In this paper, we study one kind of stochastic recursive optimal control problem for thesystems described by stochastic differential equations with delay (SDDE). In ourframework, not only the dynamics of the systems but also the recursive utility depend onthe past path segment of the state process in a general form. We give the dynamicprogramming principle for this kind of optimal control problems and show that the valuefunction is the viscosity solution of the corresponding infinite dimensionalHamilton-Jacobi-Bellman partial differential equation.

We consider a nonlinear elliptic equation of the formdiv [a(∇u)] + F[u] = 0on a domain Ω, subject to a Dirichlet boundary conditiontru = φ. We do not assume that the higher order terma satisfies growth conditions from above. We prove the existence ofcontinuous solutions either when Ω is convex and φ satisfies a one-sidedbounded slope condition, or when a is radial:\hbox{$a(\xi)=\fr{l(|\xi|)}{|\xi|} \xi$}$\mathit{a}\mathrm{\left(}\mathit{\xi }\mathrm{\right)}\mathrm{=}\frac{\mathit{l}\mathrm{\left(}\mathrm{\right|}\mathit{\xi }\mathrm{\left|}\mathrm{\right)}}{\mathrm{\left|}\mathit{\xi }\mathrm{\right|}}\mathit{\xi }$ for some increasingl:ℝ+ → ℝ+.

We examine shape optimization problems in the context of inexact sequential quadraticprogramming. Inexactness is a consequence of using adaptive finite element methods (AFEM)to approximate the state and adjoint equations (via the dual weightedresidual method), update the boundary, and compute the geometric functional. We present anovel algorithm that equidistributes the errors due to shape optimization anddiscretization, thereby leading to coarse resolution in the early stages and fineresolution upon convergence, and thus optimizing the computational effort. We discuss theability of the algorithm to detect whether or not geometric singularities such as cornersare genuine to the problem or simply due to lack of resolution – a new paradigm inadaptivity.

In this paper we study the compact and convex setsK ⊆ Ω ⊆ ℝ2 that minimize

\begin{equation*} \int_{\Omega} \dist(\x,K) \,{\rm d}\x +\lambda_1 {\rm Vol}(K)+\lambda_2 {\rm Per}(K) \end{equation*}${\mathrm{\int }}_{\mathit{\Omega }}\mathrm{dist}\mathrm{\left(}{x}\mathit{,K}\mathrm{\right)}\mathrm{d}{x}\mathrm{+}{\mathit{\lambda }}_{\mathrm{1}}\mathrm{Vol}\mathrm{\left(}\mathit{K}\mathrm{\right)}\mathrm{+}{\mathit{\lambda }}_{\mathrm{2}}\mathrm{Per}\mathrm{\left(}\mathit{K}\mathrm{\right)}$

Traffic flow is modeled by a conservation law describing the density of cars. It isassumed that each driver chooses his own departure time in order to minimize the sum of adeparture and an arrival cost. There are N groups of drivers, Thei-th group consists of κidrivers, sharing the same departure and arrival costsϕi(t),ψi(t).For any given population sizesκ1,...,κn,we prove the existence of a Nash equilibrium solution, where no driver can lower his owntotal cost by choosing a different departure time. The possible non-uniqueness, and acharacterization of this Nash equilibrium solution, are also discussed.

For robust discretizations of the Navier-Stokes equations with small viscosity, standardGalerkin schemes have to be augmented by stabilization terms due to the indefiniteconvective terms and due to a possible lost of a discrete inf-sup condition. For optimalcontrol problems for fluids such stabilization have in general an undesired effect in thesense that optimization and discretization do not commute. This is the case for thecombination of streamline upwind Petrov-Galerkin (SUPG) and pressure stabilizedPetrov-Galerkin (PSPG). In this work we study the effect of different stabilized finiteelement methods to distributed control problems governed by singular perturbed Oseenequations. In particular, we address the question whether a possible commutation error inoptimal control problems lead to a decline of convergence order. Therefore, we givea priori estimates for SUPG/PSPG. In a numerical study for a flow withboundary layers, we illustrate to which extend the commutation error affects theaccuracy.

We study the initial value problem for the drift-diffusion model arising in semiconductordevice simulation and plasma physics. We show that the corresponding stationary problem inthe whole space ℝn admits a unique stationary solution in ageneral situation. Moreover, it is proved that when n ≥ 3, a uniquesolution to the initial value problem exists globally in time and converges to thecorresponding stationary solution as time tends to infinity, provided that the amplitudeof the stationary solution and the initial perturbation are suitably small. Also, we showthe sharp decay estimate for the perturbation. The stability proof is based on the timeweighted Lp energy method.

In this paper, we study the multiplicity of solutions for a class of noncooperativep-Laplacian operator elliptic system. Under suitable assumptions, weobtain a sequence of solutions by using the limit index theory.

This paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacsoperators. We establish such an estimate for the parabolic Cauchy problem in the wholespace  [0, +∞) × ℝn and, under some periodicity and eitherellipticity or controllability assumptions, we deduce a similar estimate for the ergodicconstant associated to the operator. An interesting byproduct of the latter result will bethe local uniform convergence for some classes of singular perturbation problems.

We develop a well-posedness theory for second order systems in bounded domains whereboundary phenomena like glancing and surface waves play an important role. Attempts havepreviously been made to write a second order system consisting of nequations as a larger first order system. Unfortunately, the resulting first order systemconsists, in general, of more than 2n equations which leads to manycomplications, such as side conditions which must be satisfied by the solution of thelarger first order system. Here we will use the theory of pseudo-differential operatorscombined with mode analysis. There are many desirable properties of this approach: (1) thereduction to first order systems of pseudo-differential equations poses no difficulty andalways gives a system of 2n equations. (2) We can localize the problem,i.e., it is only necessary to study the Cauchy problem and halfplaneproblems with constant coefficients. (3) The class of problems we can treat is much largerthan previous approaches based on “integration by parts”. (4) The relation betweenboundary conditions and boundary phenomena becomes transparent.

The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we prove that three greedy algorithms converge; the last algorithm, based on the use of an a posteriori estimator, is the approach actually employed in the calculations.

We consider high order finite difference approximations to the Helmholtz equation in an exterior domain. We include a simplified absorbing boundary condition to approximate the Sommerfeld radiation condition. This yields a large, but sparse, complex system, which is not self-adjoint and not positive definite. We discretize the equation with a compact fourth or sixth order accurate scheme. We solve this large system of linear equations with a Krylov subspace iterative method. Since the method converges slowly, a preconditioner is introduced, which is a Helmholtz equation but with a modified complex wavenumber. This is discretized by a second or fourth order compact scheme. The system is solved by BICGSTAB with multigrid used for the preconditioner. We study, both by Fourier analysis and computations this preconditioned system especially for the effects of high order discretizations.

A new set of nonlocal boundary conditions is proposed for the higher modes of the 3D inviscid primitive equations. Numerical schemes using the splitting-up method are proposed for these modes. Numerical simulations of the full nonlinear primitive equations are performed on a nested set of domains, and the results are discussed.

We study order-adaptive implementations of Hermite methods for hyperbolic and singularly perturbed parabolic initial value problems. Exploiting the facts that Hermite methods allow the degree of the local polynomial representation to vary arbitrarily from cell to cell and that, for hyperbolic problems, each cell can be evolved independently over a time-step determined only by the cell size, a relatively straightforward method is proposed. Its utility is demonstrated on a number of model problems posed in 1+1 and 2+1 dimensions.

The compatibility of unsynchronized interleaved uniform sampling with Sigma-Deltaanalog-to-digital conversion is investigated. Let f be a bandlimitedsignal that is sampled on a collection of N interleaved grids {kT + Tn} k ∈ Zwith offsets \hbox{$\{T_n\}_{n=1}^N\subset [0,T]$}$\mathrm{\{}{\mathit{T}}_{\mathit{n}}{\mathrm{\}}}_{\mathit{n}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathrm{\subset }\mathrm{[}\mathrm{0}\mathit{,T}\mathrm{]}$. If the offsets Tn arechosen independently and uniformly at random from  [0,T]  and if thesample values of f are quantized with a first order Sigma-Deltaalgorithm, then with high probability the quantization error \hbox{$|f(t) - \widetilde{f}(t)|$}$\mathrm{\left|}\mathit{f}\mathrm{\right(}\mathit{t}\mathrm{)}\mathrm{-}\begin{array}{}{}_{\mathrm{􏽥}}\\ \mathit{f}\end{array}\mathrm{(}\mathit{t}\mathrm{\left)}\mathrm{\right|}$is at most of orderN-1log N.

The purpose of this paper is to apply particle methods to the numerical solution of theEPDiff equation. The weak solutions of EPDiff are contact discontinuities that carrymomentum so that wavefront interactions represent collisions in which momentum isexchanged. This behavior allows for the description of many rich physical applications,but also introduces difficult numerical challenges. We present a particle method for theEPDiff equation that is well-suited for this class of solutions and for simulatingcollisions between wavefronts. Discretization by means of the particle method is shown topreserve the basic Hamiltonian, the weak and variational structure of the originalproblem, and to respect the conservation laws associated with symmetry under the Euclideangroup. Numerical results illustrate that the particle method has superior features in bothone and two dimensions, and can also be effectively implemented when the initial data ofinterest lies on a submanifold.

Implicit sampling is a sampling scheme for particle filters, designed to move particles one-by-one so that they remain in high-probability domains. We present a new derivation of implicit sampling, as well as a new iteration method for solving the resulting algebraic equations.

We design efficient numerical schemes for approximating the MHD equations in multi-dimensions. Numerical approximations must be able to deal with the complex wave structure of the MHD equations and the divergence constraint. We propose schemes based on the genuinely multi-dimensional (GMD) framework of [S. Mishra and E. Tadmor, Commun. Comput. Phys. 9 (2010) 688–710; S. Mishra and E. Tadmor, SIAM J. Numer. Anal. 49 (2011) 1023–1045]. The schemes are formulated in terms of vertex-centered potentials. A suitable choice of the potential results in GMD schemes that preserve a discrete version of divergence. First- and second-order divergence preserving GMD schemes are tested on a series of benchmark numerical experiments. They demonstrate the computational efficiency and robustness of the GMD schemes.