A probabilistic algorithm approximating solutions of a singular PDE of porous media type

The object of this paper is a one-dimensional generalized porous media equation (PDE) with possibly discontinuous coefficient $\beta$, which is well-posed as an evolution problem in $L^1(\mathbb{R})$. In some recent papers of Blanchard et alia and Barbu et alia, the solution was represented by the solution of a non-linear stochastic differential equation in law if the initial condition is a bounded integrable function. We first extend this result, at least when $\beta$ is continuous and the initial condition is only integrable with some supplementary technical assumption. The main purpose of the article consists in introducing and implementing a stochastic particle algorithm to approach the solution to (PDE) which also fits in the case when $\beta$ is possibly irregular, to predict some long-time behavior of the solution and in comparing with some recent numerical deterministic techniques

Microscopic theory of the linear and nonlinear optical properties of TMDCs

Since the discovery of graphene, the research interest in two-dimensional materials has drastically increased. Among them, semiconducting transition-metal dichalcogenides promise great potential for future applications in optoelectronics and photonics as they combine atomic-scale thickness with pronounced light-matter coupling and sizable band gaps in the visible to near-infrared range. In this context, a quantitative and predictive description of the optical properties is of great importance. For the results summarized in this thesis, a self-consistent scheme was established to provide such a quantitative and predictive description for various semiconducting transition-metal dichalcogenide systems in the vicinity of the K/K' points. The theoretical framework combines an anisotropic dielectric model for the Coulomb potential in layered materials with gap equations for the ground-state renormalization, Dirac-Wannier equation to determine the excitonic properties, and Dirac-Bloch equations to access linear and nonlinear optical properties. The latter are formally equivalent to the semiconductor Bloch equations, that have proven to be reliable to compute the optical properties of various semiconductor systems for many years. Detailed differences arise from the relativistic framework, the massive Dirac Fermion model, that applies to transition-metal dichalcogenides. To account for the finite out-of-plane extension of the individual layers, a form factor was introduced in the Coulomb potential. The theoretical framework described above was applied in investigations on the ground-state and excitonic properties of monolayer and homogeneous-multilayer structures. For the case of an unspecified monolayer, the dielectric tuning of the renormalized bands and excitonic resonances was simulated by variation of the Coulomb coupling showing characteristics that are observed in experiments on real monolayer systems. Encouraged by the initial results, realistic monolayers were considered, i.e. MoS2, MoSe2, WS2, WSe2, whose material parameters were taken from external density-functional-theory calculations. The procedure to determine the effective-thickness parameter, entering the form factor to account for finite-thickness effects, was illustrated for a SiO2-supported MoS2 monolayer. Once this parameter was fixed for a given material, the advantage of this approach was demonstrated for MoS2, again, by predicting the K/K'-point interband transition energies and excitonic resonances for various dielectric environments and layer numbers, including the bulk limit. Comparisons to experimental findings and similar theoretical approaches were drawn for all of the stated material systems yielding almost excellent overall agreement. In particular, the results suggest a reinterpretation of the bulk exciton series of MoS 2 as a combined two-dimensional intra- and interlayer exciton series. The results strongly indicate that the applied approach captures the essential physics around the K/K' points. Stacking two materials with different band gaps adds a new element to the band-gap engineering of transition-metal dichalcogenides. Heterostructures such as bilayers WSe2/MoS2 and WSe2 /MoSe2 display type-II band alignment enabling highly efficient charge transfer which is promising for applications in photovoltaics. In a theoretical study on the stated bilayer systems, it was demonstrated that the established theoretical framework could also be applied to investigate intra- and interlayer excitons in transition-metal dichalcogenide heterostructures. For this purpose the anisotropic dielectric model for the Coulomb potential was adjusted to the hetero-bilayer environment. Based on the material parameters provided by internal density-functional-theory calculations, linear optical absorption spectra were computed revealing tightly bound interlayer excitons with binding energies comparable to those of the intralayer excitons. Computing the oscillator strength of the respective resonances yielded relatively long ratiative lifetimes for the interlayer excitons, two orders of magnitude larger than that of the intralayer excitons. The artificial strain in WSe2/MoS2 bilayer resulted in heavily misaligned spectra which is why theory-experiment comparisons were avoided for this system. For the rather unstrained WSe2/MoSe2 bilayer, intra- and interlayer excitonic resonances as well as the ratio of the intra- and interlayer exciton lifetimes compared reasonably well to experimental and theoretical findings. Among the semiconducting transition-metal dichalcogenides, monolayer MoS2 has drawn the most attention from researchers, not least because it was the first representative that displayed experimental evidence of a direct band gap. Combining the direct band gap with pronounced light-matter coupling, monolayer systems hold promise for laser applications on the atomic scale. In this context, the optical properties of suspended and SiO2-supported MoS2 monolayers were investigated in the nonlinear excitation regime for the case of initial thermal charge carriers located in the K/K' valleys. In particular, it was demonstrated that excited carriers lead to an enormous reduction of the band gap. In the range of comparable carrier densities, the computed optical spectra, excitation-induced band-gap renormalization and exciton binding energies were found to be in good agreement with earlier theoretical investigations on MoS2 , as was the predicted Mott-density. For densities beyond the Mott-transition, broadband plasma-induced optical gain energetically below the exciton resonance was observed, which has yet to be realized in experimental setups. Besides the canonical representatives discussed so far, the optical properties of a SiO2-supported MoTe2 monolayer were studied. This material system became of particular interest since room- temperature lasing had already been observed. A numerical experiment in the nonlinear excitation regime was performed. In particular, excitation conditions for achieving plasma gain in MoTe2 monolayers were identified. Within the scope of this investigation, the theoretical framework was extended beyond the quasiequilibrium regime by including Boltzmann-like carrier- and phonon-scattering rates. Whereas a Markovian treatment was sufficient within the simulation of the K/K'-point carrier-relaxation dynamics, the excitation-induced dephasing of the microscopic polarizations was treated dynamically in order to avoid unphysical behavior within the optical spectra. It was demonstrated that pump-injected charge carriers induce a huge reduction of the band gap on the timescale of the optical pulse. This observation including the magnitude of the band-gap renormalization compared well with experimental findings on monolayer MoS2 . Probing the strongly excited system at distinct time delays yielded ultrafast gain build-up on a few-picosecond timescale as a result of efficient carrier thermalization. Allowing the carriers to equilibriate within the entire Billouin zone, even larger output was predicted. This numerical experiment represents the first study proposing monolayer MoTe2 as a promising candidate to achieve plasma-induced optical gain

Boundary integral equations in Kinetic Plasma Theory

In this thesis, we use boundary integral equations (BIE) as a powerful tool to gain new insights into the dynamics of plasmas. On the theoretical side, our work provides new results regarding the oscillation of bounded plasmas. With the analytical computation of the frequencies for a general ellipsoid we contribute a new benchmark for numerical methods. Our results are validated by an extensive numerical study of several three-dimensional problems, including a particle accelerator with complex geometry and mixed boundary conditions. The use of Boundary Element Methods (BEM) reduces the dimension of the problem from three to two, thus drastically reducing the number of unknowns. By employing hierarchical methods for the computation of the occurring nonlocal sums and integral operators, our method scales linearly with the number of particles and the number of surface triangles, where the error decays exponentially in the expansion parameter. Furthermore, our method allows the pointwise evaluation of the electric field without loss of convergence order. As we are able to compute the occurring boundary integrals analytically, we can precisely predict the electric field near the boundary. This property makes our method exceptionally well suited for the numerical simulation of plasma sheaths near irregular boundaries or of plasma-surface interaction such as etching of semiconductors.In der vorliegenden Arbeit nutzen wir Randintegralgleichungen als ein mächtiges Werkzeug, um neue Einsichten in die Dynamik von Plasmen zu gewinnen. Auf theoretischer Seite entwickelt diese Arbeit neue Resultate bezüglich der Oszillation beschränkter Plasmen. Durch die ana- lytische Berechnung der Frequenzen im Fall eines allgemeinen Ellipsoids stellen wir ein neues Testbeispiel für numerische Methoden bereit. Unsere Resultate werden durch umfangreiche numerische Untersuchen dreidimensionaler Beispiele validiert, etwa einen Partikelbeschleuniger mit komplexer Geometrie und gemischten Randwerten. Mithilfe der Randelementmethode reduziert sich die Dimension des Problems von drei auf zwei, womit sich die Anzahl der Un- bekannten drastisch reduziert. Dank der Nutzung hierarchischer Methoden zur Berechnung der auftauchenden nichtlokalen Summen und Integraloperatoren skaliert unsere Methode linear mit der Anzahl der Partikel und der Anzahl der Oberflächendreiecken, wobei der Fehler exponen- tiell im Entwicklungsparameter abfällt. Des Weiteren erlaubt unsere Methode die Berechnung des elektrischen Felds ohne Verringerung der Konvergenzordnung. Da wir die auftretenden Randintegrale analytisch berechnen können, können wir präzise Aussagen über das elektrische Feld nahe des Rands treffen. Dank dieser Eigenschaft ist unsere Methode außergewöhnlich gut geeignet, um Plasmaränder nahe irregulärer Ränder oder Plasma-Oberflächen-Interaktionen, etwa das Ätzen von Halbleitern, zu simulieren

Learning Generative Models with Sinkhorn Divergences

The ability to compare two degenerate probability distributions (i.e. two probability distributions supported on two distinct low-dimensional manifolds living in a much higher-dimensional space) is a crucial problem arising in the estimation of generative models for high-dimensional observations such as those arising in computer vision or natural language. It is known that optimal transport metrics can represent a cure for this problem, since they were specifically designed as an alternative to information divergences to handle such problematic scenarios. Unfortunately, training generative machines using OT raises formidable computational and statistical challenges, because of (i) the computational burden of evaluating OT losses, (ii) the instability and lack of smoothness of these losses, (iii) the difficulty to estimate robustly these losses and their gradients in high dimension. This paper presents the first tractable computational method to train large scale generative models using an optimal transport loss, and tackles these three issues by relying on two key ideas: (a) entropic smoothing, which turns the original OT loss into one that can be computed using Sinkhorn fixed point iterations; (b) algorithmic (automatic) differentiation of these iterations. These two approximations result in a robust and differentiable approximation of the OT loss with streamlined GPU execution. Entropic smoothing generates a family of losses interpolating between Wasserstein (OT) and Maximum Mean Discrepancy (MMD), thus allowing to find a sweet spot leveraging the geometry of OT and the favorable high-dimensional sample complexity of MMD which comes with unbiased gradient estimates. The resulting computational architecture complements nicely standard deep network generative models by a stack of extra layers implementing the loss function

A GPU-accelerated Direct-sum Boundary Integral Poisson-Boltzmann Solver

In this paper, we present a GPU-accelerated direct-sum boundary integral method to solve the linear Poisson-Boltzmann (PB) equation. In our method, a well-posed boundary integral formulation is used to ensure the fast convergence of Krylov subspace based linear algebraic solver such as the GMRES. The molecular surfaces are discretized with flat triangles and centroid collocation. To speed up our method, we take advantage of the parallel nature of the boundary integral formulation and parallelize the schemes within CUDA shared memory architecture on GPU. The schemes use only $11N+6N_c$ size-of-double device memory for a biomolecule with $N$ triangular surface elements and $N_c$ partial charges. Numerical tests of these schemes show well-maintained accuracy and fast convergence. The GPU implementation using one GPU card (Nvidia Tesla M2070) achieves 120-150X speed-up to the implementation using one CPU (Intel L5640 2.27GHz). With our approach, solving PB equations on well-discretized molecular surfaces with up to 300,000 boundary elements will take less than about 10 minutes, hence our approach is particularly suitable for fast electrostatics computations on small to medium biomolecules

Near-optimal perfectly matched layers for indefinite Helmholtz problems

A new construction of an absorbing boundary condition for indefinite Helmholtz problems on unbounded domains is presented. This construction is based on a near-best uniform rational interpolant of the inverse square root function on the union of a negative and positive real interval, designed with the help of a classical result by Zolotarev. Using Krein's interpretation of a Stieltjes continued fraction, this interpolant can be converted into a three-term finite difference discretization of a perfectly matched layer (PML) which converges exponentially fast in the number of grid points. The convergence rate is asymptotically optimal for both propagative and evanescent wave modes. Several numerical experiments and illustrations are included.Comment: Accepted for publication in SIAM Review. To appear 201