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Local well-posedness and global stability of one-dimensional shallow water equations with surface tension and constant contact angle
Get PDFWe consider the one-dimensional shallow water problem with capillary surfaces and moving contact lines. An energy-based model is derived from the two-dimensional water wave equations, where we explicitly discuss the case of a stationary force balance at a moving contact line and highlight necessary changes to consider dynamic contact angles. The moving contact line becomes our free boundary at the level of shallow water equations, and the depth of the shallow water degenerates near the free boundary, which causes singularities for the derivatives and degeneracy for the viscosity. This is similar to the physical vacuum of compressible flows in the literature. The equilibrium, the global stability of the equilibrium, and the local well-posedness theory are established in this paper
Transition to Anomalous Dynamics in a Simple Random Map
No full textThe famous Bernoulli shift (or dyadic transformation) is perhaps the simplest deterministic dynamical system exhibiting chaotic dynamics. It is a piecewise linear time-discrete map on the unit interval with a uniform slope larger than one, hence expanding, with a positive Lyapunov exponent and a uniform invariant density. If the slope is less than one the map becomes contracting, the Lyapunov exponent is negative, and the density trivially collapses onto a fixed point. Sampling from these two different types of maps at each time step by randomly selecting the expanding one with probability , and the contracting one with probability , gives a prototype of a random dynamical system. Here we calculate the invariant density of this simple random map, as well as its position autocorrelation function, analytically and numerically under variation of . We find that the map exhibits a non-trivial transition from fully chaotic to completely regular dynamics by generating a long-time anomalous dynamics at a critical sampling probability , defined by a zero Lyapunov exponent. This anomalous dynamics is characterised by an infinite invariant density, weak ergodicity breaking and power law correlation decay
Primal-Dual Regression Approach for Markov Decision Processes with General State and Action Spaces
No full textWe develop a regression based primal-dual martingale approach for solving finite time horizon MDPs with general state and action space. As a result, our method allows for the construction of tight upper and lower biased approximations of the value functions, and, provides tight approximations to the optimal policy. In particular, we prove tight error bounds for the estimated duality gap featuring polynomial dependence on the time horizon, and sublinear dependence on the cardinality/dimension of the possibly infinite state and action space. From a computational point of view the proposed method is efficient since, in contrast to usual duality-based methods for optimal control problems in the literature, the Monte Carlo procedures here involved do not require nested simulations
Data-Driven Solutions of Ill-Posed Inverse Problems Arising from Doping Reconstruction in Semiconductors
No full textThe non-destructive estimation of doping concentrations in semiconductor devices is of paramount importance for many applications ranging from crystal growth to defect and inhomogeneity detection. A number of technologies (such as LBIC, EBIC and LPS) have been developed which allow the detection of doping variations via photovoltaic effects. The idea is to illuminate the sample at several positions and detect the resulting voltage drop or current at the contacts. We model a general class of such photovoltaic technologies by ill-posed global and local inverse problems based on a drift-diffusion system which describes charge transport in a self-consistent electrical field. The doping profile is included as a parametric field. To numerically solve a physically relevant local inverse problem, we present three approaches, based on least squares, multilayer perceptrons, and residual neural networks. Our data-driven methods reconstruct the doping profile for a given spatially varying voltage signal induced by a laser scan along the sample's surface. The methods are trained on synthetic data sets which are generated by finite volume solutions of the forward problem. While the linear least square method yields an average absolute error around 10%, the nonlinear networks roughly halve this error to 5%
Real-variable characterizations and their applications of matrix-weighted Triebel-Lizorkin spaces
No full textLet , , , and be an -matrix weight. In this article, the authors characterize the matrix-weighted Triebel-Lizorkin space via the Peetre maximal function, the Lusin area function, and the Littlewood-Paley -function. As applications, the authors establish the boundedness of Fourier multipliers on matrix-weighted Triebel-Lizorkin spaces under the generalized Hörmander condition. The main novelty of these results exists in that their proofs need to fully use both the doubling property of matrix weights and the reducing operator associated to matrix weights, which are essentially different from those proofs of the corresponding cases of classical Triebel-Lizorkin spaces that strongly depend on the Fefferman-Stein vector-valued maximal inequality on Lebesgue spaces
Utilizing Anatomical Information for Signal Detection in Functional Magnetic Resonance Imaging - Data
No full textThis is p-value data from a multi-subject program comprehension study mapped onto anatomical labels (APARC). The data can be used to reproduce the figures in the accompanying publication
On Loss Functionals for Physics-Informed Neural Networks for Steady-State Convection-Dominated Convection-Diffusion Problems
No full textSolutions of convection-dominated convection-diffusion problems usually possess layers, which are regions where the solution has a steep gradient. It is well known that many classical numerical discretization techniques face difficulties when approximating the solution to these problems. In recent years, physics-informed neural networks (PINNs) for approximating the solution to (initial-)boundary value problems ((I)BVPs) received a lot of interest. This paper studies various loss functionals for PINNs that are especially designed for convection-dominated convection-diffusion problems and that are novel in the context of PINNs. They are numerically compared to the vanilla and an hp-variational loss functional from the literature based on two steady-state benchmark problems whose solutions possess different types of layers. We observe that the best novel loss functionals reduce the L2(Ω) error by 17.3% for the first and 5.5% for the second problem compared to the methods from the literature
A Reproducing Kernel Hilbert Space Approach to Singular Local Stochastic Volatility McKean–Vlasov Models
No full textMotivated by the challenges related to the calibration of financial models, we consider the problem of numerically solving a singular McKean–Vlasov equation dXt=σ(t,Xt)XtvtE[vt|Xt]dWt, where W is a Brownian motion and v is an adapted diffusion process. This equation can be considered as a singular local stochastic volatility model. While such models are quite popular among practitioners, its well-posedness has unfortunately not yet been fully understood and in general is possibly not guaranteed at all. We develop a novel regularisation approach based on the reproducing kernel Hilbert space (RKHS) technique and show that the regularised model is well posed. Furthermore, we prove propagation of chaos. We demonstrate numerically that a thus regularised model is able to perfectly replicate option prices coming from typical local volatility models. Our results are also applicable to more general McKean–Vlasov equations
Massive Conscious Neighborhood-Based Crow Search Algorithm for the Pseudo-Coloring Problem
No full textThe pseudo-coloring problem (PsCP) is a combinatorial optimization challenge that involves assigning colors to elements in a way that meets specific criteria, often related to minimizing conflicts or maximizing some form of utility. A variety of metaheuristic algorithms have been developed to solve PsCP efficiently. However, these algorithms sometimes struggle with the quality of solutions, impacting their ability to achieve optimal or near-optimal results reliably. To overcome these issues, this study introduces an adapted conscious neighborhood-based crow search algorithm (CCSA) and a massive variant of CCSA specifically tailored for PsCP. The performance of CCSA and MCCSA are evaluated on real and synthetic images and compared with state-of-the-art optimizers. The results showed that the adapted CCSA and MCCSA outperformed offering an effective strategy for image pseudo-colorization