Preconditioners for state constrained optimal control problems\ud with Moreau-Yosida penalty function tube

Optimal control problems with partial differential equations play an important role in many applications. The inclusion of bound constraints for the state poses a significant challenge for optimization methods. Our focus here is on the incorporation of the constraints via the Moreau-Yosida regularization technique. This method has been studied recently and has proven to be advantageous compared to other approaches. In this paper we develop preconditioners for the efficient solution of the Newton steps associated with the fast solution of the Moreau-Yosida regularized problem. Numerical results illustrate the competitiveness of this approach. \ud \ud Copyright c 2000 John Wiley & Sons, Ltd

All-at-Once Solution if Time-Dependent PDE-Constrained Optimisation Problems

Time-dependent partial differential equations (PDEs) play an important role in applied mathematics and many other areas of science. One-shot methods try to compute the solution to these problems in a single iteration that solves for all time-steps at the same time. In this paper, we look at one-shot approaches for the optimal control of time-dependent PDEs and focus on the fast solution of these problems. The use of Krylov subspace solvers together with an efficient preconditioner allows for minimal storage requirements. We solve only approximate time-evolutions for both forward and adjoint problem and compute accurate solutions of a given control problem only at convergence of the overall Krylov subspace iteration. We show that our approach can give competitive results for a variety of problem formulations

Employer Demand for Welfare Recipients by Race

This paper uses new survey data on employers in four large metropolitan areas to examine the determinants of employer demand for welfare recipients. The results suggest a high level of demand for welfare recipients, though such demand appears fairly sensitive to business cycle conditions. A broad range of factors, including skill needs and industry, affect the prospective demand for welfare recipients among employers, while other characteristics that affect the relative supply of welfare recipients to these employers (such as location and employer use of local agencies or welfare-to-work programs) influence the extent to which such demand is realized in actual hiring. Moreover, the conditional demand for black (and to a lesser extent Hispanic) welfare recipients lags behind their representation in the welfare population and seems to be more heavily affected by employers’ location and indicators of preferences than by their skill needs or overall hiring activity. Thus, a variety of factors on the demand side of the labor market continue to limit the employment options of welfare recipients, especially those who are minorities.

On elliptic curves with an isogeny of degree 7

We show that if $E$ is an elliptic curve over $\mathbf{Q}$ with a $\mathbf{Q}$-rational isogeny of degree 7, then the image of the 7-adic Galois representation attached to $E$ is as large as allowed by the isogeny, except for the curves with complex multiplication by $\mathbf{Q}(\sqrt{-7})$. The analogous result with 7 replaced by a prime $p > 7$ was proved by the first author in [7]. The present case $p = 7$ has additional interesting complications. We show that any exceptions correspond to the rational points on a certain curve of genus 12. We then use the method of Chabauty to show that the exceptions are exactly the curves with complex multiplication. As a by-product of one of the key steps in our proof, we determine exactly when there exist elliptic curves over an arbitrary field $k$ of characteristic not 7 with a $k$-rational isogeny of degree 7 and a specified Galois action on the kernel of the isogeny, and we give a parametric description of such curves.Comment: The revision gives a complete answer to the question considered in Version 1. Version 3 will appear in the American Journal of Mathematic

Are Suburban Firms More Likely to Discriminate Against African Americans?

This paper presents a test of the hypothesis that employers in suburban locations are more likely to discriminate against African Americans than are employers located in central cities. Using a difference-in-difference framework, we compare central-city/suburban differences in racial hiring outcomes for firms where a white person is in charge of hiring (white employers, for short) to similar geographic differences in outcomes for firms where a black person is in charge of hiring (black employers). We find that both suburban black and white employers hire fewer blacks than their central-city counterparts. Moreover, the central-city/suburban hiring gap among black employers is as large as, or larger than, that of white employers. Suburban black employers, however, receive many more applications from blacks and hire more blacks than do white firms in either location.

Within Cities and Suburbs: Racial Residential Concentration and the Spatial Distribution of Employment Opportunities across Submetropolitan Areas

In this paper, we examine and compare the spatial distributions of jobs and people across submetropolitan areas using data on firms from the Multi-City Study of Urban Inequality and data on people from the U.S. Bureau of the Census. The results indicate that less-educated people and those on public assistance mostly reside in areas with high minority populations. Low-skill jobs are quite scarce in these areas, while the availability of such jobs relative to less-educated people in heavily white suburban areas is high. Large fractions of the low-skill jobs in these metropolitan areas are not accessible by public transit. Furthermore, there is significant variation within both central cities and suburbs in the ethnic composition of residents and in the availability of low-skill jobs. The ability of various minority groups to gain employment in each area depends heavily on the ethnic composition of the particular area.

A Bramble-Pasciak-like method with applications in optimization

Saddle-point systems arise in many applications areas, in fact in any situation where an extremum principle arises with constraints. The Stokes problem describing slow viscous flow of an incompressible fluid is a classic example coming from partial differential equations and in the area of Optimization such problems are ubiquitous.\ud In this manuscript we show how new approaches for the solution of saddle-point systems arising in Optimization can be derived from the Bramble-Pasciak Conjugate Gradient approach widely used in PDEs and more recent generalizations thereof. In particular we derive a class of new solution methods based on the use of Preconditioned Conjugate Gradients in non-standard inner products and demonstrate how these can be understood through more standard machinery. We show connections to Constraint Preconditioning and give the results of numerical computations on a number of standard Optimization test examples

A low-rank matrix equation method for solving PDE-constrained optimization problems

PDE-constrained optimization problems arise in a broad number of applications such as hyperthermia cancer treatment and blood flow simulation. Discretization of the optimization problem and using a Lagrangian approach result in a large-scale saddle-point system, which is challenging to solve, and acquiring a full space-time solution is often infeasible. We present a new framework to efficiently compute a low-rank approximation to the solution by reformulating the KKT system into a Sylvester-like matrix equation. This matrix equation is subsequently projected onto a small subspace via an iterative rational Krylov method, and we obtain a reduced problem by imposing a Galerkin condition on its residual. In our work we discuss implementation details and dependence on the various problem parameters. Numerical experiments illustrate the performance of the new strategy also when compared to other low-rank approaches