Quadratic Wasserstein Distance Research Articles

An optimal transport approach for 3D electrical impedance tomography

Abstract This work solves the three-dimensional inverse boundary value problem with the quadratic Wasserstein distance (W 2), which originates from the optimal transportation (OT) theory. The computation of the W 2 distance on the manifold surface is boiled down to solving the generalized Monge–Ampère equation, whose solution is directly related to the gradient of the W 2 distance. An efficient first-order method based on iteratively solving Poisson’s equation is introduced to solve the fully nonlinear elliptic equation. Combining with the adjoint-state technique, the optimization framework based on the W 2 distance is developed to solve the three-dimensional electrical impedance tomography problem. The proposed method is especially suitable for severely ill-posed and highly nonlinear inverse problems. Numerical experiments demonstrate that our method improves the stability and outperforms the traditional regularization methods.

Enhanced Multimodal Nonparametric Probabilistic Method for Model-Form Uncertainty Quantification and Digital Twinning

The nonparametric probabilistic method (NPM) is a physics-based machine learning approach for model-form (MF) uncertainty quantification (UQ), model updating, and digital twinning. It extracts from reference data information not captured by a real-time computational model and infuses it into a “hyperparameterized,” stochastic version of this model by solving an inverse statistical problem formulated using a loss function and a few hyperparameters. The loss function is designed such that the mean value and statistical fluctuations of some quantities of interest predicted using the real-time stochastic model match target values obtained from the reference data. NPM’s performance hinges upon the efficient minimization of the loss function, which is unfortunately stochastic and nonconvex and thus prone to getting the optimization procedure trapped in suboptimal local minima. The latter scenario is exacerbated when the reference data is scarce. The paper addresses these issues by adopting the squared quadratic Wasserstein distance as the measure of dissimilarity between two different sets of data and by reformulating NPM’s inverse statistical problem as a multimodal data-assimilation problem. The potential of the resulting enhanced NPM for MF-UQ, model updating, and digital twinning is demonstrated using numerical simulations relevant to applications in structural dynamics, including structural health monitoring.

Wasserstein distance-based full waveform inversion method for density reconstruction

Reliable density information is an essential factor in lithologic interpretation and reservoir evaluation. However, density reconstruction is hampered by a lack of long-wavelength information and the crosstalk between different parameters. The powerful nonlinearity in density inversion constrains the multi-parameter full waveform inversion application. This paper introduces the Wasserstein distance into the full waveform inversion for velocity and density based on the optimal transport theory. We construct an objective function with better convexity to avoid the cycle-skipping caused by the poor initial model. Furthermore, due to the sensitivity of the quadratic Wasserstein distance to low-frequency, we add the low-wavenumber component absent in the conventional density inversion. We acquire a long-wavelength density background using optimal transport inversion and then apply it as an initial model for least-squares inversion. Numerical examples show that our optimal transport inversion effectively builds a reasonable density background model for FWI. Density inversion can be improved by incorporating optimal transport and least-squares theory, leading to a reliable quantitative description for petroleum exploration and development.

Towards optimal transport for quantum densities

An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on $\mathbf{R}^d$ has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165-205] for density operators on $L^2(\mathbf{R}^d)$, and used to estimate the convergence rate of various asymptotic theories in the context of quantum mechanics. The present work proves a Kantorovich type duality theorem for this quantum variant of the Monge-Kantorovich or Wasserstein distance, and discusses the structure of optimal quantum couplings. Specifically, we prove that optimal quantum couplings involve a gradient type structure similar to the Brenier transport map (which is the gradient of a convex function), or more generally, to the subdifferential of a l.s.c. convex function as in the Knott-Smith optimality criterion (see Theorem 2.12 in [C. Villani: Topics in Optimal Transportation, Amer. Math. Soc. 2003]).

Optimal transportation for electrical impedance tomography

This work establishes a framework for solving inverse boundary problems with the geodesic-based quadratic Wasserstein distance ( W 2 W_{2} ). A general form of the Fréchet gradient is systematically derived from the optimal transportation (OT) theory. In addition, a fast algorithm based on the new formulation of OT on S 1 \mathbb {S}^{1} is developed to solve the corresponding optimal transport problem. The computational complexity of the algorithm is reduced to O ( N ) O(N) from O ( N 3 ) O(N^{3}) of the traditional method. Combining with the adjoint-state method, this framework provides a new computational approach for solving the challenging electrical impedance tomography problem. Numerical examples are presented to illustrate the effectiveness of our method.

Refinements of asymptotics at zero of Brownian self-intersection local times

In this paper, we establish some estimates related to the Gaussian densities and to Hermite polynomials in order to obtain an almost sure estimate for each term of the Itô-Wiener expansion of the self-intersection local times of the Brownian motion. In dimension [Formula: see text] the self-intersection local times of the Brownian motion can be considered as a family of measures on the classical Wiener space. We provide some asymptotics relative to these measures. Finally, we try to estimate the quadratic Wasserstein distance between these measures and the Wiener measure.

Classical uncertainty relations and entropy production in non-equilibrium statistical mechanics

We analyze Fürth’s 1933 classical uncertainty relations in the modern language of stochastic differential equations. Our interest is motivated by their application to non-equilibrium classical statistical mechanics. We show that Fürth’s uncertainty relations are a property inherent in martingales within the framework of a diffusion process. This result implies a lower bound on the fluctuations in current velocities of entropic quantifiers associated with transitions in stochastic thermodynamics. In cases of particular interest, we recover a well-known inequality for optimal mass transport relating the mean kinetic energy of the current velocity and the squared quadratic Wasserstein distance between the probability distributions of the entropy. We take advantage in particular of an unpublished suggestion by Krzysztof Gawȩdzki to derive a lower bound to the entropy production by a transition described by a Langevin–Kramers process in terms of the squared quadratic Wasserstein distance between the initial and final states of the transition. Finally, we illustrate how Fürth’s relations admit a straightforward extension to piecewise deterministic processes. We show that the results presented in this paper pertain to the characteristics exhibited by general Markov processes.

Full waveform inversion guided wave tomography with a recurrent neural network

Corrosion quantitative detection of plate or plate-like structure materials is crucial in industrial Non-Destructive Testing (NDT) for determining their remaining life. For doing that, a novel ultrasonic guided wave tomography method, incorporating recurrent neural network (RNN) into full waveform inversion (FWI) called as RNN-FWI, is proposed in this paper. When the wave equation of an acoustic model is solved by a forward model with the cyclic calculation units of an RNN, it is shown that the inversion of the forward model can be obtained iteratively by minimizing a waveform misfit function of quadratic Wasserstein distance between the modeled and measured data. It is also demonstrated that the gradient of the objective function can be obtained by automatic differentiation while the parameters of the waveform velocity model are updated by the adaptive momentum estimation algorithm (Adam). The U-Net deep image prior (DIP) is used as the velocity model regularization in each iteration. The final thickness maps of the plate or plate-like structure materials shown can be archived by the dispersion characteristics of guided waves. Both the numerical simulation and experimental results show that the proposed RNN-FWI tomography method performs better than the conventional time-domain FWI in terms of convergence rate, initial model requirement, and robustness.

Full‐waveform inversion for reverse vertical seismic profiling data based on a variant of the optimal transport theory

ABSTRACTFull‐waveform inversion has become a useful tool to construct the details of the geological structure. However, due to the strong nonlinearity of the inverse problem and inaccuracy of the initial model, full‐waveform inversion often faces severe cycle‐skipping problems and thus may exhibit poor inversion results. The optimal transport‐based full‐waveform inversion can mitigate the cycle‐skipping problem. For that, transformations of oscillatory seismic data to distributions need to satisfy positive and mass conservation conditions. In this work, we introduce a new data normalization method based on the Sigmoid function. The convexity of the objective function is improved compared to the traditional L2 norm by setting appropriate parameters. Combined with the adaptive quadratic Wasserstein distance, our approach presents strong adaptability to a poor initial model and can reduce cycle skipping. Numerical examples are provided to validate the method in a reverse vertical seismic profiling geometry, and the inverted models obtained by different normalization methods are also compared.

Wasserstein Convergence for Empirical Measures of Subordinated Diffusions on Riemannian Manifolds

Let M be a connected compact Riemannian manifold possibly with a boundary ∂M, let V ∈ C2(M) such that μ(dx) := eV (x)dx is a probability measure, where dx is the volume measure, and let L =Δ +∇V. As a continuation to Wang and Zhu (2019) where convergence in the quadratic Wasserstein distance \(\mathbb {W}_{2}\) is studied for the empirical measures of the L-diffusion process (with reflecting boundary if ∂M≠∅), this paper presents the exact convergence rate for the subordinated process. In particular, letting \((\mu _{t}^{\alpha})_{t>0}\) (α ∈ (0,1)) be the empirical measures of the Markov process generated by Lα := −(−L)α, when ∂M is empty or convex we have $ \lim _{t\to \infty } \big \{t \mathbb {E}^{x} [\mathbb {W}_{2}(\mu _{t}^{\alpha},\mu )^{2}]\big \}= \sum \limits_{i=1}^{\infty }\frac {2}{\lambda _{i}^{1+\alpha }}\ \text { uniformly\ in\ } x\in M,$ where \(\mathbb {E}^{x}\) is the expectation for the process starting at point x, {λi}i≥ 1 are non-trivial (Neumann) eigenvalues of − L. In general, $\mathbb {E}^{x} [\mathbb {W}_{2}(\mu _{t}^{\alpha},\mu )^{2}] \begin {cases} \asymp t^{-1}, &\text {if}\ d<2(1+\alpha ),\\ \asymp t^{-\frac {2}{d-2\alpha }}, &\text {if} \ d>2(1+\alpha ),\\ \preceq t^{-1}\log (1+t), &\text {if} \ d=2(1+\alpha ), \text {i.e.}\ \alpha =\frac {1}{2}, d=3\end {cases}$ holds uniformly in x ∈ M, where in the last case \(\mathbb {E}^{x} [\mathbb {W}_{1}(\mu _{t}^{\alpha},\mu )^{2}]\succeq t^{-1}\log (1+t)\) holds for \(M=\mathbb {T}^{3}\) and V = 0.

Spatiotemporal imaging with diffeomorphic optimal transportation

Motivated by the image reconstruction in spatiotemporal imaging, we introduce a concept named diffeomorphic optimal transportation (DOT), which combines the Wasserstein distance with Benamou–Brenier formula in optimal transportation and the flow of diffeomorphisms in large deformation diffeomorphic metric mapping. Using DOT, we propose a new variational model for joint image reconstruction and motion estimation, which is suitable for spatiotemporal imaging involving mass-preserving large diffeomorphic deformations. We further get its equivalent PDE-constrained optimal control formulation. The proposed model is easy-to-implement and solved by an alternating gradient descent algorithm, which is compared against existing alternatives theoretically and numerically. The performance is validated by several numerical experiments in spatiotemporal tomography, where the projection data is time-dependent sparse and/or high-noise. Moreover, we present several extensions based on DOT. Under appropriate conditions, the proposed algorithm can be adapted as a new algorithm to solve the models using quadratic Wasserstein distance.

Differential semblance optimisation based on the adaptive quadratic Wasserstein distance

Abstract As the robustness for the wave equation-based inversion methods, wave equation migration velocity analysis (WEMVA) is stable for overcoming the multipathing problem and has become popular in recent years. As a rapidly developed method, differential semblance optimisation (DSO) is convenient to implement and can automatically detect the moveout existing in common image gathers (CIGs). However, by implementing in the image domain with the target of minimising moveouts and improving coherence of the CIGs, the DSO method often suffers from imaging artefacts caused by uneven illumination and irregular observation geometry, which may produce poor velocity updates with artefact contamination. To deal with this issue, in this paper, by introducing Wiener-like filters, we modify the conventional image matching-based objective function to a new one by introducing the quadratic Wasserstein metric technique. The new misfit function measures the distance of two distributions obtained by the convolutional filters and target functions. With the new misfit function, the adjoint sources and the corresponding gradients are improved. We apply the new method to two numerical examples and one field dataset. The corresponding results indicate that the new method is robust to compensate low frequency components of velocity models.

The quadratic Wasserstein metric for inverse data matching

This work characterizes, analytically and numerically, two major effects of the quadratic Wasserstein (W2) distance as the measure of data discrepancy in computational solutions of inverse problems. First, we show, in the infinite-dimensional setup, that the W2 distance has a smoothing effect on the inversion process, making it robust against high-frequency noise in the data but leading to a reduced resolution for the reconstructed objects at a given noise level. Second, we demonstrate that, for some finite-dimensional problems, the W2 distance leads to optimization problems that have better convexity than the classical L2 and distances, making it a more preferred distance to use when solving such inverse matching problems.

Squared quadratic Wasserstein distance: optimal couplings and Lions differentiability

In this paper, we remark that any optimal coupling for the quadratic Wasserstein distanceW22(μ,ν) between two probability measuresμandνwith finite second order moments on ℝdis the composition of a martingale coupling with an optimal transport map 𝛵. We check the existence of an optimal coupling in which this map gives the unique optimal coupling betweenμand 𝛵#μ. Next, we give a direct proof thatσ↦W22(σ,ν) is differentiable atμin the Lions (Cours au Collège de France. 2008) sense iff there is a unique optimal coupling betweenμandνand this coupling is given by a map. It was known combining results by Ambrosio, Gigli and Savaré (Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005) and Ambrosio and Gangbo (Comm. Pure Appl. Math., 61:18–53, 2008) that, under the latter condition, geometric differentiability holds. Moreover, the two notions of differentiability are equivalent according to the recent paper of Gangbo and Tudorascu (J. Math. Pures Appl. 125:119–174, 2019). Besides, we give a self-contained probabilistic proof that mere Fréchet differentiability of a law invariant functionFonL2(Ω, ℙ; ℝd) is enough for the Fréchet differential atXto be a measurable function ofX.

Harmonic mappings valued in the Wasserstein space

We propose a definition of the Dirichlet energy (which is roughly speaking the integral of the square of the gradient) for mappings μ:Ω→(P(D),W2) defined over a subset Ω of Rp and valued in the space P(D) of probability measures on a compact convex subset D of Rq endowed with the quadratic Wasserstein distance. Our definition relies on a straightforward generalization of the Benamou-Brenier formula (already introduced by Brenier) but is also equivalent to the definition of Korevaar, Schoen and Jost as limit of approximate Dirichlet energies, and to the definition of Reshetnyak of Sobolev spaces valued in metric spaces.We study harmonic mappings, i.e. minimizers of the Dirichlet energy provided that the values on the boundary ∂Ω are fixed. The notion of constant-speed geodesics in the Wasserstein space is recovered by taking for Ω a segment of R. As the Wasserstein space (P(D),W2) is positively curved in the sense of Alexandrov we cannot apply the theory of Korevaar, Schoen and Jost and we use instead arguments based on optimal transport. We manage to get existence of harmonic mappings provided that the boundary values are Lipschitz on ∂Ω, uniqueness is an open question.If Ω is a segment of R, it is known that a curve valued in the Wasserstein space P(D) can be seen as a superposition of curves valued in D. We show that it is no longer the case in higher dimensions: a generic mapping Ω→P(D) cannot be represented as the superposition of mappings Ω→D.We are able to show the validity of a maximum principle: the composition F∘μ of a function F:P(D)→R convex along generalized geodesics and a harmonic mapping μ:Ω→P(D) is a subharmonic real-valued function.We also study the special case where we restrict ourselves to a given family of elliptically contoured distributions (a finite-dimensional and geodesically convex submanifold of (P(D),W2) which generalizes the case of Gaussian measures) and show that it boils down to harmonic mappings valued in the Riemannian manifold of symmetric matrices endowed with the distance coming from optimal transport.

Seismic inversion and the data normalization for optimal transport

Full waveform inversion (FWI) has recently become a favorite technique for the inverse problem of finding properties in the earth from measurements of vibrations of seismic waves on the surface. Mathematically, FWI is PDE constrained optimization where model parameters in a wave equation are adjusted such that the misfit between the computed and the measured dataset is minimized. In a sequence of papers, we have shown that the quadratic Wasserstein distance from optimal transport is to prefer as misfit functional over the standard $L^2$ norm. Datasets need however first to be normalized since seismic signals do not satisfy the requirements of optimal transport. There has been a puzzling contradiction in the results. Normalization methods that satisfy theorems pointing to ideal properties for FWI have not performed well in practical computations, and other scaling methods that do not satisfy these theorems have performed much better in practice. In this paper, we will shed light on this issue and resolve this contradiction.

Weighted Ultrafast Diffusion Equations: From Well-Posedness to Long-Time Behaviour

In this paper we devote our attention to a class of weighted ultrafast diffusion equations arising from the problem of quantisation for probability measures. These equations have a natural gradient flow structure in the space of probability measures endowed with the quadratic Wasserstein distance. Exploiting this structure, in particular through the so-called JKO scheme, we introduce a notion of weak solutions, prove existence, uniqueness, BV and H1 estimates, L1 weighted contractivity, Harnack inequalities, and exponential convergence to a steady state.

Comparison between W2 distance and Ḣ−1 norm, and Localization of Wasserstein distance

It is well known that the quadratic Wasserstein distance W2(⋅, ⋅) is formally equivalent, for infinitesimally small perturbations, to some weighted H−1 homogeneous Sobolev norm. In this article I show that this equivalence can be integrated to get non-asymptotic comparison results between these distances. Then I give an application of these results to prove that the W2 distance exhibits some localization phenomenon: if μ and ν are measures on ℝn and ϕ: ℝn → ℝ+ is some bump function with compact support, then under mild hypotheses, you can bound above the Wasserstein distance between ϕ ⋅ μ and ϕ ⋅ ν by an explicit multiple of W2(μ, ν).

Modified Massive Arratia Flow and Wasserstein Diffusion

Extending previous work by the first author we present a variant of the Arratia flow, which consists of a collection of coalescing Brownian motions starting from every point of the unit interval. The important new feature of the model is that individual particles carry mass that aggregates upon coalescence and that scales the diffusivity of each particle in an inverse proportional way. In this work we relate the induced measure‐valued process to the Wasserstein diffusion of von Renesse and Sturm. First, we present the process as a martingale solution to an SPDE similar to that of von Renesse and Sturm. Second, as our main result we show a Varadhan formula 42 for short times that is governed by the quadratic Wasserstein distance. © 2018 Wiley Periodicals, Inc.

Wasserstein Continuity of Entropy and Outer Bounds for Interference Channels

It is shown that under suitable regularity conditions, differential entropy is a Lipschitz functional on the space of distributions on $n$-dimensional Euclidean space with respect to the quadratic Wasserstein distance. Under similar conditions, (discrete) Shannon entropy is shown to be Lipschitz continuous in distributions over the product space with respect to Ornstein's $\bar d$-distance (Wasserstein distance corresponding to the Hamming distance). These results together with Talagrand's and Marton's transportation-information inequalities allow one to replace the unknown multi-user interference with its i.i.d. approximations. As an application, a new outer bound for the two-user Gaussian interference channel is proved, which, in particular, settles the "missing corner point" problem of Costa (1985).