We study the value of European security derivatives in the Black–Scholes model when the underlying asset $\xi $ is approximated by random walks ${\xi }^{(n)} $. We obtain an explicit error formula, up to a term of order $ \mathcal{O} ({n}^{- 3/ 2} )$, which is valid for general approximating schemes and general payoff functions. We show how this error formula can be used to find random walks ${\xi }^{(n)} $ for which option values converge at a speed of $ \mathcal{O} ({n}^{- 3/ 2} )$.

We review the work of the present authors to employ variational calculus to formulate continuous models for the connections between various carbon nanostructures. In formulating such a variational principle, there is some evidence that carbon nanotubes deform as in perfect elasticity, and rather like the elastica, and therefore we seek to minimize the elastic energy. The calculus of variations is utilized to minimize the curvature subject to a length constraint, to obtain an Euler–Lagrange equation, which determines the connection between two carbon nanostructures. Moreover, a numerical solution is proposed to determine the geometric parameters for the connected structures. Throughout this review, we assume that the defects on the nanostructures are axially symmetric and that the into-the-plane curvature is small in comparison to that in the two-dimensional plane, so that the problems can be considered in the two-dimensional plane. Since the curvature can be both positive and negative, depending on the gap between the two nanostructures, two distinct cases are examined, which are subsequently shown to smoothly connect to each other.

The Kohlrausch functions $\exp (- {t}^{\beta } )$, with $\beta \in (0, 1)$, which are important in a wide range of physical, chemical and biological applications, correspond to specific realizations of completely monotone functions. In this paper, using nonuniform grids and midpoint estimates, constructive procedures are formulated and analysed for the Kohlrausch functions. Sharper estimates are discussed to improve the approximation results. Numerical results and representative approximations are presented to illustrate the effectiveness of the proposed method.

A 3D-2D dimension reduction for −Δ1 is obtained. A powerlaw approximation from −Δp asp → 1 in terms of Γ-convergence, duality andasymptotics for least gradient functions has also been provided.

We consider the problem of providing optimal uncertainty quantification (UQ) – and hencerigorous certification – for partially-observed functions. We present a UQ frameworkwithin which the observations may be small or large in number, and need not carryinformation about the probability distribution of the system in operation. The UQobjectives are posed as optimization problems, the solutions of which are optimal boundson the quantities of interest; we consider two typical settings, namely parametersensitivities (McDiarmid diameters) and output deviation (or failure) probabilities. Thesolutions of these optimization problems depend non-trivially (even non-monotonically anddiscontinuously) upon the specified legacy data. Furthermore, the extreme values are oftendetermined by only a few members of the data set; in our principal physically-motivatedexample, the bounds are determined by just 2 out of 32 data points, and the remaindercarry no information and could be neglected without changing the final answer. We proposean analogue of the simplex algorithm from linear programming that uses these observationsto offer efficient and rigorous UQ for high-dimensional systems with high-cardinalitylegacy data. These findings suggest natural methods for selecting optimal (maximallyinformative) next experiments.

We consider the shape optimization problem \hbox{$\min\big\{\E(\Gamma)\ :\ \Gamma\in\A,\ \H^1(\Gamma)=l\\big\},$}min{ℰ(Γ):Γ ∈ 𝒜,ℋ1(Γ) = l}, where ℋ1 is the one-dimensional Hausdorffmeasure and𝒜is an admissible class of one-dimensional setsconnecting some prescribed set of points \hbox{$\D=\{D_1,\dots,D_k\}\subset\R^d$}𝒟 =  { D1,...,Dk }  ⊂ Rd. The cost functional ℰ(Γ) is theDirichlet energy of Γ defined through the Sobolev functions onΓ vanishing on the pointsDi. We analyze the existence of a solutionin both the families of connected sets and of metric graphs. At the end, several explicitexamples are discussed.

This paper is concerned with the internal distributed control problem for the 1DSchrödinger equation,i ut(x,t) = −uxx+α(x) u+m(u) u, that arises in quantum semiconductor models. Here m(u)is a non local Hartree–type nonlinearity stemming from the coupling with the 1D Poissonequation, and α(x) is a regular function with lineargrowth at infinity, including constant electric fields. By means of both the HilbertUniqueness Method and the contraction mapping theorem it is shown that for initial andtarget states belonging to a suitable small neighborhood of the origin, and fordistributed controls supported outside of a fixed compact interval, the model equation iscontrollable. Moreover, it is shown that, for distributed controls with compact support,the exact controllability problem is not possible.

In this paper, we investigate Nash equilibrium payoffs for nonzero-sum stochasticdifferential games with reflection. We obtain an existence theorem and a characterizationtheorem of Nash equilibrium payoffs for nonzero-sum stochastic differential games withnonlinear cost functionals defined by doubly controlled reflected backward stochasticdifferential equations.

In this paper we study the null-controllability of an artificial advection-diffusionsystem in dimension n. Using a spectral method, we prove that the controlcost goes to zero exponentially when the viscosity vanishes and the control time is largeenough. On the other hand, we prove that the control cost tends to infinity exponentiallywhen the viscosity vanishes and the control time is small enough.

In this work, depending on the relation between the Deborah, the Reynolds and the aspectratio numbers, we formally derived shallow-water type systems starting from a micro-macrodescription for non-Newtonian fluids in a thin domain governed by an elastic dumbbell typemodel with a slip boundary condition at the bottom. The result has been announced by theauthors in [G. Narbona-Reina, D. Bresch, Numer. Math. and Advanced Appl.Springer Verlag (2010)] and in the present paper, we provide a self-containeddescription, complete formal derivations and various numerical computations. Inparticular, we extend to FENE type systems the derivation of shallow-water models forNewtonian fluids that we can find for instance in [J.-F. Gerbeau, B. Perthame,Discrete Contin. Dyn. Syst. (2001)] which assume an appropriaterelation between the Reynolds number and the aspect ratio with slip boundary condition atthe bottom. Under a radial hypothesis at the leading order, for small Deborah number, wefind an interesting formulation where polymeric effect changes the drag term in the secondorder shallow-water formulation (obtained by J.-F. Gerbeau, B. Perthame). We also discussintermediate Deborah number with a fixed Reynolds number where a strong coupling is foundthrough a nonlinear time-dependent Fokker–Planck equation. This generalizes, at a formallevel, the derivation in [L. Chupin, Meth. Appl. Anal. (2009)] includingnon-linear effects (shallow-water framework).

The Coupled Cluster (CC) method is a widely used and highly successful high precisionmethod for the solution of the stationary electronic Schrödingerequation, with its practical convergence properties being similar to that of acorresponding Galerkin (CI) scheme. This behaviour has for the discrete CC method beenanalyzed with respect to the discrete Galerkin solution (the “full-CI-limit”) in[Schneider, 2009]. Recently, we globalized the CC formulation to the full continuousspace, giving a root equation for an infinite dimensional, nonlinear Coupled Clusteroperator that is equivalent the full electronic Schrödinger equation [Rohwedder, 2011]. Inthis paper, we combine both approaches to prove existence and uniqueness results,quasi-optimality estimates and energy estimates for the CC method with respect to thesolution of the full, original Schrödinger equation. The main property used is a localstrong monotonicity result for the Coupled Cluster function, and we give twocharacterizations for situations in which this property holds.

We propose a derivation of a nonequilibrium Langevin dynamics for a large particleimmersed in a background flow field. A single large particle is placed in an ideal gasheat bath composed of point particles that are distributed consistently with thebackground flow field and that interact with the large particle through elasticcollisions. In the limit of small bath atom mass, the large particle dynamics converges inlaw to a stochastic dynamics. This derivation follows the ideas of [P. Calderoni, D. Dürrand S. Kusuoka, J. Stat. Phys. 55 (1989) 649–693. D. Dürr,S. Goldstein and J. Lebowitz, Z. Wahrscheinlichkeit 62(1983) 427–448. D. Dürr, S. Goldstein and J.L. Lebowitz. Comm. Math. Phys.78 (1981) 507–530.] and provides extensions to handle the nonzerobackground flow. The derived nonequilibrium Langevin dynamics is similar to the dynamicsin [M. McPhie, P. Daivis, I. Snook, J. Ennis and D. Evans, Phys. A299 (2001) 412–426]. Some numerical experiments illustrate the useof the obtained dynamic to simulate homogeneous liquid materials under shear flow.

In this paper, we propose a numerical method to solve stochastic elliptic interfaceproblems with random interfaces. Shape calculus is first employed to derive theshape-Taylor expansion in the framework of the asymptotic perturbation approach. Given themean field and the two-point correlation function of the random interface, we can thusquantify the mean field and the variance of the random solution in terms of certain ordersof the perturbation amplitude by solving a deterministic elliptic interface problem andits tensorized counterpart with respect to the reference interface. Error estimates arederived for the interface-resolved finite element approximation in both, the physical andthe stochastic dimension. In particular, a fast finite difference scheme is proposed tocompute the variance of random solutions by using a low-rank approximation based on thepivoted Cholesky decomposition. Numerical experiments are presented to validate andquantify the method.

Discrete-velocity approximations represent a popular way for computing the Boltzmanncollision operator. The direct numerical evaluation of such methods involve a prohibitivecost, typically O(N2d + 1)where d is the dimension of the velocity space. In this paper, followingthe ideas introduced in [C. Mouhot and L. Pareschi, C. R. Acad. Sci. Paris Sér. IMath. 339 (2004) 71–76, C. Mouhot and L. Pareschi, Math.Comput. 75 (2006) 1833–1852], we derive fast summation techniquesfor the evaluation of discrete-velocity schemes which permits to reduce the computationalcost from O(N2d + 1) to O(N̅dNd log2N), N̅ ≪ N, with almost no loss of accuracy.

We provide a deterministic-control-based interpretation for a broad class of fullynonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in asmooth domain. We construct families of two-person games depending on a small parameterε which extend those proposed by Kohn and Serfaty [21]. These new games treat a Neumann boundary conditionby introducing some specific rules near the boundary. We show that the value functionconverges, in the viscosity sense, to the solution of the PDE as ε tendsto zero. Moreover, our construction allows us to treat both the oblique and the mixed typeDirichlet–Neumann boundary conditions.

We study the Fokker–Planck equation as the many-particle limit of a stochastic particlesystem on one hand and as a Wasserstein gradient flow on the other. We write thepath-space rate functional, which characterises the large deviations from the expectedtrajectories, in such a way that the free energy appears explicitly. Next we use thisformulation via the contraction principle to prove that the discrete time rate functionalis asymptotically equivalent in the Gamma-convergence sense to the functional derived fromthe Wasserstein gradient discretization scheme.

This paper deals with the numerical computation of boundary null controls for the 1D waveequation with a potential. The goal is to compute approximations of controls that drivethe solution from a prescribed initial state to zero at a large enough controllabilitytime. We do not apply in this work the usual duality arguments but explore instead adirect approach in the framework of global Carleman estimates. More precisely, we considerthe control that minimizes over the class of admissible null controls a functionalinvolving weighted integrals of the state and the control. The optimality conditions showthat both the optimal control and the associated state are expressed in terms of a newvariable, the solution of a fourth-order elliptic problem defined in the space-timedomain. We first prove that, for some specific weights determined by the global Carlemaninequalities for the wave equation, this problem is well-posed. Then, in the framework ofthe finite element method, we introduce a family of finite-dimensional approximate controlproblems and we prove a strong convergence result. Numerical experiments confirm theanalysis. We complete our study with several comments.