A COMPACT UPWIND SECOND ORDER SCHEME FOR THE EIKONAL EQUATION

We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on： 1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities; 2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil. We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities.     

Second-Order Models for Optimal Transport and Cubic Splines on the Wasserstein Space

Foundations of Computational Mathematics - On the space of probability densities, we extend the Wasserstein geodesics to the case of higher-order interpolation such as cubic spline interpolation....     

Source point discovery through high frequency asymptotic time reversal

Given local frequency domain wave data, the Numerical Micro-Local Analysis (NMLA) method (Benamou et al., 2004) [5] and its recent improved version (Benamou et al., 2011) [4] gives a pointwise numerical approximation of the number of rays, their slowness vectors and corresponding wavefront curvatures. With time domain wave data and assuming the source wavelet is given, the method also estimates the travel-time. The paper provides a non technical presentation of the improved NMLA algorithm and presents a numerical application which can be interpreted as a high frequency asymptotic version of the classical time reversal method (Borcea et al., 2003) [7]. A detailed technical presentation of the algorithm is available in Benamou et al. (2011) [4] and more numerical experiments can be found in Collino and Marmorat (2011) [15].     

A semi-lagrangian numerical method for geometric optics type problems

Linear multistep methods (LMMs) are written as irreducible general linear methods (GLMs). A-stable LMMs are shown to be algebraically stable GLMs for strictly positive definite G-matrices. Optimal order error bounds, independent of stiffness, are derived for A-stable methods, without considering one-leg methods (OLMs). As a GLM, the OLM is shown to be the transpose of the LMM. For A-stable methods, the LMM G-matrix is the inverse of the OLM G-matrix. Examples of G-symplectic LMMs are given. AMS subject classification (2000) 65L20     

Complexes cationiques η-1′,2,3-[(methylene-2γ-butyrolactone)-yl-3]-(η-cyclopentadienyl)carbonylnitrosyl molybdene: Synthese et etudes des additions nucleophiles stereoselectives

The decomposition of hydroxyethyl-, ethoxyethyl- and phenoxyethyl(aquo)cobaloximes in sulfuric acid/water mixtures to produce ethylene has been studied spectrophotometrically and manometrically. Definitive kinetic evidence for formation of an intermediate which is common to all three starting complexes and accumulates at acidities greater than 7.3 M H₂SO₄ has been obtained. A complete rate law which accounts for all of the ionizations of starting materials and intermediate has been derived and fit to the rate data. The rate-determining step for product formation is decomposition of the intermediate at all acidities: the intermediate accumulates in strong acid because of a shift in the equilibrium for its formation due to the reduced activity of water in strongly acidic media. Activation parameters for the decomposition of the intermediate, which may be formulated as an ethylene-cobaloxime(III) π-complex or as a σ-bonded ethyl carbonium ion, have been obtained. ¹H and ¹³C NMR observations of the intermediate and its deuterated analog (from 1,1,2,2-tetradeuterio-2-hydroxyethyl(aquo)cobaloxime) have led to the conclusion that it is probably a σ-bonded ethyl carbonium ion which may be stabilized by σ-π hyperconjugation.     

Mixed L 2-Wasserstein Optimal Mapping Between Prescribed Density Functions

A time-dependent minimization problem for the computation of a mixed L ²-Wasserstein distance between two prescribed density functions is introduced in the spirit of Ref. 1 for the classical Wasserstein distance. The optimum of the cost function corresponds to an optimal mapping between prescribed initial and final densities. We enforce the final density conditions through a penalization term added to our cost function. A conjugate gradient method is used to solve this relaxed problem. We obtain an algorithm which computes an interpolated L ²-Wasserstein distance between two densities and the corresponding optimal mapping.     

Domain Decomposition,Optimal Control of Systems Governed by Partial Differential Equations,and Synthesis of Feedback Laws

We present an iterative domain decomposition method for the optimal control of systems governed by linear partial differential equations. The equations can be of elliptic, parabolic, or hyperbolic type. The space region supporting the partial differential equations is decomposed and the original global optimal control problem is reduced to a sequence of similar local optimal control problems set on the subdomains. The local problems communicate through transmission conditions, which take the form of carefully chosen boundary conditions on the interfaces between the subdomains. This domain decomposition method can be combined with any suitable numerical procedure to solve the local optimal control problems. We remark that it offers a good potential for using feedback laws (synthesis) in the case of time-dependent partial differential equations. A test problem for the wave equation is solved using this combination of synthesis and domain decomposition methods. Numerical results are presented and discussed. Details on discretization and implementation can be found in Ref. 1.     

A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

Summary. The Monge-Kantorovich mass transfer problem [31] is reset in a fluid mechanics framework and numerically solved by an augmented Lagrangian method. Received August 30, 1998 / Published online September 24, 1999