Young Researchers Conference Numerical Aspects of Hyperbolic Balance Laws and Related Problems University of Verona December 15-17, 2021

Schedule

Wednesday 15/12

Thursday 16/12

Friday 17/12

Dipartimento di Informatica SALA VERDE

Chiostro S. Maria delle Vittorie SALA MESSEDAGLIA

Chiostro S. Maria delle Vittorie SALA MESSEDAGLIA

G. Bertaglia

D. Peurichard

M. Schmidtchen

M. Conte

R. Bailo

S. Tozza

BREAK

BREAK

Liu Liu

F. Chiarello

A. Bondesan

E. Iacomini

LUNCH

LUNCH

LUNCH

G. Visconti

A. Coco

E. Gaburro

P. Bacigaluppi

G. Martalò

F. Zivcovich

BREAK

BREAK

A. Alla

N. Pouradier Duteil

R. Della Marca

N. Loy

M. Menci

Y. Zhu

09:00-9:35 09:35 - 10:10 10:10 - 10:45 10:45 - 11:15 11:15 - 11:50 11:50 - 12:25

12:25 - 14:00

14:00 - 14:35 14:35 - 15:10 15:10 - 15:45 15:45 - 16:15 16:15 - 16:50 16:50 - 17:25 17:25 - 18:00

20:00

SOCIAL DINNER

Abstracts HJB-RBF based approach for the control of PDEs Alessandro Alla (PUC-Rio, Brazil) Semi-lagrangian schemes for discretization of the dynamic programming principle are based on a time discretization projected on a state-space grid. The use of a structured grid makes this approach not feasible for high-dimensional problems due to the curse of dimensionality. Here, we present a new approach for infinite horizon optimal control problems where the value function is computed using Radial Basis Functions (RBF) by the Shepard’s moving least squares approximation method on scattered grids. We propose a new method to generate a scattered mesh driven by the dynamics and the selection of the shape parameter in the RBF using an optimization routine. This mesh will help to localize the problem and approximate the dynamic programming principle in high dimension. Error estimates for the value function are also provided. Numerical tests for high dimensional problems will show the effectiveness of the proposed method. Joint work with H. Oliveira (PUC-Rio, Brazil) and G. Santin (FBK, Italy).

Recent developments of structure preserving RD schemes for hyperbolic systems with multiphase applications Paola Bacigaluppi (Politecnico di Milano, Italy) A novel structure preserving finite element-type Residual Distribution (RD) scheme for time dependent hyperbolic problems with application to multiphase flows is presented. In particular, the goal of the method is to allow for high order of accuracy in smooth regions of the flow, while ensuring robustness and a non-oscillatory behaviour in the regions of steep gradients such as across shocks. For this purpose, we consider the high order residual distribution scheme [1] with the MOOD detection criteria [2, 3] and entropy correction [4, 5]. Specifically, a candidate solution is computed at a next time level via an entropy structure preserving high-order accurate scheme. A so-called detector determines if the candidate solution reveals any spurious oscillation or numerical issue and, if so, only the troubled cells are locally recomputed via a more dissipative scheme. This allows to design a family of ”a posteriori” limited, entropy dissipative, robust as well as highly accurate, non-oscillatory schemes. The capability of the scheme to be structure preserving is shown by means of benchmarks in the context of unsteady two-phase flows problems. This is a joint work with R. Abgrall and P. Offner

References [1] Abgrall et al. High-order residual distribution scheme for the time-dependent Euler equations of fluid dynamics. Computers and Mathematics with Applications, 2019. [2] Bacigaluppi et al. A Posteriori Limited High Order and Robust Residual Distribution Schemes for Transient Simulations of Fluid Flows in Gas Dynamics. arXiv:1902.07773, 2019.

[3] Bacigaluppi et al. Assessment of a non-conservative four-equation multiphase system with phase transition. arXiv:2105.12874, 2021. [4] Abgrall. A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes. JCP, 2018 [5] Abgrall et al. Reinterpretation and Extension of Entropy Correction Terms for Residual Distribution and Discontinuous Galerkin Schemes: Application to Structure Preserving Discretization. arXiv:1908.04556, 2019.

Bound-Preserving Finite-Volume Schemes for Systems of Continuity Equations with Saturation Rafael Bailo (University of Lille, France) We propose finite volume schemes for general continuity equations which preserve positivity and global bounds arising from saturation effects in the mobility function. In the particular case of gradient flows, they dissipate the free energy at the fully discrete level. Moreover, these schemes are generalized to coupled systems of non-linear continuity equations, such as multispecies models in mathematical physics or biology, preserving the bounds and the dissipation of the energy whenever they enjoy a gradient flow structure. These results are illustrated through extensive numerical simulations; in particular, we investigate cell-cell adhesion models in mathematical biology with a mobility derived from volume constraints due to the total cell population.

Hyperbolic models for the spatial spread of infectious diseases under uncertain data: kinetic description and numerical methods Giulia Bertaglia (University of Ferrara, Italy) Standard compartmental epidemiological models describe the spread of epidemics only with respect to the temporal evolution of the infection among the population, but not taking into account spatial effects. In many cases, the concept of the average behavior of a large population is sufficient to provide useful guidance on the development of an epidemic; however, the importance of the spatial component is being increasingly recognized, especially when there is the need to consider spatially heterogeneous interventions, as in the case of the ongoing COVID-19 pandemic. To permit an effective design of confinement strategies, we introduce multiscale hyperbolic transport models for the propagation of an epidemic phenomenon described by the diffusive behavior of the non-commuting part of the population acting only over an urban scale and the spatial movement and interaction of commuters moving also on an extra-urban scale, based on kinetic equations. The presence of a group of non-commuting individuals permits to avoid unrealistic mass migration effects in which the whole population in a compartment moves indiscriminately

in the spatial domain, which can be either structured as a network, whose nodes identify cities of interest and arcs represent common mobility paths, or represent realistic 2D geographical regions. Moreover, since data of the spread of epidemics are generally highly heterogeneous and affected by a great deal of uncertainty, we choose to transmit statistical information to the problem, related to random input parameters such as the initial amount of infected people. The resulting model is solved numerically through a suitable stochastic Asymptotic-Preserving Implicit-Explicit Finite Volume Collocation Method, robust enough to deal with the presence of multiple scales and the uncertainty quantification process. Several numerical results show that the proposed methodology correctly defines the main features of the spatial spread of infectious diseases, particularly focusing on the simulation of the onset of COVID-19 pandemic in the northern Italy. This is a joint research with W. Boscheri (University of Ferrara), G. Dimarco (University of Ferrara) and L. Pareschi (University of Ferrara).

References [1] G. Bertaglia, W. Boscheri, G. Dimarco, L. Pareschi. Spatial spread of COVID-19 outbreak in Italy using multiscale kinetic transport equations with uncertainty. Math. Biosci. Eng., to appear. [2] G. Bertaglia, L. Pareschi. Hyperbolic compartmental models for epidemic spread on networks with uncertain data: application to the emergence of Covid-19 in Italy. Math. Mod. and Meth. in Appl.Scie., to appear. [3] G. Bertaglia, L. Pareschi. Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods. ESAIM Math. Model. and Numer. Anal., 55, 381–407, 2020.

A numerical scheme for the Boltzmann multi-species equation in the diffusion limit: well-posedness and main properties Andrea Bondesan (University of Graz, Austria) We consider the Boltzmann multi-species equation set in a standard diffusive scaling where the Mach and Knudsen numbers are of the same order of magnitude  > 0 small enough. For each species i of the mixture we define its macroscopic concentration and flux as the moments 0 and 1 in velocity of the distribution functions fi , solutions of the Boltzmann equation. Introducing a specific ansatz on the form of these fi we apply the moment method to recover an Euler-MaxwellStefan system describing the evolution of the associated macroscopic quantities. In a one-dimensional setting we then build [1] a suitable numerical scheme to approximate this system in different regimes of the parameter . We prove some a priori estimates (mass conservation and nonnegativity) and well-posedness of the discrete problem. We also present numerical examples where we observe that the scheme shows an asymptotic-preserving property similar to the one presented in [2].

References [1] A. Bondesan, L. Boudin and B. Grec. A numerical scheme for a kinetic model for mixtures in the diffusive limit using the moment method. Numer. Methods Partial Differential Equations, 45, 1184–1205, 2019. [2] S. Jin and Q. Li. A BGK-penalization-based asymptotic-preserving scheme for the multispecies Boltzmann equation. Numer. Methods Partial Differential Equations, 29, 1056–1080, 2013.

A statistical mechanics approach to macroscopic limits of car-following traffic dynamic Felisia Angela Chiarello (Politecnico di Torino, Italy) We will derive macroscopic traffic models from car-following vehicle dynamics by means of hydrodynamic limits of an Enskog-type kinetic description. In particular, we will consider the superposition of Follow-the-Leader (FTL) interactions and relaxation towards a traffic-dependent Optimal Velocity (OV) showing that the resulting macroscopic models depend on the relative frequency between these two microscopic processes. If FTL interactions dominate then one gets an inhomogeneous Aw-Rascle-Zhang model, whose (pseudo) pressure and stability of the uniform flow are precisely defined by some features of the microscopic FTL and OV dynamics. Conversely, if the rate of OV relaxation is comparable to that of FTL interactions then one gets aLighthill-Whitham-Richards model ruled only by the OV function.

A geometric multigrid approach for complex-shaped domains on Cartesian grids: a Local Fourier Analysis for curved boundaries Armando Coco (Oxford Brookes University, UK) Multigrid (MG) method is an iterative solver among the most efficient approaches to solve discrete PDEs. In complex-shaped time-dependent domains, it is well known that boundary effects can propagate from the boundary to the overall domain and then degrade the efficiency of the entire MG if a proper treatment is not adopted. A common practice in literature is to relax the boundary conditions with a fictitious time step and eventually add extra relaxations to keep the optimal MG convergence factor. If the shape of the domain is highly complicated, the extra computational work is not negligible. To overcome this issue, an improved technique will be proposed in this talk by adopting a local fictitious time step that varies along the boundary and that is determined by optimising the boundary relaxation parameters through a Local Fourier Analysis in order to maximise the smoothing property along the tangential direction. We present a numerical approach

where the boundary is embedded in a steady grid, avoiding the computational burden of boundary-fitted grid methods where a mesh is generated at each time step. The domain is represented by level set functions, allowing greater flexibility. The method is based on ghost-point extrapolation and shows second order accuracy and optimal multigrid convergence factor. Applications to fluid dynamics and monument degradation are presented.

A multiscale approach to glioma progression: phenotypic heterogeneity, vasculature, and acidity effects Martina Conte (Politecnico di Torino, Italy) Gliomas are malignant brain tumors arising from mutations in the glia cells of the central nervous system. Their growth and migration inside the brain is a highly complex phenomenon, influenced by a multitude of intrinsic and extrinsic factors, which are responsible for the typical features of tumor aggressiveness and invasiveness. Here, we propose a multiscale model for the description of glioma progression with a specific focus on the influence of vasculature, (hypoxia-driven) acidity, and phenotypic heterogeneity on tumor evolution. Starting from a microscopic description of the interactions between cell membrane receptors, healthy tissue, and extracellular protons, we define a system of coupled kinetic transport equations for endothelial (ECs) and tumor cells. In particular, relying on the go-or-grow hypothesis, phenotypic heterogeneity is modeled for the tumor population. Using a parabolic scaling, we derive the macroscopic system that describes tumor and ECs evolution, and we couple it with the equations accounting for proton dynamics, healthy tissue degradation, and necrotic region formation. We analyze the system evolution by performing several numerical tests that aim at assessing the role of vasculature, acidity, and phenotypic heterogeneity in tumor invasion and progression [1]. Joint work with C. Surulescu (TU Kaiserslautern).

References [1] M. Conte and C. Surulescu Mathematical modeling of glioma invasion: acidand vasculature mediated go-or-grow dichotomy and the influence of tissue anisotropy. Appl. Math. Comput., 407, 126305, 2021.

On the optimal control of SIR models: two novel frameworks Rossella Della Marca (SISSA, Italy) Optimal control theory applied to Susceptible-Infected-Recovered (SIR) models has been widely used to identify effective strategies for minimizing the impact of infectious diseases.

However, to the best of our knowledge, no attempts have been made to investigate two cases with significant real–world applications: i) the case that the infectious period is Erlang–distributed, implying that the chance for an infected individual to recover depends on the time since infection, as it has been documented for a wide class of infectious diseases [2]; ii) the case that the costs of the epidemics are related not only to epidemic size, but also to epidemic duration. Indeed, the minimization of outbreaks duration is a priority when the imposed sanitary restrictions involve travel bans (in human diseases) and export bans (in livestock diseases) [1]. For addressing the case (i), we implemented the method of stages in an SIR model [3], by splitting the infected compartment in n fictitious stages in series (n ∈ N+ ); infected individuals at any stage are isolated at the rate that jointly minimizes the costs associated to the control efforts and the epidemic size. As regards the case (ii), we built a time–optimal control problem for an SIR model using two alternative control strategies: vaccination of susceptible individuals or isolation of infected individuals. By applying the Pontryagin’s minimum principle [4], we proved that both the optimal control problems admits only bang–bang solutions with at most two switches. The analytical results are supported by extensive numerical simulations obtained by a simple ad hoc numerical scheme.

References [1] ] L. Bolzoni, E. Bonacini, R. Della Marca and M. Groppi. Optimal control of epidemic size and duration with limited resources. Math. Biosci., 315, 108232, 2019. [2] L. Bolzoni, R. Della Marca and M. Groppi. On the optimal control of SIR model with Erlang–distributed infectious period: isolation strategies. J. Math. Biol., to appear. [3] D.R. Cox and H.D. Miller. The Theory of Stochastic Processes, Chapman and Hall, London, 1965. [4] L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mishchenko. The Mathematical Theory of Optimal Processes. International Series of Monographs in Pure and Applied Mathematics, Interscience Publishers, Los Angeles, USA, 1962.

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes Elena Gaburro (INRIA - Bordeaux-Sud-West, France) In this talk, we present a new family of high order accurate direct ArbitraryLagrangianEulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes on moving Voronoi meshes that, at each time step, are regenerated and are even free to change their topology [1, 2]. The Voronoi tessellations are obtained from a set of generator points moving with the fluid flow; their shape and their connectivity are rearranged in order to maintain a high quality mesh even for long computational times, vortical phenomena and moving boundaries. Then, the old and new elements associated to the same generator point

are connected in space–time to construct the so-called space-time control volumes, whose bottom and top faces may be different polygons; also degenerating sliver elements are incorporated in order to fill the space–time holes that arise by the topology changes. The final ALE FV-DG scheme is obtained using a high order accurate fully discrete one-step ADER scheme [3], which has been adapted to Voronoi and sliver elements: by integrating over arbitrary closed space-time elements, and by using a space-time conservation formulation of the governing hyperbolic PDE system, we directly evolve the solution in time satisfying the GCL by construction and being conservative thanks to the careful treatment of the space–time holes. Finally, a particular attention will be devoted to the description of a novel physics based WENO reconstruction procedure [4].

References [1] E. Gaburro A Unified Framework for the Solution of Hyperbolic PDE Systems Using High Order Direct Arbitrary-Lagrangian–Eulerian Schemes on Moving Unstructured Meshes with Topology Change. Archives of Computational Methods in Engineering, 2021. [2] E. Gaburro, W. Boscheri, S. Chiocchetti, C. Klingenberg, V. Springel and M. Dumbser. High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes. Journal of Computational Physics, 2020. [3] M. Dumbser, D.S. Balsara, E.F. Toro, C.-D. Munz. A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. Journal of Computational Physics, 2008. [4] S. Chiocchetti, E. Gaburro. Direct Arbitrary-Lagrangian-Eulerian ADER schemes on Voronoi meshes with physics based WENO stencil selection. in preparation.

Hyperbolic stochastic Galerkin formulation for traffic flow models Elisa Iacomini (RWTH Aachen University, Germany) Although traffic models have been extensively studied, obtaining reliable forecast from these models is still challenging, since the evolution of traffic is also exposed to the presence of uncertainties. In this talk, we will investigate the propagation of uncertainties in traffic flow models, especially in macroscopic second order models applying the stochastic Galerkin approach. Hyperbolic preserving stochastic Galerkin formulations are presented in conservative form, and for smooth solutions also in the corresponding non-conservative form. This allows one to obtain stabilization results, when the system is relaxed to a first-order model [2]. We will illustrate the theoretical results with numerical simulations.

References [1] S. Gerster, M. Herty, E. Iacomini. Stability analysis of a hyperbolic stochastic Galerkin formulation for the Aw-Rascle-Zhang model with relaxation. Mathematical Biosciences and Engineering, 2021. [2] M. Herty, G. Puppo, S. Roncoroni, G. Visconti. The BGK approximation of kinetic models for traffic. Kinetic and Related Models, 2020. [3] S. Jin and R. Shu. A study of hyperbolicity of kinetic stochastic Galerkin system for the isentropic Euler equations with uncertainty. Chinese Annals of Mathematics, Serie B, 2019.

A bi-fidelity method for a class of kinetic models with uncertain parameters Liu Liu (University of Honk Kong, China) In this talk, we first introduce the bi-fidelity stochastic collocation method in uncertainty quantification, then adopt it to solve different kinetic models with random parameters and multiple scalings. We use the Boltzmann, the linear transport, and epidemic transport models with random parameters as examples to illustrate our idea, with different strategies of choosing the low-fidelity models in our bi-fidelity approximation. A formal uniform-inKnudsen number error estimate, practical error bound and numerical experiments will be presented to demonstrate the accuracy and efficiency of the proposed method. These are joint works with Giulia Bertaglia, Lorenzo Pareschi and Xueyu Zhu.

Direction-Dependent Turning Leads to Anisotropic Diffusion and Persistence Nadia Loy (Politecnico di Torino, Italy) Cells and organisms follow aligned structures in their environment, a process that cang enerate persistent migration paths. Kinetic transport equations are a popular modelling tool for describing biological movements at the mesoscopic level, yet their formulations usually assume a constant turning rate. Here we relax this simplification, extending to include a turning rate that varies according to the anisotropy of a heterogeneous environment. We extend known methods of parabolic and hyperbolic scaling and apply the results to cell movement on micropatterned domains also through numerical simulation of the transport model. We show that inclusion of orientation dependence in the turning rate can lead to persistence of motion in an otherwise fully symmetric environment, and generate enhanced diffusion in structured domains.

References [1] N.Loy, T. Hillen and K.J. Painter. Direction-Dependent Turning Leads to Anisotropic Diffusion and Persistence. European Journal of Applied Mathematics, 1–37, 2021.

Regular singularities and multiple sub-shocks in moment closure hierarchy Giorgio Martal` o (University of Parma, Italy) Although the Navier-Stokes equations are widely used, they do not satisfactorily describe the shock structure, since they do not correctly reproduce the thickness decrease of the narrow region of transition. For this reason, some moment equation models have been used, but since Grad’s work they have immediately shown some limitations; in fact, due to its hyperbolic structure, 13-moment description provides a continuous shock structure up to a certain Mach number (M = 1.65), and from this value on the solution exhibits a non-physical discontinuity (subshock). To improve such result, it was thought to consider hyperbolic models in extended thermodynamics involving a higher number of moments. Unfortunately, this approach does not allow to significantly extend the Mach number range that guarantees a continuous solution; moreover, additional singularities due to other characteristic speeds can occur. In this presentation, by means of a geometrical approach, we want to discuss such singularities for different moment closures.

Collective motions of birds: an all-leader agent-based model for turning and flocking Marta Menci (IAC-CNR, Italy) Collective motions of birds are widely known nature’s delights. In this talk we focus on a novel mathematical model specifically designed to reproduce self-organized spontaneous sudden changes of direction, not caused by external stimuli like predator’s attacks. Starting from recent experimental observations of starlings and jackdaws, we propose a minimal agent-based model for bird flocks based on a system of second-order delayed stochastic differential equations with discontinuous (both in space and time) right-hand side. The main novelty of the model is that every bird is a potential turn initiator, thus leadership is formed in a group of indistinguishable agents. From a theoretical point of view, we prove that the initial value problem associated to the model is well-posed. Numerical simulations of 3D scenarios, based on finite difference schemes, will be shown, and biological insights will be discussed.

References [1] E. Cristiani, M. Menci, M. Papi and L. Brafman. An all-leader agent-based model for turning and flocking birds J. Math. Biol., 83(4), 1–22, 2021.

A new model for the emergence of vascular networks Diane Peurichard (INRIA - Paris, France) The generation of vascular networks is a long standing problem which has been the subject of intense research in the past decades, because of its wide range of applications (tissue regeneration, wound healing, cancer treatments etc). The mechanisms involved in the formations of vascular networks are complex and despite the vast amount of research devoted to it there are still many mechanisms involved which are poorly understood. Our aim is to bring insight into the study of vascular networks by defining heuristic rules, as simple as possible, and to simulate them numerically to test their relevance in the vascularization process. We introduce a hybrid agent-based/continuum model coupling blood flow, oxygen flow, capillary network dynamics and tissues dynamics. We assimilate the tissue and extra cellular matrix as a porous medium, using Darcy’s law for describing both blood and intersticial fluid flows. Oxygen obeys a convection-diffusion reaction equation describing advection by the blood, diffusion and consumption by the tissue. We provide two different, biologically relevant geometrical settings and numerically analyze the influence of each of the capillary creation mechanism in detail. All mechanisms seem to concur towards a harmonious network but the most important ones are those involving oxygen gradient and sheer stress.

Macroscopic Limits of collective dynamics with time-varying weights Nastassia Pouradier Duteil (INRIA - Paris, France) In this talk, we will discuss two macroscopic limits of a collective dynamics model with time-varying weights. In the classical mean-field limit, the limit equation is a transport equation with source, where the (non-local) transport term corresponds to the position dynamics, and the (non-local) source term comes from the weight redistribution among the agents. We will show existence and uniqueness of the solution and convergence of the microscopic system to this mean-field equation by introducing a new empirical measure (in the position space) taking into account the weights. This mean-field limit can be derived only if the particle dynamics preserve indistinguishability. If they do not, another point of view consists of deriving the so called graph limit. We will introduce the graph limit, show the convergence of the microscopic model to the graph limit equation, and show the subordination of the mean-field limit to the graph limit equation.

Convergence of a Fully Discrete and Energy-Dissipating Finite-Volume Scheme for Aggregation-Diffusion Equations Markus Schmidtchen (University Dresden, Germany) We study an implicit finite-volume scheme for non-linear, non-local aggregationdiffusion equations which exhibit a gradient-flow structure, recently introduced. Crucially, this scheme keeps the dissipation property of an associated fully discrete energy, and does so unconditionally with respect to the time step. Our main contribution in this work is to show the convergence of the method under suitable assumptions on the diffusion functions and potentials involved. (This is a joint work with R. Bailo, J. A. Carrillo, and H. Murakawa)

Multidimensional smoothness indicators for first-order Hamilton-Jacobi equations Silvia Tozza (University of Napoli, Italy) The lack of smoothness is a common feature of weak solutions of nonlinear hyperbolic equations and is a crucial issue in their approximation. This has motivated several efforts to define appropriate indicators, based on the values of the approximate solutions, in order to detect the most troublesome regions of the domain. This information helps to adapt the approximation scheme in order to avoid spurious oscillations when using high-order schemes. In this work we propose a genuinely multidimensional extension of the WENO procedure in order to overcome the limitations of indicators based on dimensional splitting. Our aim is to obtain new regularity indicators for problems in 2D and apply them to a class of “adaptive filtered” schemes for first order evolutive Hamilton-Jacobi equations. These filtered schemes are obtained by coupling a high-order (possibly unstable) scheme and a monotone one. The mixture is governed by a filter function and by a positive switching parameter which goes to zero as the time and space steps are going to 0. The adaptivity is related to the smoothness indicators and allows to automatically tune the switching parameter in time and space. Several numerical tests on critical situations are presented and confirm the effectiveness of the proposed indicators and the efficiency of our scheme. Joint work with M. Falcone and G. Paolucci.

Quinpi: integrating conservation laws with CWENO implicit methods Giuseppe Visconti (University of Roma “La Sapienza”, Italy) Many interesting applications of hyperbolic systems of equations are stiff, and require the time step to satisfy restrictive stability conditions. One way to avoid small time steps is to use implicit time integration. Implicit integration is quite straightforward for first order schemes. High order schemes instead need also to control spurious oscillations, which requires limiting in space and time also in the implicit case. We propose a framework to simplify considerably the application of high order non oscillatory schemes through the introduction of a low order implicit predictor, which is used both to set up the nonlinear

weights of a standard high order space reconstruction, and to achieve limiting in time. In this talk, we concentrate on the case of a third order scheme, based on DIRK integration in time and CWENO reconstruction in space.

Application of Fokker Planck Type Equations in Machine Learning Yuhua Zhu (Stanford University, California (USA)) I will talk about four different applications of Fokker Planck equations in machine learning. In the first one, we approximate the probability density function of the asynchronous stochastic gradient descent (ASGD) algorithm by a Fokker Planck equation. We prove a sharp convergence rate for the algorithm, which guides the tuning for ASGD. In the second one, we demonstrate that resampling outperforms reweighting when combined with stochastic gradient algorithms, using tools from dynamical stability and stochastic asymptotics. In the third one, we improved the consensus-based optimization method, where the mean-field limit of the new method is a non-linear Fokker Planck equation. We prove that the convergence rate of the method is independent of dimensionality. In the last project, we propose a new method to alleviate the double sampling problem in model-free reinforcement learning, where the Fokker Planck equation is used to do error analysis for the algorithm.

Numerical aspects of solving the sine-Gordon equation Franco Zivcovich (Sorbonne University, France) When approaching for the first time to the task of numerically solving the sine-Gordon equation ∂tt u − ∆u + m2 u = sin(u), ∂t u(0) = x, u(0) = v, one does not immediately figure out what are the challenges that this equation poses. This is especially true for those situations where standard numerical techniques such as Fourier space discretization are not applicable and one is forced to actually deal with the very structure of this equation. In this talk, we take a detour among the pitfalls and troubles characterizing this seemingly innocuous equation. This give us the occasion to sketch and apply cutting edge techniques arising in modern numerical analysis. In particular, we use Finite Elements methods for space discretization, then compare low-regularity exponential-type integrators with imex methods for time marching.

Organizing Committee: G. Albi, F. Ferrarese, S. Grassi, M. Zanella

Scientific Committee: G. Albi, W. Boscheri, G. Dimarco, L. Pareschi, G. Toscani, A. Tosin, M. Zanella

Website:

https://youngnumaspconf.wordpress.com/

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