UCSB Applied Math/PDE/Data Science Seminar

Abstract: Inverse problems are ubiquitous. Traditionally, the goal is to infer an unknown vector or function. But what if the unknown is a probability measure? Seeking a measure that generates data consistent with given observations leads to an optimization problem over probability space. However, the complex geometric nature of this space prevents the direct use of standard arguments and solvers. We unravel some of the surprises that emerge in this setting and discuss potential solutions.

Host: Bohan Zhou

Abstract: We will describe a mathematical model of the cancer–immunity cycle to study how the timing and combination of chemotherapy and immunotherapy influence treatment outcomes under time-dependent selection pressure. The model is framed as an evolutionary game among cancer cells, healthy cells, and T-cells, forming a non-transitive rock–paper–scissors dynamic. A central idea is to synchronize treatment schedules with the intrinsic period of the underlying nonlinear system. Treating chemotherapy and immunotherapy as time-dependent control functions, we analyze how their timing and duration affect efficacy. The model predicts that these therapies do not commute: optimal regimens require immunotherapy to precede chemotherapy, with immunotherapy applied for half a cycle and chemotherapy for a quarter cycle. More broadly, the results suggest that precise timing can compensate for reduced total dose, pointing toward less toxic yet effective treatment strategies. This highlights the importance of measuring and calibrating the cancer–immunity cycle period across patients for clinically actionable protocols. Joint work with S. Mahmoodifar and K. Stuckey, PNAS (2025).

Host: Björn Birnir

Abstract: PDE-constrained optimization enables the inverse design of materials and devices, where a low-dimensional loss is minimized over a high-dimensional shape parameterization. Reverse-mode automatic differentiation efficiently handles such "wide-Jacobian" problems, yielding end-to-end differentiability. Building on the JAX ecosystem, I will report on our recent efforts in inverse design, beginning with a topology optimization algorithm that enables differentiation with respect to changes in the topology of quasi-binary structures [1]. The method, termed second-order subpixel smoothed projection, accelerates convergence for connectivity-dominated problems and is demonstrated on the design of thermal metamaterials. I will then turn to the optimization of a nanostructure for thermal energy conversion, based on the phonon Boltzmann transport equation [2] and subject to minimum feature constraints. Next, I will describe chiplet floorplan design, where, in collaboration with the MIT-IBM Watson AI Lab, we developed a framework for minimizing the maximum operating temperature under wirelength constraints [3]. Lastly, I will discuss inverse-designed metastructures that perform matrix–vector multiplications using heat as the signal carrier [4], with potential applications to automatic diagnostics and sensor fusion. I will conclude with an overview of our software and future plans.

[1] G. Romano, R. Arrieta, and S. G. Johnson, arXiv:2601.10737 (2026). 

[2] G. Romano and S. G. Johnson, Struct. Multidisc. Optim. 65, 297 (2022).

[3] G. Romano, A. Jain, N. Dehmamy, C. Chi, and X. Zhang, in Proc. 2025 IEEE 75th Electronic Components and Technology Conference (ECTC), pp. 221–227 (2025). 

[4] C. Silva and G. Romano, Phys. Rev. Appl. 25, 014073 (2026).

Host: Paul Atzberger

Abstract: This talk introduces graphs of convex sets: a new framework that blends discrete and continuous optimization with applications in decision making, robotics, and control. Mathematically, a Graph of Convex Sets (GCS) is a graph in which vertices are associated with convex optimization problems and edges couple pairs of these problems through additional convex costs and constraints. Classical problems defined over ordinary weighted graphs (such as the shortest path, the traveling salesman, and the minimum spanning tree) naturally generalize to a GCS, giving rise to a rich class of problems at the interface of combinatorial and convex optimization. I will first discuss how GCS problems can be solved efficiently, and then show how they naturally encompass multiple core problems in robotics, control, scheduling, and decision making. Finally, I will present GCSOPT, an open-source and easy-to-use Python library that enables solving real-world GCS problems in just a few lines of code.

Host: Bohan Zhou

Abstract: Shock waves in high-speed gas dynamics cause severe numerical difficulties for classical solvers and scientific machine learning. They are fundamentally a multiscale problem: While viscous effects ensure smoothness on microscopic scales, shocks manifest as macroscopic discontinuities. This talk begins with the observation that shock formation arises from the flow map reaching the boundary of the manifold of diffeomorphisms. We modify its geometry such that geodesics approach but never reach the boundary. The resulting information geometric regularization (IGR) has smooth solutions while avoiding the excessive dissipation of viscous regularizations, accelerating and simplifying the simulation of flows with shocks. We prove the existence of global strong IGR solutions in the unidimensional pressureless case and illustrate its practical utility on multidimensional examples with complex shock interactions. With S. Bryngelson and other collaborators, we use IGR to conduct the first compressible flow simulation exceeding a quadrillion degrees of freedom. The modified geometry of the diffeomorphism manifold is the information geometry of the mass density. The last part of the talk explains how this observation motivates information geometric mechanics that views the solutions of continuum mechanical PDEs as parameters of probability distributions originating from statistical physics. Replacing the Euclidean geometry of individual particles with the information geometry of statistical families promises performant numerical methods that preserve the positivity of densities and energies and readily integrate with scientific machine learning.

Host: Bohan Zhou

Abstract: Measures provide valuable insights into the long-term and global behavior of dynamical systems. In this talk, we present recent work using measure theory and optimal transport to address system identification, parameter recovery, identifiability, and prediction. We discuss PDE-constrained and transport-based approaches for learning ODEs and SDEs from slowly sampled or derivative-free data, enabling stable forward models and uncertainty quantification.

Host: Bohan Zhou