Optimizing Low-Discrepancy Sequences with an Evolutionary Algorithm

International audienceMany elds rely on some stochastic sampling of a given com- plex space. Low-discrepancy sequences are methods aim- ing at producing samples with better space-lling properties than uniformly distributed random numbers, hence allow- ing a more ecient sampling of that space. State-of-the-art methods like nearly orthogonal Latin hypercubes and scram- bled Halton sequences are congured by permutations of in- ternal parameters, where permutations are commonly done randomly. This paper proposes the use of evolutionary al- gorithms to evolve these permutations, in order to optimize a discrepancy measure. Results show that an evolution- ary method is able to generate low-discrepancy sequences of signicantly better space-lling properties compared to sequences congured with purely random permutations

Reliability training

Discussed here is failure physics, the study of how products, hardware, software, and systems fail and what can be done about it. The intent is to impart useful information, to extend the limits of production capability, and to assist in achieving low cost reliable products. A review of reliability for the years 1940 to 2000 is given. Next, a review of mathematics is given as well as a description of what elements contribute to product failures. Basic reliability theory and the disciplines that allow us to control and eliminate failures are elucidated

Convective dissolution in porous media: three-dimensional imaging experiments and numerical simulations

Convective dissolution is a phenomenon induced by a buoyant instability between two fluids resulting in characteristic finger-like mixing patterns. One important example is a key trapping mechanism during CO2 sequestration in deep saline aquifers. The Rayleigh number, Ra, which is a measure of convective vigour and the Sherwood number, Sh, which indicates the strength of mass transport are used to parameterise this process. We present a novel methodology to image convective dissolution using X-ray CT in a three-dimensional porous medium formed of glass beads with a model fluid pair MEG/water and BEG/water. 3D reconstructions allow us to visualise the spatial and temporal evolution of the plume from onset to the shutdown of convection while quantifying the macroscopic quantities such as the rate of dissolution and horizontal concentration profiles. We investigate convective dissolution with and without permeability heterogeneity over the Rayleigh range, Ra = 2000 - 5000, in a spherical, cuboidal and cylindrical geometry. Simple heterogeneity patterns such as single inclined layers and a series of discontinuous layers were included. It was concluded that the spatial configuration of the less permeable layers was the most important factor in either enhancing (due to flow focusing) or impeding (due to compartmentalisation of the plume) the rate of dissolution compared to the homogeneous case. However, linear trends in the Sherwood and Rayleigh numbers were consistency reported in both the homogeneous Sh = 0.025Ra and heterogeneous studies Sh = 0.0236Ra. 2D COMSOL numerical simulations were performed to extend the Ra range explored by the experiments and allow an investigation of permeability heterogeneity configurations not possible in the laboratory. The results confirmed the experimental findings with very similar scaling observed Sh = 0.022Ra. The results also shed light on the plume structures responsible for flow focusing. Finally, we present suggestions for future work in CO2-brine-rock systems.Open Acces

The exponent of discrepancy is at most 1.4778

Abstract. We study discrepancy with arbitrary weights in the L2 norm over the d-dimensional unit cube. The exponent p ∗ of discrepancy is defined as the smallest p for which there exists a positive number K such that for all d and all ε ≤ 1thereexistKε−p points with discrepancy at most ε. It is well known that p ∗ ∈ (1, 2]. We improve the upper bound by showing that p ∗ ≤ 1.4778842. This is done by using relations between discrepancy and integration in the average case setting with the Wiener sheet measure. Our proof is not constructive. The known constructive bound on the exponent p ∗ is 2.454. 1

Case study for model validation : assessing a model for thermal decomposition of polyurethane foam.

A case study is reported to document the details of a validation process to assess the accuracy of a mathematical model to represent experiments involving thermal decomposition of polyurethane foam. The focus of the report is to work through a validation process. The process addresses the following activities. The intended application of mathematical model is discussed to better understand the pertinent parameter space. The parameter space of the validation experiments is mapped to the application parameter space. The mathematical models, computer code to solve the models and its (code) verification are presented. Experimental data from two activities are used to validate mathematical models. The first experiment assesses the chemistry model alone and the second experiment assesses the model of coupled chemistry, conduction, and enclosure radiation. The model results of both experimental activities are summarized and uncertainty of the model to represent each experimental activity is estimated. The comparison between the experiment data and model results is quantified with various metrics. After addressing these activities, an assessment of the process for the case study is given. Weaknesses in the process are discussed and lessons learned are summarized

The Exponent of Discrepancy is At Most 1.4778...

We study discrepancy with arbitrary weights in the L 2 norm over the d dimensional unit cube. The exponent p of discrepancy is defined as the smallest p for which there exists a positive number K such that for all d and all " 1 there exist K " \Gammap points with discrepancy at most ". It is well known that p 2 (1; 2]. We improve the upper bound by showing that p 1:478841:::: : This is done by using relations between discrepancy and integration in the average case setting with the Wiener sheet measure. Our proof is not constructive. The known constructive bound on the exponent p is 2:454. 1 Introduction We study discrepancy with arbitrary weights in the L 2 norm over the d dimensional unit cube [0; 1] d . This problem is defined as finding n points from [0; 1] d which approximate the volumes of rectangles (starting from zero) with minimal error, see [8, 9] for the precise definition, history and basic properties. Discrepancy has been extensively studied in number..

Macromolecules at interfaces : a flexible theory for hard system

A statistical theory for flexible macromolecules at interfaces has been developed. The theory is based on a lattice model in which the equilibrium set of molecular conformations in a concentration profile is evaluated, using a selfconsistent procedure. In this way, the Flory-Huggins theory for polymer solutions is extended to inhomogeneous solutions of macromolecules without any additional assumption. Apart from the Flory-Huggins polymer-solvent interaction parameter χ, a similar parameter χ s is used to describe the interaction of polymer segments with a solid interface. The average number of molecules in each particular conformation can be computed, so that a very detailed picture of the interfacial structure is obtained. Thus also the train, loop, and tail size distributions of adsorbed polymer can be calculated. In principle, there are no adjustable parameters in the theory. Moreover, there are no restrictions on the system parameters such as polymer concentration, chain length, number of species in a mixture or solvent quality, although in some cases numerical problems may occur. Results are given for adsorption of homopolymers, polydisperse polymer, polyelectrolytes, and star-branched polymer, for the structure of lipid bilayers and of the amorphous phase of semicrystalline polymer, and for the interaction between surfaces due to the presence of adsorbing or nonadsorbing polymer. Available experimental data on adsorption isotherms, bound fraction, layer thickness, surface fractionation, steric stabilization, and polymer bridging agree very well with the theoretical predictions

Dynamical Constraints of Galaxy Clusters via Spectroscopic Observations

Galaxy Clusters are the largest gravitationally bound objects in the Universe, residing at the boundary between the expansive push of dark energy in the vacuum and the attractive pull of dark matter the fills the halo in which a cluster resides. By leveraging the power of spectroscopy, I used the three-dimensional information it provides about galaxies within these clusters to infer dynamical properties about the galaxy cluster and the underlying dark matter halo. The dynamical state and dynamic mass inferences are valuable to future cosmological studies that aim to use the unique nature of galaxy clusters and the role they play in constraining the properties of dark energy and dark matter. In this work I focus on transforming galaxy spectra into line-of-sight velocities which, when paired with projected sky locations, allow me to probe the gravitational potential of the total cluster system. I designed, targeted, acquired, reduced, and analyzed 4427 galaxy spectra from 22 galaxy clusters, of which 3054 passed my strict quality cuts. Of those that passed the cuts, 1679 were identified as cluster members based on radial-velocity phase-space cuts. The data was acquired using the Michigan-Magellan Fiber System (M2FS) multi-fiber spectrograph on the 6.5m Magellan Clay telescope. The reductions were performed using a fully-featured pipeline that I created and that I describe in this work. I also summarize the resulting dataset using spatial, redshift, magnitude, and signal-to-noise information for individual galaxies, and show that there is good agreement when comparing my re-observed redshifts with those in the literature. To convey the amount of information contained in this dataset, I perform an analysis on one specifically selected massive cluster, Abell S1063, which was observed twice. I use two approaches for estimating cluster masses, the first is a velocity dispersion technique that takes the distribution of velocities, reduces it to a statistical measure of the width of the distribution, and maps that spread to a mass based on a model motivated in part by theory and calibrated with simulations. The second uses the velocity-radial distance information from the cluster center to identify the escape velocity edge of the cluster, which is observed as the velocity extrema in a given radial bin. This edge is directly related to the gravitational potential and can be used to infer the total mass of the system. I compare these techniques to one another and against other mass proxies and find that the velocity dispersion measurement differs from other estimates for the system, favoring a higher mass, while the escape velocity edge technique is in good agreement with other estimates. This is expected for a galaxy cluster with substructure, which previous studies have hypothesized for this system but could not verify. I am able to visually confirm the existence of clumps using galaxies as tracers, and quantify the substructure using the Dressler-Shectman statistic, where I found a significant result with p< 0.0001.PHDPhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155027/1/kremin_1.pd