Andrei Sobolevski

A reconstruction of the initial conditions of the Universe by optimal mass transportation

Reconstructing the density fluctuations in the early Universe that evolved into the distribution of galaxies we see today is a challenge to modern cosmology. An accurate reconstruction would allow us to test cosmological models by simulating the evolution starting from the reconstructed primordial state and comparing it to observations. Several reconstruction techniques have been proposed, but they all suffer from lack of uniqueness because the velocities needed to produce a unique reconstruction usually are not known. Here we show that reconstruction can be reduced to a well-determined problem of optimization, and present a specific algorithm that provides excellent agreement when tested against data from N-body simulations. By applying our algorithm to the redshift surveys now under way, we will be able to recover reliably the properties of the primeval fluctuation field of the local Universe, and to determine accurately the peculiar velocities (deviations from the Hubble expansion) and the true positions of many more galaxies than is feasible by any other method.

Fast Transport Optimization for Monge Costs on the Circle

Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on the real line, and suppose the cost of matching two points satisfies the Monge condition. We introduce a notion of locally optimal transport plan, motivated by the weak KAM (Aubry-Mather) theory, and show that all locally optimal transport plans are conjugate to shifts and that the cost of a locally optimal transport plan is a convex function of a shift parameter. This theory is applied to a transportation problem arising in image processing: for two sets of point masses on the circle, both of which have the same total mass, find an optimal transport plan with respect to a given cost function satisfying the Monge condition. In the circular case the sorting strategy fails to provide a unique candidate solution and a naive approach requires a quadratic number of operations. For the case of

Universal Algorithms, Mathematics of Semirings and Parallel Computations

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Particle dynamics inside shocks in Hamilton-Jacobi equations.

real-valued point masses we present an O(N |log epsilon|) algorithm that approximates the optimal cost within epsilon; when all masses are integer multiples of 1/M, the algorithm gives an exact solution in O(N log M) operations.

Local Matching Indicators for Transport Problems with Concave Costs

This isaut]Grigory L. Litvinovaut]Victor P. Maslovaut]Anatoly Ya. Rodionovaut]Andrei N. Sobolevskii a survey paper on applications of mathematics of semirings to numerical analysis and computing. Concepts of universal algorithm and generic program are discussed. Relations between these concepts and mathematics of semirings are examined. A very brief introduction to mathematics of semirings (including idempotent and tropical mathematics) is presented. Concrete applications to optimization problems, idempotent linear algebra and interval analysis are indicated. It is known that some nonlinear problems (and especially optimization problems) become linear over appropriate semirings with idempotent addition (the so-called idempotent superposition principle). This linearity over semirings is convenient for parallel computations.

On Dynamics of Lagrangian Trajectories for Hamilton–Jacobi Equations

The characteristic curves of a Hamilton–Jacobi equation can be seen as action-minimizing trajectories of fluid particles. For non-smooth ‘viscosity’ solutions, which give rise to discontinuous velocity fields, this description is usually pursued only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we show that, for any convex Hamiltonian, there exists a uniquely defined canonical global non-smooth coalescing flow that extends particle trajectories and determines the dynamics inside shocks. We also provide a variational description of the corresponding effective velocity field inside shocks, and discuss the relation to the ‘dissipative anomaly’ in the limit of vanishing viscosity.

Minimum—weight perfect matching for nonintrinsic distances on the line

In this paper, we introduce a class of local indicators that enable us to compute efficiently optimal transport plans associated with arbitrary weighted distributions of

Scientific Activity of Alfred Yarbus: The Stages of Research Work, Senior and Younger Colleagues, Conditions of Investigations.

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Planar diagrams from optimization for concave potentials.

demands and

Droplet condensation and isoperimetric towers

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