Order Restricted Statistical Inference.

Adaptive finite element methods (FEM) have been widely used in applications for over 20 years now. In practice, they converge starting from coarse grids, although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators, together with a reduction rate of data oscillation (information missed by the underlying averaging process), we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance. Extensions to higher order elements and applications to saddle point problems are discussed as well.

Keywords: A posteriori error estimators, data oscillation, adaptive mesh refinement, convergence, Stokes, Uzawa

AMS Subject Classifications: 65N12, 65N15, 65N30, 65N50, 65Y20


Published: SIAM Review, 44 (2002) 631--658.

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Convergence of Adaptive Finite Element Methods

Review: Important contributions in development and improvement of the heat integration techniques

We propose a new, nonparametric method for multivariate regression subject to convexity or concavity constraints on the response function. Convexity constraints are common in economics, statistics, operations research, financial engineering and optimization, but there is currently no multivariate method that is stable and computationally feasible for more than a few thousand observations. We introduce convex adaptive partitioning (CAP), which creates a globally convex regression model from locally linear estimates fit on adaptively selected covariate partitions. CAP is a computationally efficient, consistent method for convex regression. We demonstrate empirical performance by comparing the performance of CAP to other shape-constrained and unconstrained regression methods for predicting weekly wages and value function approximation for pricing American basket options.

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Multivariate convex regression with adaptive partitioning

http://egon.cheme.cmu.edu/Papers/Escobar_controlHENS.pdf

Simultaneous synthesis of heat exchanger networks with operability considerations: Flexibility and controllability

A procedure was developed for on-line optimization and control systems of heat-exchanger networks, which features a two-level control structure, one for a constant configuration control system and the other for a supervisor on-line optimizer. The coordination between levels is achieved by adjusting the formulation of the optimization problem to meet requirements of the adopted control system. The general goal is always to work without losing stream temperature targets while keeping the highest energy integration. The operation constraints used for heat-exchanger and utility units emphasize the computation of heat-exchanger duties rather than intermediate stream temperatures. This simplifies the modeling task and provides clear links with the limits of the manipulated variables. The optimal condition is determined using LP or NLP, depending on the final problem formulation. Degrees of freedom for optimization and equation constraints for considering simple and multiple bypasses are rigorously discussed. An example used shows how the optimization problem can be adjusted to a specific network design, its expected operating space, and the control configuration. Dynamic simulations also show benefits and limitations of this procedure.

Optimizing and controlling the operation of heat-exchanger networks

Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in $H^k(\Omega)$, or some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and an adequate discrete version of the Hessian must be given. In this paper we propose a finite element description of the Hessian, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes. Using semidefinite programming codes, we show concrete examples of approximations to optimization problems.

/pdf/on-convex-functions-and-the-finite-element-method-45ds5x9027.pdf

On Convex Functions and the Finite Element Method

Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in Hk(Ω), and optimizing functionals arising from some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and functions with positive semidefinite discrete Hessian need not be convex in a discrete sense. Previous work has concentrated on non-local descriptions of convexity, making the number of constraints to grow super-linearly with the number of nodes even in dimension 2, and these descriptions are very difficult to extend to higher dimensions. In this paper we propose a finite difference approximation using positive semidefinite programs and discrete Hessians, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes. Using semidefinite programming codes, we show concrete examples of approximations to problems in two and three dimensions.

/pdf/approximating-optimization-problems-over-convex-functions-2w2oavdfoj.pdf

Approximating optimization problems over convex functions

A new nonparametric estimator of a convex regression function in any dimension is proposed and its uniform convergence properties are studied. We start by using any estimator of the regression func...

Néstor E. Aguilera

Papers

Optimizing and controlling the operation of heat-exchanger networks

On Convex Functions and the Finite Element Method

Approximating optimization problems over convex functions

On uniform consistent estimators for convex regression

Universidad Nacional Del Litoral, Santa FE, Argentina