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

Lake Monona Causeway - Madison, Wisconsin, USA

In 1965 and 1966, a 3,500’ long, hydraulically-placed, highway embankment was constructed over the very soft lakebed deposits of Lake Monona. The compressible deposits, which ranged in thickness from 40’ to 80’, included marl, organic silt and silty clay. An incipient shear failure occurred near the end of the 1965 construction season, necessitating embankment design and construction modifications. Pertinent slope stability and settlement data are summarized, as are time dependent changes in in-situ subsoil parameters. Measured settlements, which currently range from 3’ to more than 12’, agree well with basic consolidation theory and engineering predictions formulated at project inception. General observations with respect to design, construction and overall performance of the causeway embankment are also provided

Properties of short-range and long-range correlation energy density functionals from electron-electron coalescence

The combination of density functional theory with other approaches to the many-electron problem through the separation of the electron-electron interaction into a short-range and a long-range contribution is a promising method, which is raising more and more interest in recent years. In this work some properties of the corresponding correlation energy functionals are derived by studying the electron-electron coalescence condition for a modified (long-range-only) interaction. A general relation for the on-top (zero electron-electron distance) pair density is derived, and its usefulness is discussed with some examples. For the special case of the uniform electron gas, a simple parameterization of the on-top pair density for a long-range only interaction is presented and supported by calculations within the ``extended Overhauser model''. The results of this work can be used to build self-interaction corrected short-range correlation energy functionals.Comment: revised version, to appear in Phys. Rev.

Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular jacobians of genus 2 curves

This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of these six quantities is the size of the Shafarevich-Tate group. Unable to compute that, we computed the five other quantities and solved for the last one. In all 32 cases, the result is very close to an integer that is a power of 2. In addition, this power of 2 agrees with the size of the 2-torsion of the Shafarevich-Tate group, which we could compute

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

Regularization-robust preconditioners for time-dependent PDE constrained optimization problems

In this article, we motivate, derive and test �effective preconditioners to be used with the Minres algorithm for solving a number of saddle point systems, which arise in PDE constrained optimization problems. We consider the distributed control problem involving the heat equation with two diff�erent functionals, and the Neumann boundary control problem involving Poisson's equation and the heat equation. Crucial to the eff�ectiveness of our preconditioners in each case is an eff�ective approximation of the Schur complement of the matrix system. In each case, we state the problem being solved, propose the preconditioning approach, prove relevant eigenvalue bounds, and provide numerical results which demonstrate that our solvers are eff�ective for a wide range of regularization parameter values, as well as mesh sizes and time-steps

Ground state properties of heavy alkali halides

We extend previous work on alkali halides by calculations for the heavy-atom species RbF, RbCl, LiBr, NaBr, KBr, RbBr, LiI, NaI, KI, and RbI. Relativistic effects are included by means of energy-consistent pseudopotentials, correlations are treated at the coupled-cluster level. A striking deficiency of the Hartree-Fock approach are lattice constants deviating by up to 7.5 % from experimental values which is reduced to a maximum error of 2.4 % by taking into account electron correlation. Besides, we provide ab-initio data for in-crystal polarizabilities and van der Waals coefficients.Comment: accepted by Phys. Rev.

Simple model of the static exchange-correlation kernel of a uniform electron gas with long-range electron-electron interaction

A simple approximate expression in real and reciprocal spaces is given for the static exchange-correlation kernel of a uniform electron gas interacting with the long-range part only of the Coulomb interaction. This expression interpolates between the exact asymptotic behaviors of this kernel at small and large wave vectors which in turn requires, among other thing, information from the momentum distribution of the uniform electron gas with the same interaction that have been calculated in the G0W0 approximation. This exchange-correlation kernel as well as its complement analogue associated to the short-range part of the Coulomb interaction are more local than the Coulombic exchange-correlation kernel and constitute potential ingredients in approximations for recent adiabatic connection fluctuation-dissipation and/or density functional theory approaches of the electronic correlation problem based on a separate treatment of long-range and short-range interaction effects.Comment: 14 pages, 14 figures, to be published in Phys. Rev.

Inelastic Diffraction and Spectroscopy of Very Weakly Bound Clusters

We study the coherent inelastic diffraction of very weakly bound two body clusters from a material transmission grating. We show that internal transitions of the clusters can lead to new separate peaks in the diffraction pattern whose angular positions determine the excitation energies. Using a quantum mechanical approach to few body scattering theory we determine the relative peak intensities for the diffraction of the van der Waals dimers (D_2)_2 and H_2-D_2. Based on the results for these realistic examples we discuss the possible applications and experimental challenges of this coherent inelastic diffraction technique.Comment: 15 pages + 5 figures. J. Phys. B (in press

Ab initio wavefunction based methods for excited states in solids: correlation corrections to the band structure of ionic oxides

Ab initio wavefunction based methods are applied to the study of electron correlation effects on the band structure of oxide systems. We choose MgO as a prototype closed-shell ionic oxide. Our analysis is based on a local Hamiltonian approach and performed on finite fragments cut from the infinite solid. Localized Wannier functions and embedding potentials are obtained from prior periodic Hartree-Fock (HF) calculations. We investigate the role of various electron correlation effects in reducing the HF band gap and modifying the band widths. On-site and nearest-neighbor charge relaxation as well as long-range polarization effects are calculated. Whereas correlation effects are essential for computing accurate band gaps, we found that they produce smaller changes on the HF band widths, at least for this material. Surprisingly, a broadening effect is obtained for the O 2p valence bands. The ab initio data are in good agreement with the energy gap and band width derived from thermoreflectance and x-ray photoemission experiments. The results show that the wavefunction based approach applied here allows for well controlled approximations and a transparent identification of the microscopic processes which determine the electronic band structure