The ground-state energy and properties of any many-electron atom ormolecule may be rigorously computed by variationally computing thetwo-electron reduced density matrix rather than the many-electronwavefunction. While early attempts fifty years ago to compute theground-state 2-RDM directly were stymied because the 2-RDM must beconstrained to represent an N-electron wavefunction, recentadvances in theory and optimization have made direct computation ofthe 2-RDM possible. The constraints in the variational calculationof the 2-RDM require a special optimization known as a semidefiniteprogramming. Development of first-order semidefinite programmingfor the 2-RDM method has reduced the computational costs of thecalculation by orders of magnitude [Mazziotti, Phys. Rev. Lett.93 (2004) 213001]. The variational 2-RDM approach is effective atcapturing multi-reference correlation effects that are especiallyimportant at non-equilibrium molecular geometries. Recent work on2-RDM methods will be reviewed and illustrated with particularemphasis on the importance of advances in large-scale semidefiniteprogramming.


This paper derives upper and lower bounds for the $\ell^p$ -conditionnumber of the stiffness matrix resulting from the finite elementapproximation of a linear, abstract model problem. Sharp estimates interms of the meshsize h are obtained. The theoretical results areapplied to finite element approximations of elliptic PDE's invariational and in mixed form, and to first-order PDE's approximatedusing the Galerkin–Least Squares technique or bymeans of a non-standard Galerkin technique in L 1(Ω). Numerical simulations are presented to illustrate thetheoretical results.

In the article an optimal control problem subject to a stationary variational inequalityis investigated. The optimal control problem is complemented with pointwise controlconstraints. The convergence of a smoothing scheme is analyzed. There, the variationalinequality is replaced by a semilinear elliptic equation. It is shown that solutions ofthe regularized optimal control problem converge to solutions of the original one. Passingto the limit in the optimality system of the regularized problem allows to proveC-stationarity of local solutions of the original problem. Moreover, convergence rateswith respect to the regularization parameter for the error in the control are obtained,which turn out to be sharp. These rates coincide with rates obtained by numericalexperiments, which are included in the paper.

We recently derived a very general representation formulafor the boundary voltage perturbations caused by internalconductivity inhomogeneities of low volume fraction (cf. Capdeboscq and Vogelius (2003)). In this paper we show how thisrepresentation formula may be used to obtain veryaccurate estimates for the size of the inhomogeneitiesin terms of multiple boundary measurements. As demonstrated by our computational experiments, these estimates are significantly better than previously known (single measurement) estimates,even for moderate volume fractions.

We derive a posteriori error estimates for singularlyperturbed reaction–diffusion problems which yield a guaranteedupper bound on the discretization error and are fully and easilycomputable. Moreover, they are also locally efficient and robust inthe sense that they represent local lower bounds for the actualerror, up to a generic constant independent in particular of thereaction coefficient. We present our results in the framework ofthe vertex-centered finite volume method but their nature isgeneral for any conforming method, like the piecewise linear finiteelement one. Our estimates are based on a H(div)-conformingreconstruction of the diffusive flux in the lowest-orderRaviart–Thomas–Nédélec space linked with mesh dual to the originalsimplicial one, previously introduced by the last author in thepure diffusion case. They also rely on elaborated Poincaré,Friedrichs, and trace inequalities-based auxiliary estimatesdesigned to cope optimally with the reaction dominance. In order tobring down the ratio of the estimated and actual overall energyerror as close as possible to the optimal value of one,independently of the size of the reaction coefficient, we finallydevelop the ideas of local minimizations of the estimators by localmodifications of the reconstructed diffusive flux. The numericalexperiments presented confirm the guaranteed upper bound,robustness, and excellent efficiency of the derived estimates.

The first part of this paper reviews the single time scale/multiplelength scale low Mach number asymptotic analysis by Klein (1995, 2004). This theory explicitly reveals the interaction of small scale,quasi-incompressible variable density flows with long wave linearacoustic modes through baroclinic vorticity generation and asymptoticaccumulation of large scale energy fluxes. The theory is motivated byexamples from thermoacoustics and combustion. In an almost obvious way specializations of this theory to a singlespatial scale reproduce automatically the zero Mach number variabledensity flow equations for the small scales, and the linear acousticequations with spatially varying speed of sound for the large scales. Following the same line of thought we show how a large number ofwell-known simplified equations of theoretical meteorology canbe derived in a unified fashion directly from the three-dimensionalcompressible flow equations through systematic (low Mach number) asymptotics. Atmospheric flows are, however, characterized by several singularperturbation parameters that appear in addition to the Mach number,and that are defined independently of any particular length or timescale associated with some specific flow phenomenon. These are theratio of the centripetal acceleration due to the earth's rotation vs.the acceleration of gravity, and the ratio of the sound speed vs. therotational velocity of points on the equator. To systematicallyincorporate these parameters in an asymptotic approach, we couple themwith the square root of the Mach number in a particular distinguished sothat we are left with a single small asymptotic expansion parameter,ε. Of course, more familiar parameters, such as the Rossby andFroude numbers may then be expressed in terms of ε as well. Next we consider a very general asymptotic ansatz involvingmultiple horizontal and vertical as well as multiple time scales.Various restrictions of the general ansatz to only one horizontal, onevertical, and one time scale lead directly to the family of simplifiedmodel equations mentioned above. Of course, the main purpose of the general multiple scales ansatz isto provide the means to derive true multiscale models which describeinteractions between the various phenomena described by the members ofthe simplified model family. In this context we will summarize a recentsystematic development of multiscale models for the tropics (with Majda).

In this article, we analyze the stability of various numerical schemes for differential models of viscoelastic fluids. More precisely, we consider the prototypical Oldroyd-B model, for which a free energy dissipation holds, and we show under which assumptions such a dissipation is also satisfied for the numerical scheme. Among the numerical schemes we analyze, we consider some discretizations based on the log-formulation of the Oldroyd-B system proposed by Fattal and Kupferman in [J. Non-Newtonian Fluid Mech.123 (2004) 281–285], for which solutions in some benchmark problems have been obtained beyond the limiting Weissenberg numbers for the standard scheme (see [Hulsen et al.J. Non-Newtonian Fluid Mech. 127 (2005) 27–39]). Our analysis gives some tracks to understand these numerical observations.


In a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations, the construction of lower bounds for the coercivity and inf-supstability constants is essential. In [Huynh et al., C. R. Acad.Sci. Paris Ser. I Math.345 (2007) 473–478], the authors presented an efficientmethod, compatible with an off-line/on-line strategy, where the on-line computation is reduced tominimizing a linear functional under a few linear constraints. These constraints depend on nested sets of parameters obtained iteratively using a greedy algorithm. We improve here this method so that it becomes more efficient and robust due to two related properties: (i) the lower bound isobtained by a monotonic process with respect to the size of the nested sets; (ii) less eigen-problems need to be solved. This improved evaluation of the inf-sup constant is then used to consider a reduced basis approximation of a parameter dependent electromagnetic cavity problem both for the greedy construction of the elements of the basis and the subsequent validation of the reduced basis approximation. The problem we consider has resonance features for some choices of the parameters that are well captured by the methodology.


We propose a model for a medical device, called a stent, designed for the treatment of cerebral aneurysms. The stent consists of a grid, immersed in the blood flow and located at the inlet of the aneurysm. It aims at promoting a clot within the aneurysm. The blood flow is modelled by the incompressible Navier-Stokes equations and the stent by a dissipative surface term. We propose a stabilized finite element method for this model and we analyse its convergence in the case of the Stokes equations. We present numerical results for academical test cases, and on a realistic aneurysm obtained from medical imaging.

This paper is concerned with the coupling of two models for the propagation of particles in scattering media. The first model is a linear transport equation of Boltzmann type posed in the phase space (position and velocity). It accurately describes the physics but is very expensive to solve. The second model is a diffusion equation posed in the physical space. It is only valid in areas of high scattering, weak absorption, and smooth physical coefficients, but its numerical solution is much cheaper than that of transport. We are interested in the case when the domain is diffusive everywhere except in some small areas, for instance non-scattering or oscillatory inclusions. We present a natural coupling of the two models that accounts for both the diffusive and non-diffusive regions. The interface separating the models is chosen so that the diffusive regime holds in its vicinity to avoid the calculation of boundary or interface layers. The coupled problem is analyzed theoretically and numerically. To simplify the presentation, the transport equation is written in the even parity form. Applications include, for instance, the treatment of clear or spatially inhomogeneous regions in near-infra-red spectroscopy, which is increasingly being used in medical imaging for monitoring certain properties of human tissues.

We discuss the stability of "critical" or "equilibrium" shapes ofa shape-dependent energy functional. We analyze a problem arising whenlooking at the positivity of the second derivative in order to provethat a critical shape is an optimal shape. Indeed, often whenpositivity -or coercivity- holds, it does for a weaker norm than thenorm for which the functional is twice differentiable and localoptimality cannot be a priori deduced. We solve this problem for aparticular but significant example. We prove "weak-coercivity" ofthe second derivative uniformly in a "strong" neighborhood of theequilibrium shape.


Systems of mixed hyperbolic-elliptic conservation laws can serve as models for the evolution of a liquid-vapor fluid with possible sharp dynamical phase changes. We focus on the equations of ideal hydrodynamics in the isothermal case and introduce a thermodynamically consistent solution of the Riemann problem in one space dimension. This result is the basis for an algorithm of ghost fluid type to solve the sharp-interface model numerically. In particular the approach allows to resolve phase transitions sharply, i.e., without artificial smearing in the physically irrelevant elliptic region. Numerical experiments demonstrate the reliability of the method.

We study the theoretical and numericalcoupling of two hyperbolic systems of conservation laws at a fixed interface. As already proven in the scalar case, the couplingpreserves in a weak sense the continuity of the solution at the interfacewithout imposing the overall conservativity of the coupled model. We develop a detailed analysis of the coupling inthe linear case. In the nonlinear case, we either use a linearized approach or a coupling method based on the solution of a Riemann problem. We discuss both approaches in the case of the coupling of two fluid models at a material contact discontinuity, the models being the usual gas dynamics equations with different equations ofstate. We also study the coupling of two-temperature plasma fluid models and illustrate the approach by numericalsimulations.

We study a finite volume method, used to approximate the solution of the linear two dimensional convection diffusion equation, with mixed Dirichlet and Neumann boundary conditions, on Cartesian meshes refined by an automatic technique (which leads to meshes with hanging nodes). We propose an analysis through a discrete variational approach, in a discrete H 1 finite volume space. We actually prove the convergence of the scheme in a discrete H 1 norm, with an error estimate of order O(h) (on meshes of size h).

In this article we develop a posteriori error estimates for second orderlinear elliptic problems with point sources in two- and three-dimensional domains. Weprove a global upper bound and a local lower bound for the error measured in a weightedSobolev space. The weight considered is a (positive) power of the distance to the supportof the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theoryhinges on local approximation properties of either Clément or Scott–Zhang interpolationoperators, without need of modifications, and makes use of weighted estimates forfractional integrals and maximal functions. Numerical experiments with an adaptivealgorithm yield optimal meshes and very good effectivity indices.

Much of uncertainty quantification to date has focused on determining the effect of variables modeled probabilistically, and with a known distribution, on some physical or engineering system. We develop methods to obtain information on the system when the distributions of some variables are known exactly, others are known only approximately, and perhaps others are not modeled as random variables at all.The main tool used is the duality between risk-sensitive integrals and relative entropy, and we obtain explicit bounds on standard performance measures (variances, exceedance probabilities) over families of distributions whose distance from a nominal distribution is measured by relative entropy. The evaluation of the risk-sensitive expectations is based on polynomial chaos expansions, which help keep the computational aspects tractable.

A conceptual numerical strategy for rate-independent processes in the energetic formulation is proposed and its convergence is proved under various rather mild data qualifications. The novelty is that we obtain convergence ofsubsequences of space-time discretizations even in case where the limitproblem does not have a unique solution and we need noadditional assumptions on higher regularity of the limit solution.The variety of general perspectives thus obtained is illustrated on several specific examples: plasticity with isotropic hardening, damage, debonding, magnetostriction, and two models of martensitic transformation in shape-memory alloys.

This paper describes the extension of arecently developed numerical solver for the Landau-LifshitzNavier-Stokes (LLNS) equations to binary mixtures in threedimensions. The LLNS equations incorporate thermal fluctuations intomacroscopic hydrodynamics by using white-noise fluxes. Thesestochastic PDEs are more complicated in three dimensions due to thetensorial form of the correlations for the stochastic fluxes and inmixtures due to couplings of energy and concentration fluxes (e.g.,Soret effect). We present various numerical tests of systems in andout of equilibrium, including time-dependent systems, anddemonstrate good agreement with theoretical results and molecularsimulation.

We present a heterogeneous finite element method for the solution of a high-dimensionalpopulation balance equation, which depends both the physical and the internal propertycoordinates. The proposed scheme tackles the two main difficulties in the finite elementsolution of population balance equation: (i) spatial discretization with the standardfinite elements, when the dimension of the equation is more than three, (ii) spuriousoscillations in the solution induced by standard Galerkin approximation due to pureadvection in the internal property coordinates. The key idea is to split thehigh-dimensional population balance equation into two low-dimensional equations, anddiscretize the low-dimensional equations separately. In the proposed splitting scheme, theshape of the physical domain can be arbitrary, and different discretizations can beapplied to the low-dimensional equations. In particular, we discretize the physical andinternal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG)finite elements, respectively. The stability and error estimates of the Galerkin/SUPGfinite element discretization of the population balance equation are derived. It is shownthat a slightly more regularity, i.e.the mixed partial derivatives of the solution has to be bounded, is necessary for theoptimal order of convergence. Numerical results are presented to support the analysis.

We construct a Roe-type numerical scheme for approximating the solutionsof a drift-flux two-phase flow model. The model incorporates a set ofhighly complex closure laws, and the fluxes are generally not algebraic functions of the conserved variables. Hence, the classical approach of constructing a Roe solver by means of parameter vectors is unfeasible.Alternative approaches for analytically constructing the Roe solver are discussed, and a formulation of the Roe solver valid for general closure laws is derived. In particular, a fully analytical Roe matrix is obtained for the special case of the Zuber–Findlay law describing bubbly flows.First and second-order accurate versions of the scheme are demonstrated bynumerical examples.