On the identification of the contribution of several pollutant sources from local

In the present article we discuss the pollution dispersion problem from a mixed juridical scientific technological point of view, where the principal interest lies in the pertinent question, how to transform model simulation and analysis tools into an operational tool. One evaluation method relevant for control, planning, licensing and commissioning of installations that are the origin of pollution is discussed, where observational shortcomings and the role of model simulations are addressed. We briefly review deterministic and stochastic models, complement it by some aspects concerning the pollution problem in the context of urban growth and indicate the necessity for a symbiosis of legislation and scientific-technical measures. A method is proposed that allows to optimise monitoring as well as simulations combined by mutual feedback in order to create a reliable base for decisions in the context of legal actions. The authors of the present article are aware that the presented discussion is only one step in a direction where legislation and the scientific and technological sector work hand in hand in order to make progress on the subject

Stochastic Wind Profiles Determination for Radioactive Substances Released from Nuclear Power Plants

In this review we discuss a stochastic turbulent wind proï¬le based on the three-dimensional stochastic Langevin equation for Gram-Chalier probability density function and a known mean wind velocity. Its solution permits to simulate radioactive substances dispersion in a turbulent regime, which is of interest for nuclear reactor accident scenarios and their related emergency actions. We discuss the stochastic Langevin equation together with an analytical method for solving the three-dimensional and time dependent equation which is then applied to radioactive substance dispersion for a stochastic turbulence model. The solution is obtained using the Adomian Decomposition Method, which provides a direct scheme for solving the problem without the need for linearisation and any transformation. The results of the model are compared to case studies with measured data and further compared to procedures and predictions from other approaches

Hi-Fidelity Simulation of the Self-Assembly and Dynamics of Colloids and Polymeric Solutions with Long Range Interactions

Modeling the equilibrium properties and dynamic response of the colloidal and polymeric solutions provides valuable insight into numerous biological and industrial processes and facilitates development of novel technologies. To this end, the centerpiece of this research is to incorporate the long range electrostatic or hydrodynamic interactions via computationally efficient algorithms and to investigate the effect of these interactions on the self-assembly of colloidal particles and dynamic properties of polymeric solutions. Specifically, self-assembly of a new class of materials, namely bipolar Janus nano-particles, is investigated via molecular dynamic simulation in order to establish the relationship between individual particle characteristics, such as surface charge density, particle size, etc., and the final structure formation. Furthermore, the importance of incorporating the long range electrostatic interaction in achieving the corresponding final morphology is discussed. The dynamic properties of polymeric solutions are investigated via two parallel pathways. In the first approach, force-extension behavior of the flexible polyelectrolytes is probed via fine-grained Brownian dynamics simulation of the bead-rod model. The presented model accurately incorporates the excluded volume interaction in order to capture the effect of salt concentration on the force-extension response of polyelectrolyte chain as observed in the single chain experiments. It is shown that accurate incorporation of the excluded volume effect on a long chain of more than 500 Kuhn segments is necessary to reach the universal scaling both for equilibrium properties and force-extension response. Next, a new force law is extracted using a novel discrete Pade approximant from the constant-force ensemble result of the bead-rod model. The new force law is implemented in the coarse-grained meso-scale bead-spring model with hydrodynamic interactions in order to investigate the dynamics of flexible macromolecules in the athermal solvent. In the second approach the computational cost of the long range hydrodynamic interaction in dilute solution of polymeric chains with constrains is reduced via development of a new computational technique based on the conjugate gradient and Krylov subspace methods. Moreover, an algorithm for estimating the contribution of various forces to the transient polymeric stress tensor is introduced and employed in order to investigate transient dynamics of the solution of the flexible polymeric chains

DG-IMEX Method for a Two-Moment Model for Radiation Transport in the O(v/c)\mathcal{O}(v/c) Limit

We consider particle systems described by moments of a phase-space density and propose a realizability-preserving numerical method to evolve a spectral two-moment model for particles interacting with a background fluid moving with nonrelativistic velocities. The system of nonlinear moment equations, with special relativistic corrections to $\mathcal{O}(v/c)$, expresses a balance between phase-space advection and collisions and includes velocity-dependent terms that account for spatial advection, Doppler shift, and angular aberration. This model is closely related to the one promoted by Lowrie et al. (2001; JQSRT, 69, 291-304) and similar to models currently used to study transport phenomena in large-scale simulations of astrophysical environments. The method is designed to preserve moment realizability, which guarantees that the moments correspond to a nonnegative phase-space density. The realizability-preserving scheme consists of the following key components: (i) a strong stability-preserving implicit-explicit (IMEX) time-integration method; (ii) a discontinuous Galerkin (DG) phase-space discretization with carefully constructed numerical fluxes; (iii) a realizability-preserving implicit collision update; and (iv) a realizability-enforcing limiter. In time integration, nonlinearity of the moment model necessitates solution of nonlinear equations, which we formulate as fixed-point problems and solve with tailored iterative solvers that preserve moment realizability with guaranteed convergence. We also analyze the simultaneous Eulerian-frame number and energy conservation properties of the semi-discrete DG scheme and propose an "energy limiter" that promotes Eulerian-frame energy conservation. Through numerical experiments, we demonstrate the accuracy and robustness of this DG-IMEX method and investigate its Eulerian-frame energy conservation properties

A Yosida's parametrix approach to Varadhan's estimates for a degenerate diffusion under the weak Hörmander condition

We adapt and extend Yosida's parametrix method, originally introduced for the construction of the fundamental solution to a parabolic operator on a Riemannian manifold, to derive Varadhan-type asymptotic estimates for the transition density of a degenerate diffusion under the weak Hörmander condition. This diffusion process, widely studied by Yor in a series of papers, finds direct application in the study of a class of path-dependent financial derivatives known as Asian options. We obtain a Varadhan-type formula which relates the transition density p of the stochastic process with the optimal cost Ψ of a deterministic control problem associated to the diffusion. We provide a partial proof of this formula, and present numerical evidence to support the validity of an intermediate inequality that is required to complete the proof. We also derive an asymptotic expansion of the cost function Ψ, expressed in terms of elementary functions, which is useful in order to design efficient approximation formulas for the transition density

Numerical methods for the sign problem in Lattice Field Theory.

The great majority of algorithms employed in the study of lattice field theory are based on Monte Carlo's importance sampling method, i.e. on probability interpretation of the Boltzmann weight. Unfortunately in many theories of interest one cannot associated a real and positive weight to every configuration, that is because their action is explicitly complex or because the weight is multiplied by some non positive term. In this cases one says that the theory on the lattice is affected by the sign problem. An outstanding example of sign problem preventing a quantum field theory to be studied, is QCD at finite chemical potential. Whenever the sign problem is present, standard Monte Carlo methods are problematic to apply and, in general, new approaches are needed to explore the phase diagram of the complex theory. Here we will review three of the main candidate methods to deal with the sign problem, namely complex Langevin dynamics, Lefschetz thimbles and density of states method. We will first study complex Langevin dynamics, combined with the gauge cooling method, on the one-dimensional Polyakov line model, and then we will apply it to pure gauge Yang-Mills theory with a topological theta-term. It follows a comparison between complex Langevin dynamics and the Lefschetz thimbles method on three toy models, which are the quartic model, the U(1) one-link model with a mu dependent determinant, and the SU(2) non abelian one-link model with complex beta parameter. Lastly, we introduce the density of state method, based on the LLR algorithm, and we will employ it in the study of the relativistic Bose gas at finite chemical potential

Complex paths around the sign problem

The Monte Carlo evaluation of path integrals is one of a few general purpose methods to approach strongly coupled systems. It is used in all branches of physics, from QCD and nuclear physics to the correlated electron systems. However, many systems of great importance (dense matter inside neutron stars, the repulsive Hubbard model away from half filling, and dynamical and nonequilibrium observables) are not amenable to the Monte Carlo method as it currently stands due to the so-called sign problem. A new set of ideas recently developed to tackle the sign problem based on the complexification of field space and the Picard-Lefshetz theory accompanying it is reviewed. The mathematical ideas underpinning this approach, as well as the algorithms developed thus far, are described together with nontrivial examples where the method has already been proved successful. Directions of future work, including the burgeoning use of machine learning techniques, are delineated

From the nano- to the macroscale – bridging scales for the moving contact line problem

The moving contact line problem is one of the main unsolved fundamental problems in fluid mechanics, with relevant physical phenomena spanning multiple scales, from the molecular to the macroscopic scale. In this thesis, at the macroscale, it is shown that classical asymptotic analysis is applicable at the moving contact line. This allows for a direct matching procedure between the inner (nanoscale) region and the outer (macroscale) region, therefore simplifying the analysis presented to date in the literature. At the mesoscale, a unified derivation for single and binary fluid diffuse interface models is presented, consolidating two models present in the literature. Results from an asymptotic analysis of the sharp interface limit of the binary fluid diffuse interface model are compared with numerical computations of the inner region in the vicinity of a moving contact line. Finally, the nanoscale structure of the density profile in the vicinity of the con- tact line is studied using density functional theory (DFT). At equilibrium, an effective disjoining pressure is extracted and results are compared with coarse-grained Hamiltonian theory. A derivation of Navier-Stokes like dynamic DFT equations is presented. Results for the moving contact line are compared with predictions from molecular kinetic theory. Computations for both DFT and diffuse interface approaches are performed using pseudospectral methods mapped to unbounded domains. The numerical scheme is presented, and the inclusion of hard-sphere effects via a fundamental measure theory is discussed.Open Acces

Computational Inverse Problems for Partial Differential Equations

The problem of determining unknown quantities in a PDE from measurements of (part of) the solution to this PDE arises in a wide range of applications in science, technology, medicine, and finance. The unknown quantity may e.g. be a coefficient, an initial or a boundary condition, a source term, or the shape of a boundary. The identification of such quantities is often computationally challenging and requires profound knowledge of the analytical properties of the underlying PDE as well as numerical techniques. The focus of this workshop was on applications in phase retrieval, imaging with waves in random media, and seismology of the Earth and the Sun, a further emphasis was put on stochastic aspects in the context of uncertainty quantification and parameter identification in stochastic differential equations. Many open problems and mathematical challenges in application fields were addressed, and intensive discussions provided an insight into the high potential of joining deep knowledge in numerical analysis, partial differential equations, and regularization, but also in mathematical statistics, homogenization, optimization, differential geometry, numerical linear algebra, and variational analysis to tackle these challenges