Complexity bounds on supermesh construction for quasi-uniform meshes

Projecting fields between different meshes commonly arises in computational physics. This operation requires a supermesh construction and its computational cost is proportional to the number of cells of the supermesh $n$. Given any two quasi-uniform meshes of $n_A$ and $n_B$ cells respectively, we show under standard assumptions that n is proportional to $n_A + n_B$. This result substantially improves on the best currently available upper bound on $n$ and is fundamental for the analysis of algorithms that use supermeshes

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

Resisting globalization: voting power indices and the national interest in EU decision making

The European integration process can be regarded as an aspect of the overall process of globalization and at the same time as an attempt to reconcile the move towards standardization with the defence of national diversity. A central role in the effort to exploit the benefits of integration without hampering the national interest is the collective decision-making process in the EU shared by the Council and the Parliament, where national representatives struggle for the conquest of the \u201clion\u2019s share\u201d in the division of the surplus stemming from the compromise on common policies. The paper focuses on the rationale behind the various Indices of Voting Power (Penrose, Banzhaf, Shapley and Shubik, and Holler) which are the formal tools meant to understanding the coalition formation and assessing the probability of reaching an agreement. In the EU Council, although ministers tend to reach a broad consensus and explicit voting is rare, decision may be taken by qualified majority voting where countries are assigned weights reflecting their size. This practice has often been objected on the grounds that the potential emergence of permanent coalitions would prevent countries from being adequately represented, and eventually a different rule was proposed. The pros and cons of the newly proposed system are analysed through the computation of the Indices of Voting Power for the EU Council and Parliament

Technological choices under institutional constraints: measuring the impact on earnings dispersion

The paper investigates the relationship between labour market and redistributive institutions in Europe. The hypothesis is put forward of a changing balance between decreasing labour market regulation and increasing income redistribution. Econometric estimates conducted in the paper show that more redistribution has been traded off for regulation thus allowing for higher wage inequality

Depositional architecture of a mixed travertine-terrigenous system in a fault-controlled continental extensional basin (Messinian, Southern Tuscany, Central Italy)

The extensional Neogene Albegna Basin (Southern Tuscany, Italy) includes several thermogene travertine units dating from the Miocene to Holocene time. During the late Miocene (Messinian), a continental fault-controlled basin (of nearly 500-km2 width) was filled by precipitated travertine and detrital terrigenous strata, characterized by a wedge-shaped geometry that thinned northward, with a maximum thickness of nearly 70m. This mixed travertine-terrigenous succession was investigated in terms of lithofacies types, depositional environment and architecture and the variety of precipitated travertine fabrics. Deposited as beds with thickness ranging from centimetres to a few decimetres, carbonates include nine travertine facies types: F1) clotted peloidal micrite and microsparite boundstone, F2) raft rudstone/floatstone, F3) sub-rounded radial coated grain grainstone, F4) coated gas bubble boundstone, F5) crystalline dendrite cementstone, F6) laminated boundstone, F7) coated reed boundstone and rudstone, F8) peloidal skeletal grainstone and F9) calci-mudstone and microsparstone. Beds of terrigenous deposits with thickness varying from a decimetre to > 10 m include five lithofacies: F10) breccia, F11) conglomerate, F12) massive sandstone, F13) laminated sandstone and F14) claystone. The succession recorded the following three phases of evolution of the depositional setting: 1) At the base, a northward-thinning thermogene travertine terraced slope (Phase I, travertine slope lithofacies association, F1-F6) developed close to the extensional fault system, placed southward with respect to the travertine deposition. 2) In Phase II, the accumulation of travertines was interrupted by the deposition of colluvial fan deposits with a thickness of several metres (colluvial fan lithofacies association, F10 and F12), which consisted of massive breccias, adjacent to the alluvial plain lithofacies association (F11-F14) including massive claystone and sandstone and channelized conglomerates. Travertine lenses, of 2-3-m thickness, appeared intermittently alternating with the colluvial fan breccias. 3) In the third phase, the filled fault-controlled basin evolved into an alluvial plain with ponds rich in coated reed travertines, which record the influence of freshwater (travertine flat lithofacies association, F7-F9). This study shows the stratigraphic architecture and sedimentary evolution of a continental succession, wherein the hydrothermal activity and consequent travertine precipitation were driven by the extensional tectonic regime, with faults acting as fluid paths for the thermal water. Fault activity created the accommodation space for travertine and colluvial fan accumulation. Erosion of the uplifted footwall blocks provided the source of sediments for the colluvial fan breccias, which alternated with the thermogene travertine precipitation. Climatic oscillations might have led to the recharge of the aquifer that fed the hydrothermal vents. The studied continental succession in an extensional basin provides valuable information about the interplay between thermogene travertine and alluvial/colluvial deposition, which in turn might improve the understanding of similar fault-controlled continental depositional systems in outcrops and the subsurface

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

Multi-output multilevel best linear unbiased estimators via semidefinite programming

Multifidelity forward uncertainty quantification (UQ) problems often involve multiple quantities of interest and heterogeneous models (e.g., different grids, equations, dimensions, physics, surrogate and reduced-order models). While computational efficiency is key in this context, multi-output strategies in multilevel/multifidelity methods are either sub-optimal or non-existent. In this paper we extend multilevel best linear unbiased estimators (MLBLUE) to multi-output forward UQ problems and we present new semidefinite programming formulations for their optimal setup. Not only do these formulations yield the optimal number of samples required, but also the optimal selection of low-fidelity models to use. While existing MLBLUE approaches are single-output only and require a non-trivial nonlinear optimization procedure, the new multi-output formulations can be solved reliably and efficiently. We demonstrate the efficacy of the new methods and formulations in practical UQ problems with model heterogeneity.Comment: 22 pages, 5 figures, 3 table