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Stokes problem with slip boundary conditions using stabilized finite elements combined with Nitsche
Get PDFWe discuss how slip conditions for the Stokes equation can be handled using Nitsche method, for a stabilized finite element discretization. Emphasis is made on the interplay between stabilization and Nitsche terms. Well-posedness of the discrete problem and optimal convergence rates, in natural norm for the velocity and the pressure, are established, and illustrated with various numerical experiments. The proposed method fits naturally in the context of a finite element implementation while being accurate, and allows an increased flexibility in the choice of the finite element pairs
Efficient Option Pricing in the Rough Heston Model using Weak Simulation Schemes
No full textWe provide an efficient and accurate simulation scheme for the rough Heston model in the standard () as well as the hyper-rough regime (). The scheme is based on low-dimensional Markovian approximations of the rough Heston process derived in [Bayer and Breneis, arXiv:2309.07023], and provides weak approximation to the rough Heston process. Numerical experiments show that the new scheme exhibits second order weak convergence, while the computational cost increases linear with respect to the number of time steps. In comparison, existing schemes based on discretization of the underlying stochastic Volterra integrals such as Gatheral's HQE scheme show a quadratic dependence of the computational cost. Extensive numerical tests for standard and path-dependent European options and Bermudan options show the method's accuracy and efficiency
Ontologies for Models and Algorithms in Applied Mathematics and Related Disciplines
No full textIn applied mathematics and related disciplines, the modeling-simulation-optimization workflow is a prominent scheme, with mathematical models and numerical algorithms playing a crucial role. For these types of mathematical research data, the Mathematical Research Data Initiative has developed, merged and implemented ontologies and knowledge graphs. This contributes to making mathematical research data FAIR by introducing semantic technology and documenting the mathematical foundations accordingly. Using the concrete example of microfracture analysis of porous media, it is shown how the knowledge of the underlying mathematical model and the corresponding numerical algorithms for its solution can be represented by the ontologies
Mathematical Modeling of Blood Flow in the Cardiovascular System
No full textThis chapter gives a short overview of the mathematical modeling of blood flow circulation at different resolutions, from the large vessel scale (three-dimensional, one-dimensional, and zero-dimensional modeling) to microcirculation, including aspects related to the biomechanics of vascular tissues and tissue perfusion. The chapter focuses first on the different mathematical formulations, discussing the underlying physical laws, the link to clinical data, and related challenges. Next, recent applications related to medical imaging are then discussed, highlighting the potential of computer simulation and of the interplay between modeling, imaging, and experiments in order to improve clinical diagnosis and treatment. The chapter ends presenting some current challenges and perspectives
Self-repellent Brownian bridges in an interacting Bose gas
Get PDFWe consider a model of -dimensional interacting quantum Bose gas, expressed in terms of an ensemble of interacting Brownian bridges in a large box and undergoing the influence of all the interactions between the legs of each of the Brownian bridges. We study the thermodynamic limit of the system and give an explicit formula for the limiting free energy and a necessary and sufficient criterion for the occurrence of a condensation phase transition. For and sufficiently small interaction, we prove that the condensate phase is not empty. The ideas of proof rely on the similarity of the interaction to that of the self-repellent random walk, and build on a lace expansion method conducive to treating it paths undergoing mutual repellence within each bridge
Statistics for the triangle density in ERGM and its mean-field approximation
Get PDFWe consider the edge-triangle model (or Strauss model), and focus on the asymptotic behavior of the triangle density when the size of the graph increases to infinity. In the analyticity region of the free energy, we prove a law of large numbers for the triangle density. Along the critical curve, where analyticity breaks down, we show that the triangle density concentrates with high probability in a neighborhood of its typical value. A predominant part of our work is devoted to the study of a mean-field approximation of the edge-triangle model, where explicit computations are possible. In this setting we can go further, and additionally prove a standard and non-standard central limit theorem at the critical point, together with many concentration results obtained via large deviations and statistical mechanics techniques. Despite a rigorous comparison between these two models is still lacking, we believe that they are asymptotically equivalent in many respects, therefore we formulate conjectures on the edge-triangle model, partially supported by simulations, based on the mean-field investigation
Hölder regularity for domains of fractional powers of elliptic operators with mixed boundary conditions
No full textThis work is about global Hölder regularity for solutions to elliptic partial differential equations subject to mixed boundary conditions on irregular domains. There are two main results. In the first, we show that if the domain of the realization of an elliptic differential operator in a negative Sobolev space with integrability q > d embeds into a space of Hölder continuous functions, then so do the domains of suitable fractional powers of this operator. The second main result then establishes that the premise of the first is indeed satisfied. The proof goes along the classical techniques of localization, transformation and reflection which allows to fall back to the classical results of Ladyzhenskaya or Kinderlehrer. One of the main features of our approach is that we do not require Lipschitz charts for the Dirichlet boundary part, but only an intriguing metric/measure-theoretic condition on the interface of Dirichlet- and Neumann boundary parts. A similar condition was posed in a related work by ter Elst and Rehberg in 2015 [10], but the present proof is much simpler, if only restricted to space dimension up to 4
Curvature effects in pattern formation: Well-posedness and optimal control of a sixth-order Cahn--Hilliard equation
Get PDFThis work investigates the well-posedness and optimal control of a sixth-order Cahn--Hilliard equation, a higher-order variant of the celebrated and well-established Cahn--Hilliard equation. The equation is endowed with a source term, where the control variable enters as a distributed mass regulator. The inclusion of additional spatial derivatives in the sixth-order formulation enables the model to capture curvature effects, leading to a more accurate depiction of isothermal phase separation dynamics in complex materials systems. We provide a well-posedness result for the aforementioned system when the corresponding nonlinearity of double-well shape is regular and then analyze a corresponding optimal control problem. For the latter, existence of optimal controls is established, and the first-order necessary optimality conditions are characterized via a suitable variational inequality. These results aim at contributing to improve the understanding of the mathematical properties and control aspects of the sixth-order Cahn--Hilliard equation, offering potential applications in the design and optimization of materials with tailored microstructures and properties
Polarized frequency combs in a mode-locked VECSEL
Get PDFIn this paper, we present a detailed and rigorous derivation of the delay differential equations of the spin-flip model for vertical external cavity lasers with a semiconductor saturable absorption mirror. This model describe mode-locked semiconductor lasers in the ring-resonator geometry with unidirectional lasing. This contribution completes a previous communication [Vladimirov et al. Opt. Lett., 45, 252 (2020)], and we further complete the analytical derivation by taking into account phase and amplitude anisotropies and the resulting different delay times for orthogonal linear polarizations. We show evidence of the coexistence of two linearly polarized frequency combs generation with slightly different repetition rates due to the birefringence-induced time-of- flight difference