Plenary talks

Title: An overview of structure preserving approximation of nonlinear conservation equations

Abstract: Realistic physical models of conservation equations are highly nonlinear systems of partial differential equations (PDE). Proving existence and uniqueness of solutions to these PDE systems is very often beyond reach, and in some cases the models are known to be ill-posed. In absence of a strong mathematical framework guaranteeing some form of compactness and well-posedness, one is lead to construct approximation techniques that (in addition to being consistent, of course) preserve key structures of the PDE system. For instance if the solution map to a problem is known to map pointwise data to a convex subset of $\mathbb{R}^m$, then a structure preserving approximation method would be one that ensures that the approximate solution map does the same. This concept is a multi-dimensional generalization of the maximum principle. Techniques satisfying this type of property are called invariant-domain preserving. It also often desirable to make sure that approximate solutions exactly solve a problem in some canonical situations. For instance, if a fluid flow is in hydrostatic equilibrium, one would like the approximate solution to satisfy this equilibrium as well. If the free surface of a lake is a rest, one would like the approximate model to deliver a solution that is also at rest. Approximation techniques that satisfy this type of property are said to be well-balanced. Many other forms of structures can be identified and preserved: involutions (think of Gauss laws of magnetism); asymptotic limits with respect to some model parameter; etc. In this talk I will give a brief overview of the state of the art on structure preserving approximation techniques for nonlinear conservation equations.

Title: Damage and Fracture in Quasi-brittle Materials

Abstract: Quasi-brittle materials are characterized by progressive damage and crack propagation, leading to a softening response after the peak load is reached. Examples include concrete as well as many types of rocks, including shale, granite, and sandstone. In this talk a field theory is presented for predicting damage and fracture in quasi-brittle materials. The approach taken here is formulated using a nonlocal constitutive law together with a phase field that is nonlocal in space and time. The displacement field inside the material is shown to be uniquely determined by an initial boundary value problem. The fracture set is characterized by the evolving phase field. The theory satisfies energy balance, with positive energy dissipation rate in accordance with the laws of thermodynamics. Notably, these properties are not imposed but follow directly from the evolution equation by multiplying the equation of motion by the velocity and integrating by parts. 

The model requires  a material’s elastic moduli, the strain at the onset of nonlinearity, the ultimate tensile strength, and the fracture toughness.  Here the characteristic length scale L is derived from the failure energy using geometric measure theory and is proportional to the ratio of fracture toughness to material strength consistent with the quasi-brittle length scale proposed by G. Irwin,  Proc. 7' Sagamore Research Conf. on Mechanics & Metals Behavior of Sheet Material (1960).  The characteristic time scale follows directly from the dimensional analysis of the field theory itself. Here the characteristic time scale is proportional to the time it takes a shear wave to travel a distance L. The constant of proportionality is the square root of the the ratio between elastic and inertial force.

Numerical simulations show quantitative and qualitative agreement with experiments, including three-point bending tests on concrete. The model successfully captures the cyclic load–deflection response of crack mouth opening displacement, the structural size-effect related to ultimate load and specimen size, and fracture nucleating from corner singularities in L-shaped domains. It provides dynamic results identical to the traditional phase field methods of  M.J. Borden, C.V. Verhoosel, M.A. Scott, T.J. Hughes, and C.M. Landis, CMAME  (2012). These results are recently reported in S. Coskun, D. Damircheli, and R. Lipton, JMPS (2025).

Title: Building a Gaussian Process statistical and quantitative learning framework for scientific applications

Abstract: High Performance Computing (HPC) application codes often contain many tuning parameters and require a large number of computational resources. It is desirables to have a machine learning framework to automate the parameter optimization process with minimal code executions on the actual HPC systems. Another need comes from building a trustworthy digital twin for a physical system, where we want to quantify the uncertainties of the simulation model with respect to the physical phenomena. For these optimization and UQ purposes, a powerful tool is Bayesian statistical learning framework which can treat the application as a black-box function and use Gaussian Process regression to compute the mean function and the variance in distribution. To this end, we have been developing a public domain software called GPTune. For parameter tuning we implemented a number of learning methods, including  multi-task learning, transfer learning, multi-objective tuning, and multi-fidelity tuning. For uncertainty quantification, we have been developing novel kernels to make them scientific-domain aware, fast linear algebra algorithms to enable large scale GP with millions of data points, and novel Gaussian Process to handle uncertainties in both input data and modeling. In this talk, we will illustrate the versatility of the GPTune software when it is applied to HPC codes ranging from mathematical libraries to complex simulation codes, as well as large-scale scientific apparatuses and instruments.

Title: Local-to-Global: Network Sheaves and Cohomological Inference

Abstract. How do local constraints determine global structure? Sheaf theory provides a precise framework for answering this question, extending classical tools from topology into the setting of networks. A network sheaf encodes algebraic data on vertices and edges together with consistency conditions, while cohomology quantifies the global degrees of freedom that remain after imposing all local constraints.

These ideas admit powerful analytic tools. In particular, one can define sheaf Laplacians (operators generalizing graph Laplacians) that enable a Hodge theory on networks. This decomposition separates local data into gradient-like, harmonic, and curl-like components, revealing global obstructions and redundancies in a way that is both algebraically rigorous and computationally tractable.

I will introduce the basics of network sheaves and cohomology with minimal prerequisites, then show how sheaf Laplacians and Hodge theory provide unifying methods across diverse applications: equilibrium analysis in graphic statics, consensus and polarization in opinion dynamics, data fusion in sensing systems, and more. The emphasis will be on conceptual clarity and practical computation, highlighting how local-to-global principles become concrete analytic tools for applied mathematics.