Deflation for semismooth equations

Variational inequalities can in general support distinct solutions. In this paper we study an algorithm for computing distinct solutions of a variational inequality, without varying the initial guess supplied to the solver. The central idea is the combination of a semismooth Newton method with a deflation operator that eliminates known solutions from consideration. Given one root of a semismooth residual, deflation constructs a new problem for which a semismooth Newton method will not converge to the known root, even from the same initial guess. This enables the discovery of other roots. We prove the effectiveness of the deflation technique under the same assumptions that guarantee locally superlinear convergence of a semismooth Newton method. We demonstrate its utility on various finite- and infinite-dimensional examples drawn from constrained optimization, game theory, economics and solid mechanics.Comment: 24 pages, 3 figure

Efficient white noise sampling and coupling for multilevel Monte Carlo with non-nested meshes

When solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. Here, we present a new sampling technique that can be used to efficiently compute white noise samples in a finite element method and multilevel Monte Carlo (MLMC) setting. The key idea is to exploit the finite element matrix assembly procedure and factorize each local mass matrix independently, hence avoiding the factorization of a large matrix. Moreover, in a MLMC framework, the white noise samples must be coupled between subsequent levels. We show how our technique can be used to enforce this coupling even in the case of non-nested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments. We observe optimal convergence rates for the finite element solution of the elliptic SPDEs of interest in 2D and 3D and we show convergence of the sampled field covariances. In a MLMC setting, a good coupling is enforced and the telescoping sum is respected.Comment: 28 pages, 10 figure

Exploiting Kronecker structure in exponential integrators: fast approximation of the action of φ\varphi-functions of matrices via quadrature

In this article, we propose an algorithm for approximating the action of $\varphi-$functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the $\varphi-$functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive \emph{a priori} bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art.Comment: 20 pages, 3 figures, 7 table

Fast uncertainty quantification of tracer distribution in the brain interstitial fluid with multilevel and quasi Monte Carlo

Efficient uncertainty quantification algorithms are key to understand the propagation of uncertainty -- from uncertain input parameters to uncertain output quantities -- in high resolution mathematical models of brain physiology. Advanced Monte Carlo methods such as quasi Monte Carlo (QMC) and multilevel Monte Carlo (MLMC) have the potential to dramatically improve upon standard Monte Carlo (MC) methods, but their applicability and performance in biomedical applications is underexplored. In this paper, we design and apply QMC and MLMC methods to quantify uncertainty in a convection-diffusion model of tracer transport within the brain. We show that QMC outperforms standard MC simulations when the number of random inputs is small. MLMC considerably outperforms both QMC and standard MC methods and should therefore be preferred for brain transport models.Comment: Multilevel Monte Carlo, quasi Monte Carlo, brain simulation, brain fluids, finite element method, biomedical computing, random fields, diffusion-convectio

Spherical GEMs for parallax-free detectors

We developed a method to make GEM foils with a spherical geometry. Tests of this procedure and with the resulting spherical \textsc{gem}s are presented. Together with a spherical drift electrode, a spherical conversion gap can be formed. This would eliminate the parallax error for detection of x-rays, neutrons or UV photons when a gaseous converter is used. This parallax error limits the spatial resolution at wide scattering angles. The method is inexpensive and flexible towards possible changes in the design. We show advanced plans to make a prototype of an entirely spherical triple-GEM detector, including a spherical readout structure. This detector will have a superior position resolution, also at wide angles, and a high rate capability. A completely spherical gaseous detector has never been made before.Comment: Contribution to the 2009 IEEE Nuclear Science Symposium, Orlando, Florid

Population Dynamics of Native Parasitoids Associated with the Asian Chestnut Gall Wasp (Dryocosmus kuriphilus)

Native parasitoids may play an important role in biological control. They may either support or hinder the effectiveness of introduced nonnative parasitoids released for pest control purposes. Results of a three-year survey (2011–2013) of the Asian chestnut gall wasp (ACGW) Dryocosmus kuriphilus Yasumatsu (Hymenoptera: Cynipidae) populations and on parasitism rates by native indigenous parasitoids (a complex of chalcidoid hymenopterans) in Italian chestnut forests are given. Changes in D. kuriphilus gall size and phenology were observed through the three years of study. A total of 13 species of native parasitoids were recorded, accounting for fluctuating parasitism rates. This variability in parasitism rates over the three years was mainly due to the effect of Torymus flavipes (Walker) (Hymenoptera: Torymidae), which in 2011 accounted for 75% of all parasitoid specimens yet decreased drastically in the following years. This strong fluctuation may be related to climatic conditions. Besides, our data verified that parasitoids do not choose host galls based on their size, though when they do parasitize smaller ones, they exploit them better. Consequently, ACGWs have higher chances of surviving parasitism if they are inside larger galls

Classification of Two-Dimensional Gas Chromatography Data

Gas chromatography (GC) is a popular tool for chemical analysis. Some samples are so complex that a single column does not have enough power to separate all of the analytes. In this instance a higher resolution GC method, known as comprehensive two-dimensional gas chromatography (GCxGC), is used. DSTL want to be able to use data from GCxGC to attribute samples to a particular region or cultivar. However, the nature of the data means that several difficulties must be overcome before being able to do this: noise from sample, peak mis-alignment, and low quantity of samples. In this report, we investigate several methods to overcome such difficulties, and then classify the data. We are very successful in telling apart blanks from seeds, but obtain limited success when trying to classify between seeds. The method that shows the most promise is k-Nearest Neighbours classification by Wasserstein distance. However, this is still quite sensitive to the noise created by the solvent in the sample. Thus, we suggest that more blank runs be obtained, so that the ‘ground truth’ behaviour of the solvent is better understood, allowing us to remove the effect of the solvent from seed data. We also hope that the methods explored here will be more successful on the full raw data than they were on the limited ‘peaks’ data available to us for the purpose of this study