Discretization of interior point methods for state constrained elliptic optimal control problems: optimal error estimates and parameter adjustment

An adjustment scheme for the relaxation parameter of interior point approaches to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced. The method is based on error estimates of an associated finite element discretization of the relaxed problems and optimally selects the relaxation parameter in dependence on the mesh size of discretization. The finite element analysis for the relaxed problems is carried out and a numerical example is presented which confirms our analytical findings.     

A stepsize control algorithm for SDEs with small noise based on stochastic Runge–Kutta Maruyama methods

A variable stepsize control algorithm for solution of stochastic differential equations (SDEs) with a small noise parameter ?? is presented. In order to determine the optimal stepsize for each stage of the algorithm, an estimate of the global error is introduced based on the local error of the Stochastic Runge?CKutta Maruyama (SRKM) methods. Based on the relation of the stepsize and the small noise parameter, the local mean-square stochastic convergence order can be different from stage to stage. Using this relation, a strategy for producing and controlling the stepsize in the numerical integration of SDEs is proposed. Numerical experiments on several standard SDEs with small noise are presented to illustrate the effectiveness of this approach.     

Global error estimates for a uniparametric family of stiffly accurate Runge-Kutta collocation methods on singularly perturbed problems

An error analysis for a newly defined uniparametric family of stiffly accurate Runge-Kutta collocation methods when applied to initial value problems for singularly perturbed differential equations is carried out. The so-called SAFERK methods possess a first internal stage of explicit type and are based on collocation nodes. Sharp convergence results are obtained for these methods through the analysis of a sequence of higher index Differential Algebraic Equations. A numerical test with the Van der Pol oscillator reveals that the proposed error estimates are realistic whenever the stepsize h is large enough compared to the stiffness parameter ε.     

Aggregation error for location models: survey and analysis

Location problems occurring in urban or regional settings may involve many tens of thousands of “demand points,” usually individual private residences. In modeling such problems it is common to aggregate demand points to obtain tractable models. We survey aggregation approaches to a large class of location models, consider and compare various aggregation error measures, identify some effective (and ineffective) aggregation error measures, and discuss some open research areas.     

Minimization and error estimates for a class of the nonlinear Schrödinger eigenvalue problems

It is shown that the nonlinear eigenvalue problem can be transformed into a constrained functional problem. The corresponding minimal function is a weak solution of this nonlinear problem. In this paper, one type of the energy functional for a class of the nonlinear Schrödinger eigenvalue problems is proposed, the existence of the minimizing solution is proved and the error estimate is given out.     

Global optimization of truss topology with discrete bar areas—Part II: Implementation and numerical results

A classical problem within the field of structural optimization is to find the stiffest truss design subject to a given external static load and a bound on the total volume. The design variables describe the cross sectional areas of the bars. This class of problems is well-studied for continuous bar areas. We consider here the difficult situation that the truss must be built from pre-produced bars with given areas. This paper together with Part I proposes an algorithmic framework for the calculation of a global optimizer of the underlying non-convex mixed integer design problem. In this paper we use the theory developed in Part I to design a convergent nonlinear branch-and-bound method tailored to solve large-scale instances of the original discrete problem. The problem formulation and the needed theoretical results from Part I are repeated such that this paper is self-contained. We focus on the implementation details but also establish finite convergence of the branch-and-bound method. The algorithm is based on solving a sequence of continuous non-convex relaxations which can be formulated as quadratic programs according to the theory in Part I. The quadratic programs to be treated within the branch-and-bound search all have the same feasible set and differ from each other only in the objective function. This is one reason for making the resulting branch-and-bound method very efficient. The paper closes with several large-scale numerical examples. These examples are, to the knowledge of the authors, by far the largest discrete topology design problems solved by means of global optimization.     

Classification with Gaussians and convex loss Ⅱ:improving error bounds by noise conditions

We continue our study on classification learning algorithms generated by Tikhonov regularization schemes associated with Gaussian kernels and general convex loss functions. Our main purpose of this paper is to improve error bounds by presenting a new comparison theorem associated with general convex loss functions and Tsybakov noise conditions. Some concrete examples are provided to illustrate the improved learning rates which demonstrate the effect of various loss functions for learning algorithms. In our ...     

Global optimization of truss topology with discrete bar areas—Part I: theory of relaxed problems

This two part paper considers the classical problem of finding a truss design with minimal compliance subject to a given external force and a volume bound. The design variables describe the cross-section areas of the bars. While this problem is well-studied for continuous bar areas, we treat here the case of discrete areas. This problem is of major practical relevance if the truss must be built from pre-produced bars with given areas. As a special case, we consider the design problem for a single bar area, i.e., a 0/1-problem. In contrast to heuristic methods considered in other approaches, Part I of the paper together with Part II present an algorithmic framework for the calculation of a global optimizer of the underlying large-scaled mixed integer design problem. This framework is given by a convergent branch-and-bound algorithm which is based on solving a sequence of nonconvex continuous relaxations. The main issue of the paper and of the approach lies in the fact that the relaxed nonlinear optimization problem can be formulated as a quadratic program (QP). Here the paper generalizes and extends the available theory from the literature. Although the Hessian of this QP is indefinite, it is possible to circumvent the non-convexity and to calculate global optimizers. Moreover, the QPs to be treated in the branch-and-bound search tree differ from each other just in the objective function. In Part I we give an introduction to the problem and collect all theory and related proofs for the treatment of the original problem formulation and the continuous relaxed problems. The implementation details and convergence proof of the branch-and-bound methodology and the large-scale numerical examples are presented in Part II.     

Regularity and Variationality of Solutions to Hamilton—Jacobi Equations. Part II: Variationality,Existence, Uniqueness

We formulate an Hamilton–Jacobi partial differential equation

High order local linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise

An approach for the construction of A-stable high order explicit strong schemes for stochastic differential equations (SDEs) with additive noise is proposed. We prove that such schemes also have the dynamical property that we call Random A-stability (RA-stability), which ensures that, for linear equations with stationary solutions, the numerical scheme has a random attractor that converges to the exact one as the step size decreases. Basically, the proposed integrators are obtained by splitting, at each time step, the solution of the original equation into two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary SDE. The first one is solved by the Local Linearization scheme in such a way that A-stability is guaranteed, while the second one is approximated by any extant scheme, preferably an explicit one that yields high order of convergence with low computational cost. Numerical integrators constructed in this way are called High Order Local Linearization (HOLL) methods. Various efficient HOLL schemes are elaborated in detail, and their performance is illustrated through computer simulations. Furthermore, mean-square convergence of the introduced methods is studied.     

Optimal error estimate of upwind scheme on Shishkin-type meshes for singularly perturbed parabolic problems with discontinuous convection coefficients

This paper analyzes the implicit upwind finite difference scheme on Shishkin-type meshes (including the classical piecewise-uniform Shishkin mesh and the Bakhalov-Shishkin mesh) for a class of singularly perturbed parabolic convection-diffusion problems exhibiting strong interior layers. Suitable conditions on the mesh-generating functions are derived and are found to be sufficient for the convergence of the method, uniformly with respect to the perturbation parameter. Utilizing these conditions, it is shown that the method converges uniformly in the discrete supremum norm with an optimal error bound. Numerical results are presented to validate the theoretical results.     

A new self-adaptive alternating direction method for variational inequality problems with linear equality and inequality constraints

In this paper, we present a new self-adaptive alternating direction method for solving a class of variational inequality problems with both linear equality and inequality constraints without the need to add any extra slack variables. The method is simple because it needs only to perform some projections and function evaluations. In addition, to further enhance its efficiency, we adopt a self-adaptive strategy to adjust parameter μ at each iteration. Convergence of the proposed method is proved under certain conditions. Numerical experience illustrates the efficiency of the new method.     

Uniform blow-up rates and asymptotic estimates of solutions for diffusion systems with weighted localized sources

This paper investigates the global existence of the nonnegative solution and the finite time blow-up of solutions of nonlinear parabolic equation with a more complicated source term, which is a product of localized source, local source, and weight function; we also study the blow-up rate of solution to this problem.     

Constrained Extremum Problems and Image Space Analysis—Part II: Duality and Penalization

In the light, as said in Part I, of showing the main feature of image space analysis—to unify and generalize the several topics of optimization—we continue, in Part II, considering duality and penalization. In the literature, they are distinct sectors of optimization and, as far as we know, they have nothing in common. Here it is shown that they can be derived by the same “root.”     

Generalized Tractability for Multivariate Problems Part II: Linear Tensor Product Problems, Linear Information, and Unrestricted Tractability

We continue the study of generalized tractability initiated in our previous paper “Generalized tractability for multivariate problems, Part I: Linear tensor product problems and linear information”, J. Complex. 23:262–295, 2007. We study linear tensor product problems for which we can compute linear information which is given by arbitrary continuous linear functionals. We want to approximate an operator S _d given as the d-fold tensor product of a compact linear operator S ₁ for d=1,2,…, with ‖S ₁‖=1 and S ₁ having at least two positive singular values. Let n(ε,S _d ) be the minimal number of information evaluations needed to approximate S _d to within ε∈[0,1]. We study generalized tractability by verifying when n(ε,S _d ) can be bounded by a multiple of a power of T(ε ⁻¹,d) for all (ε ⁻¹,d)∈Ω⊆[1,∞)×ℕ. Here, T is a tractability function which is non-decreasing in both variables and grows slower than exponentially to infinity. We study the exponent of tractability which is the smallest power of T(ε ⁻¹,d) whose multiple bounds n(ε,S _d ). We also study weak tractability, i.e., when . In our previous paper, we studied generalized tractability for proper subsets Ω of [1,∞)×ℕ, whereas in this paper we take the unrestricted domain Ω ^unr=[1,∞)×ℕ. We consider the three cases for which we have only finitely many positive singular values of S ₁, or they decay exponentially or polynomially fast. Weak tractability holds for these three cases, and for all linear tensor product problems for which the singular values of S ₁ decay slightly faster than logarithmically. We provide necessary and sufficient conditions on the function T such that generalized tractability holds. These conditions are obtained in terms of the singular values of S ₁ and mostly asymptotic properties of T. The tractability conditions tell us how fast T must go to infinity. It is known that T must go to infinity faster than polynomially. We show that generalized tractability is obtained for T(x,y)=x ^1+ln y . We also study tractability functions T of product form, T(x,y)=f ₁(x)f ₂(x). Assume that a _i =lim inf  _x→∞(ln ln f _i (x))/(ln ln x) is finite for i=1,2. Then generalized tractability takes place iff