# Registered Data

## [00941] Numerical methods for Hamilton-Jacobi equations and their applications

**Session Date & Time**:- 00941 (1/4) : 3D (Aug.23, 15:30-17:10)
- 00941 (2/4) : 3E (Aug.23, 17:40-19:20)
- 00941 (3/4) : 4C (Aug.24, 13:20-15:00)
- 00941 (4/4) : 4D (Aug.24, 15:30-17:10)

**Type**: Proposal of Minisymposium**Abstract**: Hamilton-Jacobi equations arise both in modeling front propagation and in decision-making processes (optimal control & differential games). As a result, they have a broad range of applications including seismic imaging, robotic navigation, photolithography, transportation engineering, optimization of medical treatment policies, and data science. Practical usefulness of HJ-based approaches hinges on efficient and accurate numerical methods, which often need to handle anisotropy, degenerate diffusion, and possible discontinuity of viscosity solutions. High-dimensional problems and HJ-based inverse problems pose additional computational challenges. This mini-symposium will focus on recent advances in numerics for HJ PDEs and their innovative use in a variety of applications.**Organizer(s)**: Samuel Potter, Alexander Vladimirsky, Hasnaa Zidani**Classification**:__49Lxx__,__49Mxx__,__65Nxx__,__49Nxx__,__86A22__**Speakers Info**:**Alexander Vladimirsky**(Cornell University)- MingYi Wang (Cornell University)
- Adriano Festa (Politecnico di Torino)
- Peter Dower (University of Melbourne)
- Jérôme Fehrenbach (Institut de Mathématiques de Toulouse)
- Jeff Calder (University of Minnesota)
- Qi Gong (UC Santa Cruz)
- Jerome Darbon (Brown University)
- Marianne Akian (Ecole Polytechnique)
- Yong-Tao Zhang (University of Notre Dame)
- Olivier Bokanowski (University Paris Cité)
- Songting Luo (Iowa State University)
- Umaid Bin Waheed (King Fahd University of Petroleum and Minerals)

**Talks in Minisymposium**:**[01729] HJ equations in optimizing system-level performance objectives of Evolutionary Game Theory models****Author(s)**:**Alexander Vladimirsky**(Cornell University)

**Abstract**: Evolutionary Game Theory models time-dependent competitions of "types" or "strategies" in a population. EGT can be used to model the natural selection in biological systems or evolving behavioral patterns among humans. Controlling EGT-models to optimize some system-level performance measures can be accomplished through solving HJ equations. We illustrate this by optimizing drug therapies with an EGT-based model of cancer dynamics. Parts of this talk reflect joint work with Mark Gluzman, MingYi Wang, and Jake Scott.

**[02691] Maximizing the probability of desirable outcomes in Hamilton-Jacobi framework****Author(s)**:**MingYi Wang**(Cornell University)

**Abstract**: We introduce new tools for robust stochastic control of indefinite-horizon processes. In particular, we maximize the probability of keeping the (random) cumulative cost under any specific threshold. Our approach yields a 2nd-order HJB equation and “threshold-aware” optimal policies recovered for all initial configurations and a range of threshold values. We illustrate this method using examples from drug therapy optimization and sailboat path-planning. Joint work with A. Vladimirsky, J. Scott, and REU students at Cornell University.

**[03127] Data assimilation for the eikonal equation on a manifold****Author(s)**:**Jerome Fehrenbach**(Institut de Mathematiques de Toulouse)- Lisl Weynans (University of Bordeaux)

**Abstract**: We propose a method to determine the source(s) and principal direction of a front propagation on an anisotropic surface, from indirect nonlinear measurements. This model aims at describing the propagation of electrical waves at the surface of the heart. The framework of variational data assimilation leads to minimize a quadratic cost-function on a manifold. The Gauus-Newton algorithm is implemented using the Exp_x map on this manifold.

**[03171] Hamilton-Jacobi equations on graphs with applications to data depth and semi-supervised learning****Author(s)**:**Jeff Calder**(University of Minnesota)- Mahmood Ettehad (Institute for Mathematics and its Applications (IMA))

**Abstract**: Shortest path graph distances are widely used in data science and machine learning, however, they can be highly sensitive to corruption in graph structures. In this talk we study a family of Hamilton-Jacobi equations on graphs called the p-eikonal equation. We show that p=1 is a provably robust distance-type function on a graph and converges in the continuum limit to a geodesic density weighted distance function. We present applications to data depth and semi-supervised learning.

**[03495] Efficient high frequency wave propagation with small sampling density****Author(s)**:**Songting Luo**(Iowa State University)- Qing Huo Liu (Duke University)

**Abstract**: In this talk, we will present a few approaches for simulating high frequency wave propagation with low sampling density. One approach is based on WKBJ approximation, which leads to Hamilton-Jacobi type equations for the phase and amplitude. Such equations will be solved efficiently by well-established schemes and their solutions will be used for building the wave. In order to resolve the difficulty of capturing the caustics in the WKBJ approximation, we will further transform the problem into a fixed-point iteration problem that can be solved by operator-splitting based pseudospectral methods, which leads to another approach. Both approaches have low sampling densities that ensure the efficiency, verified by numerical experiments.

**[03517] A system of of Hamilton-Jacobi equations characterizing geodesic centroidal tessellations****Author(s)**:**Adriano Festa**(Politecnico di Torino)

**Abstract**: We introduce a class of systems of Hamilton-Jacobi equations characterizing geodesic centroidal tessellations, i.e. tessellations of domains with respect to geodesic distances where generators and centroids coincide. Typical examples are given by geodesic centroidal Voronoi tessellations and geodesic centroidal power diagrams. An appropriate version of the Fast Marching method on unstructured grids allows computing the solution of the Hamilton-Jacobi system and therefore the associated tessellations. We propose various numerical examples to illustrate the features of the technique.

**[03577] Leveraging Multi-time Hamilton-Jacobi PDEs for Certain Scientific Machine Learning Problems****Author(s)**:**Jerome Darbon**(Brown University)- Paula Chen (Brown University)
- Tingwei Meng (UCLA)
- Zongren Zou (Brown University)
- George Em Karniadakis (Brown University)

**Abstract**: We establish a novel theoretical connection between specific optimization problems arising in machine learning and the multi-time Hopf formula, which corresponds to a representation of the solution to certain multi-time HJ PDEs. Through this connection, we increase the interpretability of the training process of certain machine learning applications by showing that when we solve these learning problems, we also solve a multi-time HJ PDE and, by extension, its corresponding optimal control problem.

**[03669] Sparse-grid WENO fast sweeping methods for Eikonal equations****Author(s)**:- Zachary Miksis (University of Notre Dame)
**Yong-Tao Zhang**(University of Notre Dame)

**Abstract**: Fixed-point WENO fast sweeping methods are a class of explicit iterative methods for efficiently solving steady-state hyperbolic PDEs. For multidimensional nonlinear problems such as Eikonal equations, high-order fixed-point WENO fast sweeping methods still require quite a large amount of computational costs. In this talk, I shall present our recent work on applying sparse-grid techniques, an effective approximation tool for multidimensional problems, to fixed-point WENO fast sweeping methods for reducing their computational costs.

**[03726] Neural networks for first order HJB equations and application to front propagation with obstacle terms****Author(s)**:**Olivier Bokanowski**(LJLL, University Paris Cité)**Olivier Bokanowski**(LJLL, University Paris Cité)- Averil Prost (Insa Rouen)
- Xavier Warin (EDF)

**Abstract**: We propose deep neural network schemes for Bellman's dynamic programming principlefor some deterministic optimal control problems,corresponding also to some first-order Hamilton-Jacobi-Bellman equations with an obstacle term.We give an error analysis in an average norm, which is new in this deterministic context. We give several academic numerical examples on front propagation problems with obstacles in order to show the relevance of the approach.

**[03786] Data-Driven Learning Method for Optimal Feedback Control****Author(s)**:**Qi Gong**(University of California, Santa Cruz)

**Abstract**: Computing optimal feedback controls for nonlinear systems generally requires solving Hamilton-Jacobi-Bellman (HJB) equations, which, in high dimensions, is a well-known challenging problem due to the curse of dimensionality. In this talk, we present a model-based data-driven method to approximate solutions to HJB equations for high dimensional nonlinear systems. To accomplish this, we model solutions to HJB equations with neural networks trained on data generated without any state space discretization. Training is made more effective and efficient by leveraging the known physics of the problem and generating training data in an adaptive fashion. We further develop different neural networks approximation structures to improve robustness during learning and enhance closed-loop stability of the learned controller.

**[04710] Tropical and multi-level numerical methods for solving optimal control problems****Author(s)**:**Marianne Akian**(Inria and CMAP, Ecole polytechnique)- Stephane Louis Gaubert (INRIA and CMAP, Ecole polytechnique)
- Shanqing Liu (CMAP, Ecole polytechnique and Inria)

**Abstract**: We develop and study several numerical approximations and algorithms for computing the solution of Hamilton-Jacobi partial differential equations satisfied by the value function of deterministic optimal control problems over a finite dimensional space. These algorithms combine tropical numerical method or fast-marching method, with a multi-level discretization in a neighborhood of the optimal trajectories. For regular problems, the complexity as a function of precision, can be as for a one-dimensional problem.

**[04939] Exploiting Hamilton-Jacobi-Bellman equations in the representation and evolution of conservative dynamics.****Author(s)**:**Peter Dower**(University of Melbourne)- William McEneaney (University of California San Diego)

**Abstract**: Connections between Hamilton's action principle and optimal control are explored in the representation and evolution of conservative dynamics. By associating these dynamics with a Hamilton-Jacobi-Bellman equation or its characteristic system, dynamic programming and verification results are used to construct a fundamental solution and solve inverse problems. On longer time horizons, where the action is not “least”, analogous developments are shown to follow using a notion of “stationary” control. Various applications are highlighted.