Approximation of Parametric Derivatives by the Empirical Interpolation Method

We introduce a general a priori convergence result for the approximation of parametric derivatives of parametrized functions. We consider the best approximations to parametric derivatives in a sequence of approximation spaces generated by a general approximation scheme, and we show that these approximations are convergent provided that the best approximation to the function itself is convergent. We also provide estimates for the convergence rates. We present numerical results with spaces generated by a particular approximation scheme—the Empirical Interpolation Method—to confirm the validity of the general theory.     

On the application of mosaic-skeleton approximations of matrices for the acceleration of computations in the vortex method for the three-dimensional Euler equations

We study the possibility of using fast matrix multiplication methods for the approximation of the velocity field when solving the system of differential equations describing the vorticity transport in an ideal incompressible fluid in Lagrangian coordinates. We suggest a numerical scheme that permits effectively using the fast matrix multiplication (the method of mosaic-skeleton approximations). We show that the functions used for the computation of the velocity field and moving grids appearing in the solution of the problem permit one to use the above-mentioned method. We prove the convergence of the resulting numerical solution to the exact solution with regard of the error contributed by the use of the algorithm for approximate fast multiplication of matrices by vectors.     

Stabilized finite element schemes with LBB-stable elements for incompressible flows

We study stabilized FE approximations of SUPG type to the incompressible Navier–Stokes problem. Revisiting the analysis for the linearized model, we show that for conforming LBB-stable elements the design of the stabilization parameters for many practical flows differs from that commonly suggested in literature and initially designed for the case of equal-order approximation. Then we analyze a reduced SUPG scheme often used in practice for LBB-stable elements. To provide the reduced scheme with appropriate stability estimates we introduce a modified LBB condition which is proved for a family of FE approximations. The analysis is given for the linearized equations. Numerical experiments for some linear and nonlinear benchmark problems support the theoretical results.     

An Adaptive Hybrid Spectral Method for Stochastic Helmholtz Problems

The implementation of an adaptive hybrid spectral method for Helmholtz equations with random parameters is addressed in this work. New error indicators for generalized polynomial chaos for stochastic approximations and spectral element methods for physical approximations are developed, and systematic adaptive strategies are proposed associated with these error indicators. Numerical results show that these error indicators provide effective estimates for the approximation errors, and the overall adaptive procedure results in efficient approximation method for the stochastic Helmholtz equations.     

Discrete Approximations of Controlled Stochastic Systems with Memory: A Survey

This survey article considers discrete approximations of an optimal control problem in which the controlled state equation is described by a general class of stochastic functional differential equations with a bounded memory. Specifically, three different approximation methods, namely (i) semidiscretization scheme; (ii) Markov chain approximation; and (iii) finite difference approximation, are investigated. The convergence results as well as error estimates are established for each of the approximation methods.     

Approximations and Randomization to Boost CSP Techniques

In recent years we have seen an increasing interest in combining constraint satisfaction problem (CSP) formulations and linear programming (LP) based techniques for solving hard computational problems. While considerable progress has been made in the integration of these techniques for solving problems that exhibit a mixture of linear and combinatorial constraints, it has been surprisingly difficult to successfully integrate LP-based and CSP-based methods in a purely combinatorial setting. Our approach draws on recent results on approximation algorithms based on LP relaxations and randomized rounding techniques, with theoretical guarantees, as well on results that provide evidence that the runtime distributions of combinatorial search methods are often heavy-tailed. We propose a complete randomized backtrack search method for combinatorial problems that tightly couples CSP propagation techniques with randomized LP-based approximations. We present experimental results that show that our hybrid CSP/LP backtrack search method outperforms the pure CSP and pure LP strategies on instances of a hard combinatorial problem.     

Uniform analysis of a stabilized hybrid finite element method for Reissner-Mindlin plates

This paper presents a low order stabilized hybrid quadrilateral finite element method for ReissnerMindlin plates based on Hellinger-Reissner variational principle,which includes variables of displacements,shear stresses and bending moments.The approach uses continuous piecewise isoparametric bilinear interpolations for the approximations of the transverse displacement and rotation.The stabilization achieved by adding a stabilization term of least-squares to the original hybrid scheme,allows independent approximations of the stresses and moments.The stress approximation adopts a piecewise independent 4-parameter mode satisfying an accuracy-enhanced condition.The approximation of moments employs a piecewise-independent 5-parameter mode.This method can be viewed as a stabilized version of the hybrid finite element scheme proposed in [Carstensen C,Xie X,Yu G,et al.A priori and a posteriori analysis for a locking-free low order quadrilateral hybrid finite element for Reissner-Mindlin plates.Comput Methods Appl Mech Engrg,2011,200:1161-1175],where the approximations of stresses and moments are required to satisfy an equilibrium criterion.A priori error analysis shows that the method is uniform with respect to the plate thickness t.Numerical experiments confirm the theoretical results.     

A robust linearization scheme for nonlinear diffusion in type-II superconductors

We study a nonlinear partial differential equation strongly motivated by the modelling of processes in type-II superconductors in a bounded domain along with appropriate boundary condition. We design a robust and efficient linear approximation scheme based on fix-point principle. The convergence of approximations to a weak solution of the original problem is proved. We also present some numerical examples to show the efficiency of the proposed method.     

Modeling Stochastic Dominance as Infinite-Dimensional Constraint Systems via the Strassen Theorem

We use the Strassen theorem to solve stochastic optimization problems with stochastic dominance constraints. First, we show that a dominance-constrained problem on general probability spaces can be expressed as an infinite-dimensional optimization problem with a convenient representation of the dominance constraints provided by the Strassen theorem. This result generalizes earlier work which was limited to finite probability spaces. Second, we derive optimality conditions and a duality theory to gain insight into this optimization problem. Finally, we present a computational scheme for constructing finite approximations along with a convergence rate analysis on the approximation quality.     

A parallelizable dynamic fleet management model with random travel times

In this paper, we present a stochastic model for the dynamic fleet management problem with random travel times. Our approach decomposes the problem into time-staged subproblems by formulating it as a dynamic program and uses approximations of the value function. In order to deal with random travel times, the state variable of our dynamic program includes all individual decisions over a relevant portion of the history. We show how to approximate the value function in a tractable manner under this new high-dimensional state variable.Under our approximation scheme, the subproblem for each time period decomposes with respect to locations, making our model very appealing for large-scale applications. Numerical work shows that the proposed approach provides high-quality solutions and performs significantly better than standard benchmark methods.     

The effect of transformations on the approximation of univariate (convex) functions with applications to Pareto curves

In the literature, methods for the construction of piecewise linear upper and lower bounds for the approximation of univariate convex functions have been proposed. We study the effect of the use of transformations on the approximation of univariate (convex) functions. In this paper, we show that these transformations can be used to construct upper and lower bounds for nonconvex functions. Moreover, we show that by using such transformations of the input variable or the output variable, we obtain tighter upper and lower bounds for the approximation of convex functions than without these approximations. We show that these transformations can be applied to the approximation of a (convex) Pareto curve that is associated with a (convex) bi-objective optimization problem.     

Risk-averse feasible policies for large-scale multistage stochastic linear programs

We develop tractable semidefinite programming based approximations for distributionally robust individual and joint chance constraints, assuming that only the first- and second-order moments as well as the support of the uncertain parameters are given. It is known that robust chance constraints can be conservatively approximated by Worst-Case Conditional Value-at-Risk (CVaR) constraints. We first prove that this approximation is exact for robust individual chance constraints with concave or (not necessarily concave) quadratic constraint functions, and we demonstrate that the Worst-Case CVaR can be computed efficiently for these classes of constraint functions. Next, we study the Worst-Case CVaR approximation for joint chance constraints. This approximation affords intuitive dual interpretations and is provably tighter than two popular benchmark approximations. The tightness depends on a set of scaling parameters, which can be tuned via a sequential convex optimization algorithm. We show that the approximation becomes essentially exact when the scaling parameters are chosen optimally and that the Worst-Case CVaR can be evaluated efficiently if the scaling parameters are kept constant. We evaluate our joint chance constraint approximation in the context of a dynamic water reservoir control problem and numerically demonstrate its superiority over the two benchmark approximations.     

Two‐grid methods for semilinear interface problems

In this article, we consider two‐grid finite element methods for solving semilinear interface problems in d space dimensions, for d = 2 or d = 3. We consider semilinear problems with discontinuous diffusion coefficients, which includes problems containing subcritical, critical, and supercritical nonlinearities. We establish basic quasioptimal a priori error estimates for Galerkin approximations. We then design a two‐grid algorithm consisting of a coarse grid solver for the original nonlinear problem, and a fine grid solver for a linearized problem. We analyze the quality of approximations generated by the algorithm and show that the coarse grid may be taken to have much larger elements than the fine grid, and yet one can still obtain approximation quality that is asymptotically as good as solving the original nonlinear problem on the fine mesh. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Two issues in using mixtures of polynomials for inference in hybrid Bayesian networks

We discuss two issues in using mixtures of polynomials (MOPs) for inference in hybrid Bayesian networks. MOPs were proposed by Shenoy and West for mitigating the problem of integration in inference in hybrid Bayesian networks. First, in defining MOP for multi-dimensional functions, one requirement is that the pieces where the polynomials are defined are hypercubes. In this paper, we discuss relaxing this condition so that each piece is defined on regions called hyper-rhombuses. This relaxation means that MOPs are closed under transformations required for multi-dimensional linear deterministic conditionals, such as Z = X + Y, etc. Also, this relaxation allows us to construct MOP approximations of the probability density functions (PDFs) of the multi-dimensional conditional linear Gaussian distributions using a MOP approximation of the PDF of the univariate standard normal distribution. Second, Shenoy and West suggest using the Taylor series expansion of differentiable functions for finding MOP approximations of PDFs. In this paper, we describe a new method for finding MOP approximations based on Lagrange interpolating polynomials (LIP) with Chebyshev points. We describe how the LIP method can be used to find efficient MOP approximations of PDFs. We illustrate our methods using conditional linear Gaussian PDFs in one, two, and three dimensions, and conditional log-normal PDFs in one and two dimensions. We compare the efficiencies of the hyper-rhombus condition with the hypercube condition. Also, we compare the LIP method with the Taylor series method.     

Inference of statistical bounds for multistage stochastic programming problems

We discuss in this paper statistical inference of sample average approximations of multistage stochastic programming problems. We show that any random sampling scheme provides a valid statistical lower bound for the optimal (minimum) value of the true problem. However, in order for such lower bound to be consistent one needs to employ the conditional sampling procedure. We also indicate that fixing a feasible first-stage solution and then solving the sampling approximation of the corresponding (T–1)-stage problem, does not give a valid statistical upper bound for the optimal value of the true problem.Supported, in part, by the National Science Foundation under grant DMS-0073770.     

The hypervolume based directed search method for multi-objective optimization problems

We present a new hybrid evolutionary algorithm for the effective hypervolume approximation of the Pareto front of a given differentiable multi-objective optimization problem. Starting point for the local search (LS) mechanism is a new division of the decision space as we will argue that in each of these regions a different LS strategy seems to be most promising. For the LS in two out of the three regions we will utilize and adapt the Directed Search method which is capable of steering the search into any direction given in objective space and which is thus well suited for the problem at hand. We further on integrate the resulting LS mechanism into SMS-EMOA, a state-of-the-art evolutionary algorithm for hypervolume approximations. Finally, we will present some numerical results on several benchmark problems with two and three objectives indicating the strength and competitiveness of the novel hybrid.     

Data-Driven Robust Chance Constrained Problems: A Mixture Model Approach

This paper discusses the mixture distribution-based data-driven robust chance constrained problem. We construct a data-driven mixture distribution-based uncertainty set from the perspective of simultaneously estimating higher-order moments. Then, we derive a reformulation of the data-driven robust chance constrained problem. As the reformulation is not a convex programming problem, we propose new and tight convex approximations based on the piecewise linear approximation method. We establish the theoretical foundation for these approximations. Finally, numerical results show that the proposed approximations are practical and efficient.     

The dynamical behavior of the discontinuous Galerkin method and related difference schemes

We study the dynamical behavior of the discontinuous Galerkin finite element method for initial value problems in ordinary differential equations. We make two different assumptions which guarantee that the continuous problem defines a dissipative dynamical system. We show that, under certain conditions, the discontinuous Galerkin approximation also defines a dissipative dynamical system and we study the approximation properties of the associated discrete dynamical system. We also study the behavior of difference schemes obtained by applying a quadrature formula to the integrals defining the discontinuous Galerkin approximation and construct two kinds of discrete finite element approximations that share the dissipativity properties of the original method.

     

Efficient sample sizes in stochastic nonlinear programming

We consider a class of stochastic nonlinear programs for which an approximation to a locally optimal solution is specified in terms of a fractional reduction of the initial cost error. We show that such an approximate solution can be found by approximately solving a sequence of sample average approximations. The key issue in this approach is the determination of the required sequence of sample average approximations as well as the number of iterations to be carried out on each sample average approximation in this sequence. We show that one can express this requirement as an idealized optimization problem whose cost function is the computing work required to obtain the required error reduction. The specification of this idealized optimization problem requires the exact knowledge of a few problems and algorithm parameters. Since the exact values of these parameters are not known, we use estimates, which can be updated as the computation progresses. We illustrate our approach using two numerical examples from structural engineering design.     

An optimal Gauss–Markov approximation for a process with stochastic drift and applications

We consider a linear stochastic differential equation with stochastic drift. We study the problem of approximating the solution of such equation through an Ornstein–Uhlenbeck type process, by using direct methods of calculus of variations. We show that general power cost functionals satisfy the conditions for existence and uniqueness of the approximation. We provide some examples of general interest and we give bounds on the goodness of the corresponding approximations. Finally, we focus on a model of a neuron embedded in a simple network and we study the approximation of its activity, by exploiting the aforementioned results.