Publications of Qinglan Xia

Qinglan Xia's Publications

Optimal paths related to transport problems. Communications in Contemporary Mathematics. Vol. 5, No. 2 (2003) 251-279.

This is my first paper on optimal transport paths. In this article, I set up the general framework for studying ramified optimal transportation. The main concept here is optimal transport paths between probabilty measures. Main results include the existence theorem for optimal transport path; and the space of probability measures becomes a length space with respect to the d_a distance introduced here.

Interior regularity of optimal transport paths. Calculus of Variations and Partial Differential Equations. Vol. 20, No. 3 (2004) 283-299.

In this article, I first showed that a transport path is a rectifiable real coefficient flat 1-chain in terms of geometric measure theory. Then, away from the support of initial measures, I showed that locally an optimal transport path is just a weighted directed tree (i.e. unions of directed segments with associated weights) satisfying a balance equation, and thus has very nice interior regularity.

Boundary regularity of optimal transport paths. Submitted to Advances in Calculus of Variations.

In this article, I showed the boundary regularity of optimal transport paths. That is, on the support of initial measures (which is of course not necessarily one dimensional), each super-level set of an optimal transport path is locally concentrated on a finite union of one dimensional chord-arc curves. These curves still satisfy a balance equations. Roughly speaking, one may take a tree leaf in nature, then the vein structure on the leaf can be viewed as optimal transport path from the leaf (the Lebesgue measure on the two dimesional domain) to the root of the leaf (i.e. a Dirac measure).

The formation of a tree leaf. ESAIM Control Optim. Calc. Var. 13 (2007), no. 2, 359--377.

In this article, we apply the theory of optimal transport path to understand the dynamic formation of tree leaves. Numerical simmulation of tree leaves is provided here. This article can be viewed as a feedback to the nature after we "stealing" the ideas from tree leaves to prove the boundary regularity of optimal transport paths.

An application of optimal transport paths to urban transport networks. Discrete and Continuous Dynmical Systems, Supp. Volume, 2005, 904-910.

In this article, we apply the theory of optimal transport path to model urban transport network. An optimal transport path can be viewed as a urban transport network. Edges with high weights can be viewed freeways, while edges with low weights can be viewed as streets, roads etc.

The geodesic problem in quasimetric spaces, Journal of Geometric Analysis: Volume 19, Issue2 (2009), Page 452-479.

In this article, we showed that on the space of atomic probabilty measures, the d_a metrics introduced in the first paper are indeed the intrinsic metrics induced by a family of generalized metrics on the space of atomic probabilty measures. These family of generalized metrics (called quasimetrics) satisfy a relaxed triangle inequlaity d(x,y)<=c(d(x,z)+d(z,y)) for some c>=1. Optimal transport paths between atomic probabilty measures are exactly geodesics in these intrinsic metric spaces of atomic probability measures. Indeed, we showed that many classical results (e.g. Ascoli-Arzel\`{a} theorem) about functions in metric spaces are still hold in quasimetrics.

Numerical simulation of optimal transport paths, the Second International Conference on Computer Modeling and Simulation (ICCMS 2010). Vol 1, (2010) pp521-525.

Here, we provide 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.

Diffusion-limited aggregation driven by optimal transportation. (With Douglas Unger). Fractals. Vol. 18, No.2 (2010) 1-7.

This is a joint work with an REU student in summer 2008. It can be viewed as an application of the idea of ramified optimal transportation to Fractals (Diffusion-limited aggregation).
Abstract: In this article, we combine the DLA model of Witten and Sander with ideas from optimal transportation. We propose a modification of the DLA model in which the probability of sticking is inversely proportional to the additional transport cost from the point to the root. We used a family of cost functions parametrized by a parameter a as studied in ramified optimal transportation. a < 0 promotes growth near the root whereas a > 0 promotes growth at the tips of the cluster. a= 0 is a phase transition point and corresponds to standard DLA. What makes this model interesting is that when a is negative enough (e.g. a < −2) the final cluster is an one dimensional curve. On the other hand, when a is positive enough (e.g. a > 2) we get a nearly two dimensional disk. Thus our model encompasses the full range of fractal dimension from 1 to 2.

The exchange value embedded in a transport system. (With Shaofeng Xu) arXiv:1001.5232. To appear in Applied Mathematics and Optimization (2010).

This is a joint work with a graduate student in Economics. It can be viewed an application of ramified optimal transportation to Economics.
Abstract: 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.

R amified optimal transportation in geodesic metric spaces. arXiv:0907.5596. Accepted by Advances in Calculus of Variations. (2010)

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.

On the transport dimension of measures. (With Anna Vershynina). SIAM J. MATH. ANAL. Vol. 41, No. 6,(2010) pp. 2407-2430.

This is a joint work with a graduate student at UC Davis.
In this article, we define the transport dimension of probability measures on a Euclidean space 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 Rm. 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 the distance from it to any finite atomic probability measure.

Regularity of minimizers of quasi perimeters with a volume constraint. Interfaces and Free Boundaries. Volume 7, Issue 3, 2005, pp: 339-352

I studied a variational problem in this article. Among all sets of finite perimeters with a fixed given volume, I considered minimizing functionals which are sums of the perimeter and some noising term. I showed that when the noising term is somehow controlled by the perimeter, then the boundary of each minimizer of the variation problem has the C^{1,\alpha} regularity . This problem ismotivated by the study of mud cracking using the methods of Monge-Kantorovich transportation.

Intersection homology theory via rectifiable currents. Calculus of Variations and Partial Differential Equations. Vol. 19, No. 4 (2004), 421-443.

This article comes from my Ph..D. dissertation. I developed a setting for treating variational problems on stratified pseudomanifolds with singularities, such as complex projective varieties. I show that the intersection homology theory introduced by MacPherson and Gorsky using (subanalytic) geometric chains can be redefined using rectifiable currents with integer multiplicities. In variational problems, one needs to take various limits (e.g. of minimizing sequences), but a basic problem is that a limit of geometric intersection chains may fail to be a geometric chain; and even if it is, it may not satisfy the important perversity conditions of the approximating chains concerning intersection with singular set. This motivates me the use of rectifiable currents with a suitably modified mass norm. Also, I introduced a suitably modified mass on rectifiable currents such that all rectifiable currents with finite modified mass and finite boundary modified mass automatically satisfy the given perversity conditions. Then, I showed that there exists a modified mass minimizer in every intersection homology class on a compact stratified subanalytic pseudomanifold. Moreover, I showed that these mass minimizers are in fact almost minimizing currents. Thus, by a lemma of Almgren, I achieved a partial regularity theorem for these mass minimizers.

Conformal deformation of a closed Riemannian submanifold to a minimal submanifold. (with Xu, Senlin) Journal of Mathematical Study, Vol 31 (1998), no. 2, 109--115. A summary version is also published on Chinese Science Bulletin, Vol 43 (1998), no. 6, 527.

In this article, we showed that after using a suitable conformal deformation of the ambient space, any closed Riemannian submanifold can be conformly transformed into a minimal submanifold in the conformally transformed ambient space.

On the spectrum of Clifford hypersurface. (with Xu, Senlin) Journal of Mathematical Study. Vol 29 (1996), no. 4, 5--9.

In this article, I studied the spectrum of Clifford hypersurface.