to minimize overall cost of transporting commodity from factories to households. This problem differentiates itself from existing ramified transportation literature in that the distribution of production among factories is not fixed but endogenously determined as observed in many allocation practices. It's shown that due to the transport economy of scale in ramified transportation, each optimal allocation plan corresponds equivalently to an optimal assignment map from households to factories. This optimal assignment map provides a natural partition of both households and allocation paths. We develop methods of marginal transportation analysis and projectional analysis to study properties of optimal assignment maps. These properties are then related to the search for an optimal assignment map in the context of state matrix.

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arXiv2010

The Exchange Value Embedded In A Transport System

This paper shows that a well designed transport system has an embedded exchange value by serving as a market for potential exchange between consumers. Under suitable conditions, one can improve the welfare of consumers in the system simply by allowing some exchange of goods between consumers during transportation without incurring additional transportation costs. We propose an explicit valuation formula to measure this exchange value for a given compatible transport system. This value is always nonnegative and bounded from above. Criteria based on transport structures, preferences and prices are provided to determine the existence of a positive exchange value. Finally, we study a new optimal transport problem with an objective taking into account of both transportation cost and exchange value.

math.OC↗

arXiv2009

On the transport dimension of measures

In this article, we define the transport dimension of probability measures on $\mathbb{R}^m$ using ramified optimal transportation theory. We show that the transport dimension of a probability measure is bounded above by the Minkowski dimension and below by the Hausdorff dimension of the measure. Moreover, we introduce a metric, called "the dimensional distance", on the space of probability measures on $\mathbb{R}^m$. This metric gives a geometric meaning to the transport dimension: with respect to this metric, we show that the transport dimension of a probability measure equals to the distance from it to any finite atomic probability measure.

math.OC↗

arXiv2009

Ramified optimal transportation in geodesic metric spaces

An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels, draining and irrigation systems. Here, we extend the study of ramified optimal transportation between probability measures from Euclidean spaces to a geodesic metric space. We investigate the existence as well as the behavior of optimal transport paths under various properties of the metric such as completeness, doubling, or curvature upper boundedness. We also introduce the transport dimension of a probability measure on a complete geodesic metric space, and show that the transport dimension of a probability measure is bounded above by the Minkowski dimension and below by the Hausdorff dimension of the measure. Moreover, we introduce a metric, called "the dimensional distance", on the space of probability measures. This metric gives a geometric meaning to the transport dimension: with respect to this metric, the transport dimension of a probability measure equals to the distance from it to any finite atomic probability measure.

math.MG↗

arXiv2009

The geodesic problem in quasimetric spaces

In this article, we study the geodesic problem in a generalized metric space, in which the distance function satisfies a relaxed triangle inequality $d(x,y)\leq σ(d(x,z)+d(z,y))$ for some constant $σ\geq 1$, rather than the usual triangle inequality. Such a space is called a quasimetric space. We show that many well-known results in metric spaces (e.g. Ascoli-Arzelà theorem) still hold in quasimetric spaces. Moreover, we explore conditions under which a quasimetric will induce an intrinsic metric. As an example, we introduce a family of quasimetrics on the space of atomic probability measures. The associated intrinsic metrics induced by these quasimetrics coincide with the $d_α$ metric studied early in the study of branching structures arisen in ramified optimal transportation. An optimal transport path between two atomic probability measures typically has a "tree shaped" branching structure. Here, we show that these optimal transport paths turn out to be geodesics in these intrinsic metric spaces.

math.MG↗

arXiv2008

Numerical simulation of optimal transport paths

This article provides numerical simulation of an optimal transport path from a single source to an atomic measure of equal total mass. We first construct an initial transport path, and then modify the path as much as possible by using both local and global minimization algorithms.

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