Unnormalized optimal transport

Gangbo, Wilfrid; Li, Wuchen; Osher, Stanley; Michael, Puthawala,

doi:10.1016/j.jcp.2019.108940

Cited by 50 publications

(45 citation statements)

References 29 publications

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“…So, multilevel unnormalized transport is implemented for faster calculation (Liu et al, 2019). A detailed study of this method can be found in (Gangbo et al, 2019;Chambolle and Pock, 2011;Li et al, 2018).…”

Section: Unnormalized Optimal Transportmentioning

confidence: 99%

“…The above classical approach from Monge and Kantrovich is in the normalized density space, i.e., the masses of two density functions or histogram is equal. Gangbo et al (2019) have described a formulation in which the masses of two density functions are unequal, refer to it as unnormalized or unbalanced optimal transport. With this approach, it is possible to examine two images with different intensities.…”

Section: Introductionmentioning

confidence: 99%

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Front Tracking of Shock and Combustion Waves by an Optimal Transport Framework

Bhattarai¹,

Lysaker²,

Bjerketvedt³

2021

Linköping Electronic Conference Proceedings

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This paper presents a framework for front tracking of shock waves from high-speed video recordings. The video was corrupted with noise and artifact that made front tracking a challenging task. To our knowledge, the implementation of Unnormalized Optimal Transport for determination of velocity is novel. An Unnormalized Optimal Transport framework was implemented in Python to determine the route of propagation from the recorded high-speed video. The obtained route was further computed for front tracking by two methods; Divergence and Transport method. These methods were investigated with both synthetic images and the recorded high-speed video frames. For both, the Transport method provided better results than the Divergence method. Therefore, the shock wave velocity and the shock angles were calculated from Unnormalized Optimal Transport combined with the Transport method. Preliminary results indicate that our findings are in good agreement with sensor-based measurements. This framework verified that the triple point height is increasing, and the oblique shockwave moves faster than the normal shock wave.

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Section: Unnormalized Optimal Transportmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Front Tracking of Shock and Combustion Waves by an Optimal Transport Framework

Bhattarai¹,

Lysaker²,

Bjerketvedt³

2021

Linköping Electronic Conference Proceedings

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show abstract

“…In this section, we form (5) as a finite dimensional minimization problem using the discretization developed in [8,9,15,20]. The discretized problem is shown to be smooth and strictly convex, so that Newton's method can be applied.…”

Section: Algorithmmentioning

confidence: 99%

Computations of Optimal Transport Distance with Fisher Information Regularization

Yin

Osher

2017

J Sci Comput

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We propose a fast algorithm to approximate the optimal transport distance. The main idea is to add a Fisher information regularization into the dynamical setting of the problem, originated by Benamou and Brenier. The regularized problem is shown to be smooth and strictly convex, thus many classical fast algorithms are available. In this paper, we adopt Newton's method, which converges to the minimizer with a quadratic rate. Several numerical examples are provided.

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“…Classical examples are equations involving probability density functions (PDFs) such as the Fokker-Planck equation [48], the Liouville equation [56,15,14], and the Boltzmann equation [12,37,9]. More recently, high-dimensional PDEs have also become central to many new areas of application such as optimal mass transport [26,57], random dynamical systems [55,56], mean field games [19,51], and functional-differential equations [54,53]. Computing the numerical solution to high-dimensional PDEs is an extremely challenging problem which has attracted substantial research efforts in recent years.…”

Section: Introductionmentioning

confidence: 99%

Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs

Dektor

Venturi

2020

Journal of Computational Physics

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We develop new dynamically orthogonal tensor methods to approximate multivariate functions and the solution of high-dimensional time-dependent nonlinear partial differential equations (PDEs). The key idea relies on a hierarchical decomposition of the approximation space obtained by splitting the independent variables of the problem into disjoint subsets. This process, which can be conveniently be visualized in terms of binary trees, yields series expansions analogous to the classical Tensor-Train and Hierarchical Tucker tensor formats. By enforcing dynamic orthogonality conditions at each level of binary tree, we obtain coupled evolution equations for the modes spanning each subspace within the hierarchical decomposition. This allows us to effectively compute the solution to high-dimensional time-dependent nonlinear PDEs on tensor manifolds of constant rank, with no need for rank reduction methods. We also propose new algorithms for dynamic addition and removal of modes within each subspace. Numerical examples are presented and discussed for high-dimensional hyperbolic and parabolic PDEs in bounded domains.

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Unnormalized optimal transport

Cited by 50 publications

References 29 publications

Front Tracking of Shock and Combustion Waves by an Optimal Transport Framework

Front Tracking of Shock and Combustion Waves by an Optimal Transport Framework

Computations of Optimal Transport Distance with Fisher Information Regularization

Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs

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