This project studies evolutionary problems in several fields, with the tools of numerical analysis, scientific computing and mathematical analysis. The common denominator is the need to address complexity in evolutionary PDEs.
We will study evolution equations in random media, well posedness at junctions and the construction of solutions on networks. We will consider hyperbolic and parabolic equations as limits of kinetic models or arising from control problems, Mean Field Games or population dynamics.
Particular care will be devoted to the construction of new numerical methods to solve multiscale PDEs and to the use of accurate numerical computations to gain insight into the solution of complex models, in order to drive rigorous mathematical analysis from simple toy problems to more complex equations.
Numerical methods developed within this project range from Monte Carlo methods for kinetic problems, implicit methods for hyperbolic PDEs, Semi Lagrangian schemes, Reduced basis methods for PDEs in high dimensional spaces.
The theory and the new numerical techniques will be applied to several problems with a strong impact, such as vehicular traffic models, in particular considering the effects of autonomous vehicles, and problems from biology, in particular cancer growth in healthy tissues, and epidemiological models.
1. Hamilton Jacobi.
The proposed research is a continuation of the study begun in [D1,DaKo20]. The homogenization result we proved in [DaKo20] can be seen as a generalization of the work [KoYZ20], where the analysis was based on probabilistic techniques, and only applied to functions G of the form G(p):=|p|^{2}-c|p| with c>0 constant. In [DaKo20] we brought in a new view-point by making a more substantial use of PDE tools, which allowed us to simplify the arguments and to extend the homogenization results to functions G of more general form. We are confident that a further refinement of these arguments could allow us to extend the results to non-convex G and potentials V of fairly general form.
2. Kinetic models and MFG
We will model the interplay between human driven and automated cars, as well as the optimization of multilane flow, using kinetic models for mixtures and microscopic models, as in [P14, P13]. Models of this type can have an important impact on the future of traffic. Further, the study of unstable phenomena through forward-backward diffusion is new in convection-diffusion equations. These phenomena are well known in image segmentation [PeMa90], much less in transport equations.
We plan to study first order time-dependent MFGs on junctions and on networks. Progresses in this field will have a positive impact in applications, especially in deep learning models. Few results are known so far. The second order case has been recently treated in [ADLT20], where existence and uniqueness of solutions have been proved.
3. Control.
The new coupling between the Tree Structure Algorithm and Model Reduction tries to take all the advantages from its two building blocks since the TSA allows to drop the interpolation step in the approximation of the value function and the coupling with MR reduces the number of variables, thus yielding a substantial reduction in computational complexity.
4. Network analysis.
Both the total communicability and the structural robustness of a network depend on the conditioning of the Perron root. Exploring with the aid of approximate pseudospectra the sensitivity of such network measures should help identify edges whose perturbation or removal may significantly enhance communicability or robustness. This approach is new and promising.
HJ and CL on networks.
Our aim is to link time dependent HJ equations and conservation laws with the same Hamiltonians in a direct geometrical way, without passing through the addition of a vanishing viscosity. The idea is to better understand the role of flow limiters at the vertices of a network, which are necessary for the well posedness of HJ equations and CL, see [GNPT07]. Accordingly, we wish to investigate the connections and possible combinations of numerical schemes for HJ and CL, based on the links between flux limiters in HJ and Riemann solvers in LC.
Epidemiology on networks.
From one side, rigorous results can be obtained using mathematical analysis techniques for small networks, exploring the existence of underlying Lyapunov functionals. At the same time, numerical solutions can provide insights, which are out-of-reach for a complete theoretical approach, in the case of large networks starting from real and reliable data.
5. Hyperbolic and parabolic problems.
We propose to design implicit schemes for stiff hyperbolic systems based on linear predictors. They would considerably increase the efficiency of present high order solvers, as [AHZK20], and be relevant in applications, such as low Mach flow [P5,P15] or steady state computations. In fact, the Jacobian of the non linear system would be simplified, thanks to the linear predictor.
Bounded domains.
The methods in [Rouy92, Abgr03,AcFa12] can be applied to approximate HJB equations in bounded domains in particular cases. One goal is to extend SL schemes to bounded domains by mimicking the behaviour of the optimal stochastic differential equation, and reflecting in a proper way the discrete characteristics inside the domain.
We will also try to obtain convergence results for the extension of filtered schemes to second order problems with boundary conditions.
Front propagation.
Existence and stability results for traveling waves can be expected for specific models with a simplified underlying dynamics, such as Fisher-KPP, Allen-Cahn. Further, the problem can be explored also through numerical solutions, especially as the size of the system increases, preventing the possibility of studying the system analytically.
6. Population dynamics.
The interaction of competing species on planar domains has become the object of an extensive investigation ([CTVe06], [L2],[LaMo21]). We plan to deal with analogous problems on bounded regions in space, analyzing the behavior of the competing species and the shape of the surfaces which bound the spatial niches of the populations. Further, we plan to study a modified model with the addition of drift terms.