Finite Volume approximation of a two-phase two fluxes degenerate Cahn-Hilliard model

We study a time implicit Finite Volume scheme for degenerate Cahn-Hilliard model proposed in [W. E and P. Palffy-Muhoray. Phys. Rev. E , 55:R3844–R3846, 1997] and studied mathematically by the authors in [C. Canc\`es, D. Matthes, and F. Nabet. Arch. Ration. Mech. Anal. , 233(2):837-866, 2019]. The scheme is shown to preserve the key properties of the continuous model, namely mass conservation, positivity of the concentrations, the decay of the energy and the control of the entropy dissipation rate. This allows to establish the existence of a solution to the nonlinear algebraic system corresponding to the scheme. Further, we show thanks to compactness arguments that the approximate solution converges towards a weak solution of the continuous problems as the discretization parameters tend to 0. Numerical results illustrate the behavior of the numerical model.

Multilevel Monte Carlo finite volume methods for random conservation laws with discontinuous flux

We consider conservation laws with discontinuous flux where the initial datum, the flux function, and the discontinuous spatial dependency coefficient are subject to randomness. We establish a notion of random adapted entropy solutions to these equations and prove well-posedness provided that the spatial dependency coefficient is piecewise constant with finitely many discontinuities. In particular, the setting under consideration allows the flux to change across finitely many points in space whose positions are uncertain. We propose a single- and multilevel Monte Carlo method based on a finite volume approximation for each sample. Our analysis includes convergence rate estimates of the resulting Monte Carlo and multilevel Monte Carlo finite volume methods as well as error versus work rates showing that the multilevel variant outperforms the single-level method in terms of efficiency. We present numerical experiments motivated by two-phase reservoir simulations for reservoirs with varying geological properties.

Modeling multi-class disordered traffic flow subject to varying vehicle composition using the concept of traversable distance

Disordered traffic stream at the microscopic level can be described as a permeable medium. Each vehicle is considered to traverse through a series of lateral gaps created by other vehicles. We develop a multi-class traffic flow model that considers such viable and accessible gaps for individual vehicle classes to traverse downstream. The model accounts for the varying shares of different vehicle classes. The concept of traversable distance and modified equilibrium speed functions are used to model the interplay among multiple vehicle classes. Using a higher order finite volume approximation method, evolution for a two-class traffic stream is shown. This model replicates prominent empirical characteristics exhibited by multi-class disordered traffic such as overtaking and creeping. Varying shares of smaller vehicles affect queue formation and discharge characteristics, and have significant impacts on roadway capacity. The model also could compute dynamic class-specific travel times at different vehicular compositions more realistically.

Sampling the feasible sets of SDPs and volume approximation

We present algorithmic, complexity, and implementation results on the problem of sampling points in the interior and the boundary of a spectrahedron, that is the feasible region of a semidefinite program. Our main tool is random walks. We define and analyze a set of primitive geometric operations that exploits the algebraic properties of spectrahedra and the polynomial eigenvalue problem and leads to the realization of a broad collection of efficient random walks. We demonstrate random walks that experimentally show faster mixing time than the ones used previously for sampling from spectrahedra in theory or applications, for example Hit and Run. Consecutively, the variety of random walks allows us to sample from general probability distributions, for example the family of log-concave distributions which arise frequently in numerous applications. We apply our tools to compute (i) the volume of a spectrahedron and (ii) the expectation of functions coming from robust optimal control. We provide a C++ open source implementation of our methods that scales efficiently up to dimension 200. We illustrate its efficiency on various data sets.

Applications of Artificial Intelligence to Prostate Multiparametric MRI (mpMRI): Current and Emerging Trends

Prostate carcinoma is one of the most prevalent cancers worldwide. Multiparametric magnetic resonance imaging (mpMRI) is a non-invasive tool that can improve prostate lesion detection, classification, and volume quantification. Machine learning (ML), a branch of artificial intelligence, can rapidly and accurately analyze mpMRI images. ML could provide better standardization and consistency in identifying prostate lesions and enhance prostate carcinoma management. This review summarizes ML applications to prostate mpMRI and focuses on prostate organ segmentation, lesion detection and segmentation, and lesion characterization. A literature search was conducted to find studies that have applied ML methods to prostate mpMRI. To date, prostate organ segmentation and volume approximation have been well executed using various ML techniques. Prostate lesion detection and segmentation are much more challenging tasks for ML and were attempted in several studies. They largely remain unsolved problems due to data scarcity and the limitations of current ML algorithms. By contrast, prostate lesion characterization has been successfully completed in several studies because of better data availability. Overall, ML is well situated to become a tool that enhances radiologists’ accuracy and speed.

Well-posedness of a non-local model for material flow on conveyor belts

In this paper, we focus on finite volume approximation schemes to solve a non-local material flow model in two space dimensions. Based on the numerical discretisation with dimensional splitting, we prove the convergence of the approximate solutions, where the main difficulty arises in the treatment of the discontinuity occurring in the flux function. In particular, we compare a Roe-type scheme to the well-established Lax–Friedrichs method and provide a numerical study highlighting the benefits of the Roe discretisation. Besides, we also prove the L1-Lipschitz continuous dependence on the initial datum, ensuring the uniqueness of the solution.