Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations

A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we show how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach

The influence of religious and cosmological beliefs on the solar architecture of the ancient world

In the earliest civilizations of the Ancient World, sun worship developed in parallel with an understanding of the movement of the stars. That was the origin of an architecture that expressed a number of religious and cosmological beliefs. Studies of ancient archaeological remains have revealed that astronomical orientations strongly influenced the construction of some of the most important ancient architectural monuments. Besides its religious role, the sun regulated the culture of the Ancient World in many of its more practical aspects. For instance, the observation of solar and lunar cycles allowed people to anticipate seasonal change. This provided man with a means of organizing and improving agricultural and livestock activities and, in turn, influenced the construction of the large civil and religious buildings. The complex relationships that developed between cosmology, sun worship, early mathematics, and the orientation of buildings with respect to the position of the sun, also decisively influenced the birth and development of what has come to be known as passive solar design. This article describes some of these influences dating from the megalithic period to the development of Mesopotamian and Egyptian architecture.Postprint (published version

The Ursinus Weekly, October 5, 1914

The season\u27s lecture course • Religious work during week • Ursinus 7, Lafayette 7 • Sun dials • Attendance at German universities • Literary societies • College directory • Reserves tie Bridgeton High • Mass meetinghttps://digitalcommons.ursinus.edu/weekly/2636/thumbnail.jp

Fast derivatives of likelihood functionals for ODE based models using adjoint-state method

We consider time series data modeled by ordinary differential equations (ODEs), widespread models in physics, chemistry, biology and science in general. The sensitivity analysis of such dynamical systems usually requires calculation of various derivatives with respect to the model parameters. We employ the adjoint state method (ASM) for efficient computation of the first and the second derivatives of likelihood functionals constrained by ODEs with respect to the parameters of the underlying ODE model. Essentially, the gradient can be computed with a cost (measured by model evaluations) that is independent of the number of the ODE model parameters and the Hessian with a linear cost in the number of the parameters instead of the quadratic one. The sensitivity analysis becomes feasible even if the parametric space is high-dimensional. The main contributions are derivation and rigorous analysis of the ASM in the statistical context, when the discrete data are coupled with the continuous ODE model. Further, we present a highly optimized implementation of the results and its benchmarks on a number of problems. The results are directly applicable in (e.g.) maximum-likelihood estimation or Bayesian sampling of ODE based statistical models, allowing for faster, more stable estimation of parameters of the underlying ODE model.Comment: 5 figure

Spartan Daily, June 13, 1934

Volume 22, Issue 143https://scholarworks.sjsu.edu/spartandaily/2179/thumbnail.jp

Modeling of pulsed laser guide stars for the Thirty Meter Telescope project

The Thirty Meter Telescope (TMT) has been designed to include an adaptive optics system and associated laser guide star (LGS) facility to correct for the image distortion due to Earth's atmospheric turbulence and achieve diffraction-limited imaging. We have calculated the response of mesospheric sodium atoms to a pulsed laser that has been proposed for use in the LGS facility, including modeling of the atomic physics, the light-atom interactions, and the effect of the geomagnetic field and atomic collisions. This particular pulsed laser format is shown to provide comparable photon return to a continuous-wave (cw) laser of the same average power; both the cw and pulsed lasers have the potential to satisfy the TMT design requirements for photon return flux.Comment: 16 pages, 20 figure

The two pillars of design theory: Method of analysis and rhetoric

Since the 1960’s, there have been many initiatives for promoting theoretical understanding on design. However, in spite of definite progress, there are several puzzles and anomalies in the current theoretical landscape of design. We present an interpretation of the evolution of design theorizing that throws new light on these puzzles and anomalies. We contend that in the modern time, there have been two hidden inspirational sources for design theorizing, namely the method of analysis and rhetoric. As they concern different situations, their prescriptions in many ways differ. However, without explicit recourse to the sources, these mutually incompatible prescriptions, along with associated concepts and terms, have been mixed up in a confusing way. It is proposed that these two influences to design theory and practice have to be disentangled, to be able to act like two pillars supporting the phenomenon of designing. We hypothesize that actual design assignments require the mobilization of ideas and support from both pillar