Asymptotic expansions for high-contrast elliptic equations

In this paper, we present a high-order expansion for elliptic equations in high-contrast media. The background conductivity is taken to be one and we assume the medium contains high (or low) conductivity inclusions. We derive an asymptotic expansion with respect to the contrast and provide a procedure to compute the terms in the expansion. The computation of the expansion does not depend on the contrast which is important for simulations. The latter allows avoiding increased mesh resolution around high conductivity features. This work is partly motivated by our earlier work in \cite{ge09_1} where we design efficient numerical procedures for solving high-contrast problems. These multiscale approaches require local solutions and our proposed high-order expansion can be used to approximate these local solutions inexpensively. In the case of a large-number of inclusions, the proposed analysis can help to design localization techniques for computing the terms in the expansion. In the paper, we present a rigorous analysis of the proposed high-order expansion and estimate the remainder of it. We consider both high and low conductivity inclusions

The cost of continuity: performance of iterative solvers on isogeometric finite elements

In this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized Galerkin weak form. Specifically, we compare performance of linear solvers when discretizing using $C^0$ B-splines, which span traditional finite element spaces, and $C^{p-1}$ B-splines, which represent maximum continuity. We provide theoretical estimates for the increase in cost of the matrix-vector product as well as for the construction and application of black-box preconditioners. We accompany these estimates with numerical results and study their sensitivity to various grid parameters such as element size $h$ and polynomial order of approximation $p$. Finally, we present timing results for a range of preconditioning options for the Laplace problem. We conclude that the matrix-vector product operation is at most \slfrac{33p^2}{8} times more expensive for the more continuous space, although for moderately low $p$, this number is significantly reduced. Moreover, if static condensation is not employed, this number further reduces to at most a value of 8, even for high $p$. Preconditioning options can be up to $p^3$ times more expensive to setup, although this difference significantly decreases for some popular preconditioners such as Incomplete LU factorization

Computational complexity and memory usage for multi-frontal direct solvers in structured mesh finite elements

The multi-frontal direct solver is the state-of-the-art algorithm for the direct solution of sparse linear systems. This paper provides computational complexity and memory usage estimates for the application of the multi-frontal direct solver algorithm on linear systems resulting from B-spline-based isogeometric finite elements, where the mesh is a structured grid. Specifically we provide the estimates for systems resulting from $C^{p-1}$ polynomial B-spline spaces and compare them to those obtained using $C^0$ spaces.Comment: 8 pages, 2 figure

On Stochastic Error and Computational Efficiency of the Markov Chain Monte Carlo Method

In Markov Chain Monte Carlo (MCMC) simulations, the thermal equilibria quantities are estimated by ensemble average over a sample set containing a large number of correlated samples. These samples are selected in accordance with the probability distribution function, known from the partition function of equilibrium state. As the stochastic error of the simulation results is significant, it is desirable to understand the variance of the estimation by ensemble average, which depends on the sample size (i.e., the total number of samples in the set) and the sampling interval (i.e., cycle number between two consecutive samples). Although large sample sizes reduce the variance, they increase the computational cost of the simulation. For a given CPU time, the sample size can be reduced greatly by increasing the sampling interval, while having the corresponding increase in variance be negligible if the original sampling interval is very small. In this work, we report a few general rules that relate the variance with the sample size and the sampling interval. These results are observed and confirmed numerically. These variance rules are derived for the MCMC method but are also valid for the correlated samples obtained using other Monte Carlo methods. The main contribution of this work includes the theoretical proof of these numerical observations and the set of assumptions that lead to them

Against Notice Skepticism in Privacy (and Elsewhere)

What follows is an exploration of innovative new ways to deliver privacy notice. Unlike traditional notice that relies upon text or symbols to convey information, emerging strategies of “visceral” notice leverage a consumer’s very experience of a product or service to warn or inform. A regulation might require that a cell phone camera make a shutter sound so people know their photo is being taken. Or a law could incentivize websites to be more formal (as opposed to casual) wherever they collect personal information, as formality tends to place people on greater guard about what they disclose. The thesis of this Article is that, for a variety of reasons, experience as a form of privacy disclosure is worthy of further study before we give in to calls to abandon notice as a regulatory strategy in privacy and elsewhere. In Part I, the Article examines the promise of radical new forms of experiential or visceral notice based in contemporary design psychology. This Part also compares and contrasts visceral notice to other regulator strategies that seek to “nudge” or influence consumer or citizen behavior. Part II discusses why the further exploration of visceral notice and other notice innovation is warranted. Part III explores potential challenges to visceral notice—for instance, from the First Amendment—and lays out some thoughts on the best regulatory context for requiring or incentivizing visceral notice. In particular, this Part highlights the potential of safe harbors and goal-based rules, i.e., rules that look to the outcome of a notice strategy rather than dictate precisely how notice must be delivered. This Article uses online privacy as a case study for several reasons. First, notice is among the only affirmative obligations that companies face with respect to privacy—online privacy is a quintessential notice regime. Second, the Internet is a context in which notice is widely understood to have failed, but where the nature of digital services means that viable regulatory alternatives are few and poor. Finally, the fact that websites are entirely designed environments furnishes unique opportunities for the sorts of untraditional interventions explored in Part I

People Can Be So Fake: A New Dimension to Privacy and Technology Scholarship

This article updates the traditional discussion of privacy and technology, focused since the days of Warren and Brandeis on the capacity of technology to manipulate information. It proposes a novel dimension to the impact of anthropomorphic or social design on privacy. Technologies designed to imitate people-through voice, animation, and natural language-are increasingly commonplace, showing up in our cars, computers, phones, and homes. A rich literature in communications and psychology suggests that we are hardwired to react to such technology as though a person were actually present. Social interfaces accordingly capture our attention, improve interactivity, and can free up our hands for other tasks. At the same time, technologies that imitate people have the potential to implicate long-standing privacy values. One of the well-documented effects on users of interfaces and devices that emulate people is the sensation of being observed and evaluated. Their presence can alter our attitude, behavior, and physiological state. Widespread adoption of such technology may accordingly lessen opportunities for solitude and chill curiosity and self-development. These effects are all the more dangerous in that they cannot be addressed through traditional privacy protections such as encryption or anonymization. At the same time, the unique properties of social technology also present an opportunity to improve privacy, particularly online