This work focuses on finding optimal dividend payment and capital injection policies to maximize the present value of the difference between the cumulative dividend payment and the possible capital injections with delays. Starting from the classical Cramér–Lundberg process, using the dynamic programming approach, the value function obeys a quasi-variational inequality. With delays in capital injections, the company will be exposed to the risk of financial ruin during the delay period. In addition, the optimal dividend payment and capital injection strategy should balance the expected cost of the possible capital injections and the time value of the delay period. In this paper, the closed-form solution of the value function and the corresponding optimal policies are obtained. Some limiting cases are also discussed. A numerical example is presented to illustrate properties of the solution. Some economic insights are also given.

It has been known for a long time that the equivariant $2+1$ wave map into the $2$-sphere blows up if the initial data are chosen appropriately. Here, we present numerical evidence for the stability of the blow-up phenomenon under explicit violations of equivariance.

There is much interest within the mathematical biology and statistical physics community in converting stochastic agent-based models for random walkers into a partial differential equation description for the average agent density. Here a collection of noninteracting biased random walkers on a one-dimensional lattice is considered. The usual master equation approach requires that two continuum limits, involving three parameters, namely step length, time step and the random walk bias, approach zero in a specific way. We are interested in the case where the two limits are not consistent. New results are obtained using a Fokker–Planck equation and the results are highly dependent on the simulation update schemes. The theoretical results are confirmed with examples. These findings provide insight into the importance of updating schemes to an accurate macroscopic description of stochastic local movement rules in agent-based models when the lattice spacing represents a physical object such as cell diameter.

A stochastic algorithm for bound-constrained global optimization is described. The method can be applied to objective functions that are nonsmooth or even discontinuous. The algorithm forms a partition on the search region using classification and regression trees (CART), which defines a region where the objective function is relatively low. Further points are drawn directly from the low region before a new partition is formed. Alternating between partition and sampling phases provides an effective method for nonsmooth global optimization. The sequence of iterates generated by the algorithm is shown to converge to an essential global minimizer with probability one under mild conditions. Nonprobabilistic results are also given when random sampling is replaced with points taken from the Halton sequence. Numerical results are presented for both smooth and nonsmooth problems and show that the method is effective and competitive in practice.

The adaptive stabilization is investigated for a class of coupled PDE-ODE systems withmultiple uncertainties. The presence of the multiple uncertainties and the interactionbetween the sub-systems makes the systems to be considered more general andrepresentative, and moreover it may result in the ineffectiveness of the conventionalmethods on this topic. Motivated by the existing literature, an infinite-dimensionalbacksteppping transformation with new kernel functions is first introduced to change theoriginal system into a target system, from which the control design and performanceanalysis of the original system will become quite convenient. Then, by certaintyequivalence principle and Lyapunov method, an adaptive stabilizing controller issuccessfully constructed, which guarantees that all the closed-loop system states arebounded while the original system states converging to zero. A simulation example isprovided to validate the proposed method.

In this article, given a reference feasible trajectory of an optimal control problem, we say that the quadratic growth property for bounded strong solutions holds if the cost function of the problem has a quadratic growth over the set of feasible trajectories with a bounded control and with a state variable sufficiently close to the reference state variable. Our sufficient second-order optimality conditions in Pontryagin form ensure this property and ensure a fortiori that the reference trajectory is a bounded strong solution. Our proof relies on a decomposition principle, which is a particular second-order expansion of the Lagrangian of the problem.

We consider the variational problem

        inf{αλ1(Ω) + βλ2(Ω) + (1 − α − β)λ3(Ω) | Ω open in ℝn, |Ω| ≤ 1},

for α, β ∈ [0, 1], α + β ≤ 1, whereλk(Ω)is the ktheigenvalue of the Dirichlet Laplacian acting in L2(Ω) and|Ω| isthe Lebesgue measure of Ω. We investigate for which values of α, β every minimiser isconnected.

In this paper, we employ the reduced basis method as a surrogate model for the solutionof linear-quadratic optimal control problems governed by parametrized elliptic partialdifferential equations. We present a posteriori error estimation and dualprocedures that provide rigorous bounds for the error in several quantities of interest:the optimal control, the cost functional, and general linear output functionals of thecontrol, state, and adjoint variables. We show that, based on the assumption of affineparameter dependence, the reduced order optimal control problem and the proposed boundscan be efficiently evaluated in an offline-online computational procedure. Numericalresults are presented to confirm the validity of our approach.

Regularity results for minimal configurations of variational problems involving both bulkand surface energies and subject to a volume constraint are established. The bulk energiesare convex functions with p-power growth, but are otherwise not subjected toany further structure conditions. For a minimal configuration (u,E), Hölder continuity ofthe function u is proved as well as partial regularity of theboundary of the minimal set E. Moreover, full regularity of the boundary of theminimal set is obtained under suitable closeness assumptions on the eigenvalues of thebulk energies.

In this paper we are concerned with questions of multiplicity and concentration behaviorof positive solutions of the elliptic problem

$$ (P_{\var})\hspace*{4cm} \left\{ \begin{array}{rcl} \mathcal{L}_{\var}u=f(u) \ \ \mbox{in} \ \ \R^{3},\\[1.5mm] u>0 \ \ \mbox{in} \ \ \R^{3},\\[1.5mm] u \in H^{1}(\R^3), \end{array} \right. $$$\mathrm{\left(}{\mathit{P}}_{\mathit{\epsilon }}\mathrm{\right)}\left\{\begin{array}{c}\\ {\mathrm{ℒ}}_{\mathit{\epsilon }}\mathit{u}\mathrm{=}\mathit{f}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathrm{in}{\mathrm{IR}}^{\mathrm{3}}\mathit{,}\\ \mathit{u}\mathit{>}\mathrm{0}\mathrm{in}{\mathrm{IR}}^{\mathrm{3}}\mathit{,}\\ \mathit{u}\mathrm{\in }{\mathit{H}}^{\mathrm{1}}\mathrm{\left(}{\mathrm{IR}}^{\mathrm{3}}\mathrm{\right)}\mathit{,}\end{array}\right.$

where ε is a small positiveparameter, f : ℝ → ℝ is a continuous function, $$ \mathcal{L}_{\var} $$${\mathrm{ℒ}}_{\mathit{\epsilon }}$ is a nonlocaloperator defined by

$$ \mathcal{L}_{\var}u=M\left(\dis\frac{1}{\var}\int_{\R^{3}}|\nabla u|^{2}+\frac{1}{\var^{3}}\dis\int_{\R^{3}}V(x)u^{2}\right)\left[-\var^{2}\Delta u + V(x)u \right], $$${\mathrm{ℒ}}_{\mathit{\epsilon }}\mathit{u}\mathrm{=}\mathit{M}\left(\frac{\mathrm{1}}{\mathit{\epsilon }}{\mathrm{\int }}_{{\mathrm{IR}}^{\mathrm{3}}}\mathrm{|}\mathrm{\nabla }\mathit{u}{\mathrm{|}}^{\mathrm{2}}\mathrm{+}\frac{\mathrm{1}}{{\mathit{\epsilon }}^{\mathrm{3}}}{\mathrm{\int }}_{{\mathrm{IR}}^{\mathrm{3}}}\mathit{V}\mathrm{(}\mathit{x}\mathrm{)}{\mathit{u}}^{\mathrm{2}}\right)\mathrm{[}\mathrm{-}{\mathit{\epsilon }}^{\mathrm{2}}\mathrm{\Delta }\mathit{u}\mathrm{+}\mathit{V}\mathrm{(}\mathit{x}\mathrm{)}{\mathit{u}}^{\mathrm{]}}\mathit{,}$

M : IR+ → IR+ andV : IR3 → IR are continuous functions whichverify some hypotheses.

The problem of distributing two conducting materials with a prescribed volume ratio in aball so as to minimize the first eigenvalue of an elliptic operator with Dirichletconditions is considered in two and three dimensions. The gap ε between the twoconductivities is assumed to be small (low contrast regime). The main result of the paperis to show, using asymptotic expansions with respect to ε and to small geometricperturbations of the optimal shape, that the global minimum of the first eigenvalue in lowcontrast regime is either a centered ball or the union of a centered ball and of acentered ring touching the boundary, depending on the prescribed volume ratio between thetwo materials.

We introduce a new variational method for the numerical homogenization of divergence formelliptic, parabolic and hyperbolic equations with arbitrary rough (L∞)coefficients. Our method does not rely on concepts of ergodicity or scale-separation buton compactness properties of the solution space and a new variational approach tohomogenization. The approximation space is generated by an interpolation basis (overscattered points forming a mesh of resolution H) minimizing the L2 norm of thesource terms; its (pre-)computation involves minimizing 𝒪(H−d)quadratic (cell) problems on (super-)localized sub-domains of size 𝒪(H ln(1/H)).The resulting localized linear systems remain sparse and banded. The resultinginterpolation basis functions are biharmonic for d ≤ 3, and polyharmonic ford ≥ 4, forthe operator −div(a∇·) and can be seen as a generalization ofpolyharmonic splines to differential operators with arbitrary rough coefficients. Theaccuracy of the method (𝒪(H)in energy norm and independent from aspect ratios of the mesh formed by the scatteredpoints) is established via the introduction of a new class ofhigher-order Poincaré inequalities. The method bypasses (pre-)computations on the fulldomain and naturally generalizes to time dependent problems, it also provides a naturalsolution to the inverse problem of recovering the solution of a divergence form ellipticequation from a finite number of point measurements.

A multiscale spectral generalized finite element method (MS-GFEM) is presented for the solution of large two and three dimensional stress analysis problems inside heterogeneous media. It can be employed to solve problems too large to be solved directly with FE techniques and is designed for implementation on massively parallel machines. The method is multiscale in nature and uses an optimal family of spectrally defined local basis functions over a coarse grid. It is proved that the method has an exponential rate of convergence. To fix ideas we describe its implementation for a two dimensional plane strain problem inside a fiber reinforced composite. Here fibers are separated by a minimum distance however no special assumption on the fiber configuration such as periodicity or ergodicity is made. The implementation of MS-GFEM delivers the discrete solution operator using the same order of operations as the number of fibers inside the computational domain. This implementation is optimal in that the number of operations for solution is of the same order as the input data for the problem. The size of the MS-GFEM matrix used to represent the discrete inverse operator is controlled by the scale of the coarse grid and the convergence rate of the spectral basis and can be of order far less than the number of fibers. This strategy is general and can be applied to the solution of very large FE systems associated with the discrete solution of elliptic PDE.

In this paper, multiscale finite element methods (MsFEMs) and domain decomposition techniques are developed for a class of nonlinear elliptic problems with high-contrast coefficients. In the process, existing work on linear problems [Y. Efendiev, J. Galvis, R. Lazarov, S. Margenov and J. Ren, Robust two-level domain decomposition preconditioners for high-contrast anisotropic flows in multiscale media. Submitted.; Y. Efendiev, J. Galvis and X. Wu, J. Comput. Phys. 230 (2011) 937–955; J. Galvis and Y. Efendiev, SIAM Multiscale Model. Simul. 8 (2010) 1461–1483.] is extended to treat a class of nonlinear elliptic operators. The proposed method requires the solutions of (small dimension and local) nonlinear eigenvalue problems in order to systematically enrich the coarse solution space. Convergence of the method is shown to relate to the dimension of the coarse space (due to the enrichment procedure) as well as the coarse mesh size. In addition, it is shown that the coarse mesh spaces can be effectively used in two-level domain decomposition preconditioners. A number of numerical results are presented to complement the analysis.

This paper analyzes the random fluctuations obtained by a heterogeneous multi-scalefirst-order finite element method applied to solve elliptic equations with a randompotential. Several multi-scale numerical algorithms have been shown to correctly capturethe homogenized limit of solutions of elliptic equations with coefficients modeled asstationary and ergodic random fields. Because theoretical results are available in thecontinuum setting for such equations, we consider here the case of a second-order ellipticequations with random potential in two dimensions of space. We show that the randomfluctuations of such solutions are correctly estimated by the heterogeneous multi-scalealgorithm when appropriate fine-scale problems are solved on subsets that cover the wholecomputational domain. However, when the fine-scale problems are solved over patches thatdo not cover the entire domain, the random fluctuations may or may not be estimatedaccurately. In the case of random potentials with short-range interactions, the varianceof the random fluctuations is amplified as the inverse of the fraction of the mediumcovered by the patches. In the case of random potentials with long-range interactions,however, such an amplification does not occur and random fluctuations are correctlycaptured independent of the (macroscopic) size of the patches. These results areconsistent with those obtained in [9] for moregeneral equations in the one-dimensional setting and provide indications on the loss inaccuracy that results from using coarser, and hence computationally less intensive,algorithms.

We propose a multiscale model reduction method for partial differential equations. Themain purpose of this method is to derive an effective equation for multiscale problemswithout scale separation. An essential ingredient of our method is to decompose theharmonic coordinates into a smooth part and a highly oscillatory part so that the smoothpart is invertible and the highly oscillatory part is small. Such a decomposition plays akey role in our construction of the effective equation. We show that the solution to theeffective equation is in H2, and can be approximated by a regularcoarse mesh. When the multiscale problem has scale separation and a periodic structure,our method recovers the traditional homogenized equation. Furthermore, we provide erroranalysis for our method and show that the solution to the effective equation is close tothe original multiscale solution in the H1 norm. Numerical results are presentedto demonstrate the accuracy and robustness of the proposed method for several multiscaleproblems without scale separation, including a problem with a high contrastcoefficient.

We present a parallel preconditioning method for the iterative solution of thetime-harmonic elastic wave equation which makes use of higher-order spectral elements toreduce pollution error. In particular, the method leverages perfectly matched layerboundary conditions to efficiently approximate the Schur complement matrices of a blockLDLTfactorization. Both sequential and parallel versions of the algorithm are discussed andresults for large-scale problems from exploration geophysics are presented.

This work is concerned with the asymptotic analysis of a time-splitting scheme for theSchrödinger equation with a random potential having weak amplitude, fast oscillations intime and space, and long-range correlations. Such a problem arises for instance in thesimulation of waves propagating in random media in the paraxial approximation. Thehigh-frequency limit of the Schrödinger equation leads to different regimes depending onthe distance of propagation, the oscillation pattern of the initial condition, and thestatistical properties of the random medium. We show that the splitting scheme capturesthese regimes in a statistical sense for a time stepsize independent of the frequency.

We study the approximation of harmonic functions by means of harmonic polynomials intwo-dimensional, bounded, star-shaped domains. Assuming that the functions possessanalytic extensions to a δ-neighbourhood of the domain, we proveexponential convergence of the approximation error with respect to the degree of theapproximating harmonic polynomial. All the constants appearing in the bounds are explicitand depend only on the shape-regularity of the domain and on δ. We applythe obtained estimates to show exponential convergence with rate O(exp(-b√N)), N being the number of degrees offreedom and b > 0, of a hp-dGFEM discretisation ofthe Laplace equation based on piecewise harmonic polynomials. This result is animprovement over the classical rate O(exp(-b 3√N)), and is due to the use of harmonic polynomialspaces, as opposed to complete polynomial spaces.