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Thesis (M.S.)--Boston Universit

A New Approximation of the Schur Complement in Preconditioners for PDE Constrained Optimization

Saddle point systems arise widely in optimization problems with constraints. The utility of Schur complement approximation is now broadly appreciated in the context of solving such saddle point systems by iteration. In this short manuscript, we present a new Schur complement approximation for PDE constrained optimization, an important class of these problems. Block diagonal and block triangular preconditioners have previously been designed to be used to solve such problems along with MINRES and non-standard Conjugate Gradients respectively; with appropriate approximation blocks these can be optimal in the sense that the time required for solution scales linearly with the problem size, however small the mesh size we use. In this paper, we extend this work to designing such preconditioners for which this optimality property holds independently of both the mesh size and of the Tikhonov regularization parameter \beta that is used. This also leads to an effective symmetric indefinite preconditioner that exhibits mesh and \beta-independence. We motivate the choice of these preconditioners based on observations about approximating the Schur complement obtained from the matrix system, derive eigenvalue bounds which verify the effectiveness of the approximation, and present numerical results which show that these new preconditioners work well in practice

Fast iterative solvers for convection-diffusion control problems

In this manuscript, we describe effective solvers for the optimal control of stabilized convection-diffusion problems. We employ the local projection stabilization, which we show to give the same matrix system whether the discretize-then-optimize or optimize-then-discretize approach for this problem is used. We then derive two effective preconditioners for this problem, the �first to be used with MINRES and the second to be used with the Bramble-Pasciak Conjugate Gradient method. The key components of both preconditioners are an accurate mass matrix approximation, a good approximation of the Schur complement, and an appropriate multigrid process to enact this latter approximation. We present numerical results to demonstrate that these preconditioners result in convergence in a small number of iterations, which is robust with respect to the mesh size h, and the regularization parameter β, for a range of problems

Robust Iterative Solution of a Class of Time-Dependent Optimal Control Problems

The fast iterative solution of optimal control problems, and in particular PDE-constrained optimization problems, has become an active area of research in applied mathematics and numerical analysis. In this paper, we consider the solution of a class of time-dependent PDE-constrained optimization problems, specifically the distributed control of the heat equation. We develop a strategy to approximate the (1,1)-block and Schur complement of the saddle point system that results from solving this problem, and therefore derive a block diagonal preconditioner to be used within the MINRES algorithm. We present numerical results to demonstrate that this approach yields a robust solver with respect to step-size and regularization parameter

Using Proxies for the Short Rate: When are Three Months Like an Instant?

The dynamics of the unobservable "short" or "instantaneous" rate of interest are frequently estimated using a proxy variable. We show the biases resulting from this practice (the "proxy" problem) are related to the derivatives of the proxy with respect to the short rate and the (inverse) function from the proxy to the short rate. Analytic results show that the proxy problem is not economically significant for single- factor affine models, for parameter values consistent with US data. In addition, for the two-factor affine model of Longstaff and Schwartz (1992), the proxy problem is only economically significant for pricing discount bonds with maturities of more than 5 years. We also describe two different procedures which can be used to assess the magnitude of the proxy problem in more general interest rate models. Numerical evaluation of a nonlinear single-factor model suggests that the proxy problem can significantly affect both estimates of the diffusion function and discount bond prices.interest rates, proxies, term structure

A tunable cavity-locked diode laser source for terahertz photomixing

An all solid-state approach to the precise frequency synthesis and control of widely tunable terahertz radiation by differencing continuous-wave diode lasers at 850 nm is reported in this paper. The difference frequency is synthesized by three fiber-coupled external-cavity laser diodes. Two of the lasers are Pound-Drever-Hall locked to different orders of a Fabry-Perot (FP) cavity, and the third is offset-frequency locked to the second of the cavity-locked lasers using a tunable microwave oscillator. The first cavity-locked laser and the offset-locked laser produce the difference frequency, whose value is accurately determined by the sum of an integer multiple of the free spectral range of the FP cavity and the offset frequency. The dual-frequency 850-nm output of the three laser system is amplified to 500 mW through two-frequency injection seeding of a single semiconductor tapered optical amplifier. As proof of precision frequency synthesis and control of tunability, the difference frequency is converted into a terahertz wave by optical-heterodyne photomixing in low-temperature-grown GaAs and used for the spectroscopy of simple molecules. The 3-dB spectral power bandwidth of the terahertz radiation is routinely observed to be ≾1 MHz. A simple, but highly accurate, method of obtaining an absolute frequency calibration is proposed and an absolute calibration of 10^(-7) demonstrated using the known frequencies of carbon monoxide lines between 0.23-1.27 THz

Fast iterative solution of reaction-diffusion control problems arising from chemical processes

PDE-constrained optimization problems, and the development of preconditioned iterative methods for the efficient solution of the arising matrix system, is a field of numerical analysis that has recently been attracting much attention. In this paper, we analyze and develop preconditioners for matrix systems that arise from the optimal control of reaction-diffusion equations, which themselves result from chemical processes. Important aspects in our solvers are saddle point theory, mass matrix representation and effective Schur complement approximation, as well as the outer (Newton) iteration to take account of the nonlinearity of the underlying PDEs

Floristic Composition and Conservation Status of Fens in Iowa

Over 200 extant fens of varying condition were documented during an extensive inventory conducted in Iowa between 1986 and 1991. Approximately half of the extant fens support endangered, threatened, special concern, or other rare plant species. Approximately 25 fens are outstanding conservation prospects with intact vegetation, high species richness, and rare species. Nearly 40% of all potential fen sites have been destroyed by cultivation or damage; another 30% remain unknown due to lack of a field visit, but most appear on aerial photographs to be very small, disturbed fragments. In addition to their traditionally recognized range in northwest Iowa, fens were found to be numerous and widespread in eastern Iowa. Most (95%) of the extant fens occurred on private lands; these were variously affected by grazing (65%), cropfield edge effects (33%), potential expansion of woody plants (20% ), drainage by tile lines or ditches (10% ), excavation for ponds (2%), and mining of nearby sand and gravel deposits (2%)