We study bounding approximations for a multistage stochastic program with expected value constraints. Two simpler approximate stochastic programs, which provide upper and lower bounds on the original problem, are obtained by replacing the original stochastic data process by finitely supported approximate processes. We model the original and approximate processes as dependent random vectors on a joint probability space. This probabilistic coupling allows us to transform the optimal solution of the upper bounding problem to a near-optimal decision rule for the original problem. Unlike the scenario tree based solutions of the bounding problems, the resulting decision rule is implementable in all decision stages, i.e., there is no need for dynamic reoptimization during the planning period. Our approach is illustrated with a mean-risk portfolio optimization model.
In this paper, the algorithm for the discretization of 3D domains into tetrahedral boundary conforming meshes is presented. The algorithm is based on the Delaunay triangulation with special point ordering. The conformity of the resulting mesh with the initial triangulation of the domain boundary is ensured a priori thus the boundary recovery postprocessing step is eliminated. The constrained Delaunay triangulation of the boundary points is obtained using modified Watson's point insertion algorithm. The actual appearance of boundary faces in the final triangulation is achieved by the form of an oriented graph, of the violation of the empty-sphere property of all boundary faces. The cycle dependencies (closed loops in the graph) are eliminated by using the nodal perturbations, by classification of some of the violations as safe and (as the last resort) by forming a new tetrahedron using the advancing front technique. Once all the cyclic dependencies are eliminated, the point insertion process controlled by the dependency graph is started and the constrained Delaunay triangulation of the boundary points is built. In the next phase, additional points are inserted in the interior of the domain, while preserving the boundary constraints, to make the elements of appropriate size with aspect ratio close to one. The resulting mesh is then subjected to optimization in terms of the combination of Laplacian smoothing and topological transformations, in order to remove the potential slivers and to improve the overall mesh quality. and Obsahuje seznam literatury
We solve the Dirichlet problem for line integrals of holomorphic functions in the unit ball: For a function $u$ which is lower semi-continuous on $\partial \mathbb{B}^{n}$ we give necessary and sufficient conditions in order that there exists a holomorphic function $f\in \mathbb{O}(\mathbb{B}^{n})$ such that \[ u(z)=\int _{|\lambda |<1}\left|f(\lambda z)\right|^{2}\mathrm{d}{\mathfrak L}^{2}(\lambda ).\].
We solve the following Dirichlet problem on the bounded balanced domain $\Omega $ with some additional properties: For $p>0$ and a positive lower semi-continuous function $u$ on $\partial \Omega $ with $u(z)=u(\lambda z)$ for $|\lambda |=1$, $z\in \partial \Omega $ we construct a holomorphic function $f\in \Bbb O(\Omega )$ such that $u(z)=\int _{\Bbb Dz}|f|^pd \frak L_{\Bbb Dz}^2$ for $z\in \partial \Omega $, where $\Bbb D=\{\lambda \in \Bbb C\:|\lambda |<1\}$.
This paper deals with the three-point boundary value problem for the nonlinear singularly perturbed second-order systems. Especially, we focus on an analysis of the solutions in the right endpoint of considered interval from an appearance of the boundary layer point of view. We use the method of lower and upper solutions combined with analysis of the integral equation associated with the class of nonlinear systems considered here.
The paper deals with the multivalued boundary value problem x ′ ∈ A(t,x)x + F(t,x) for a.a. t ∈ [a, b], Mx(a) + Nx(b) = 0, in a separable, reflexive Banach space E. The nonlinearity F is weakly upper semicontinuous in x. We prove the existence of global solutions in the Sobolev space W1,p([a, b], E) with 1 < p < ∞ endowed with the weak topology. We consider the case of multiple solutions of the associated homogeneous linearized problem. An example completes the discussion.
In this paper we investigate the problem of existence and asymptotic behavior of solutions for the nonlinear boundary value problem εy ′′ + ky = f(t, y), t ∈ ha,bi, k < 0, 0 < ε ≪ 1 satisfying three point boundary conditions. Our analysis relies on the method of lower and upper solutions and delicate estimations.
We study the method of layer potentials for manifolds with boundary and cylindrical ends. The fact that the boundary is non-compact prevents us from using the standard characterization of Fredholm or compact pseudo-differential operators between Sobolev spaces, as, for example, in the works of Fabes-Jodeit-Lewis and Kral-Wedland . We first study the layer potentials depending on a parameter on compact manifolds. This then yields the invertibility of the relevant boundary integral operators in the global, non-compact setting. As an application, we prove a well-posedness result for the non-homogeneous Dirichlet problem on manifolds with boundary and cylindrical ends. We also prove the existence of the Dirichlet-to-Neumann map, which we show to be a pseudodifferential operator in the calculus of pseudodifferential operators that are “almost translation invariant at infinity”.
We use the method of quasilinearization to boundary value problems of ordinary differential equations showing that the corresponding monotone iterations converge to the unique solution of our problem and this convergence is quadratic.