Portland State University
Applied and Computational Mathematics Seminar

No seminar meeting on October 3

Title: Sublinear methods for discretizing partial differential equations using data science techniques

Abstract: This talk is not about neural networks. 

Finding an approximate solution to Poisson’s equation in D dimensions on a grid with nD grid points can be done in time O(nD) using multigrid. In this talk we will show that when the solution is low rank it is possible to do this in time O(n) using techniques from data science. 

More precisely we will discuss how the SVD factorization, CUR and matrix Cross approximation can be used to design numerical methods for PDE in two dimensions and how generalizations of these factorizations to tensor networks can break the curse of dimensionality when solving PDE in D dimensions. Our discussion will be guided by examples of increasing complexity, starting with linear PDE and explicit time stepping, and moving towards non-linear PDE and implicit time stepping. 

This work is joint with Prof. Yingda Cheng and described in arXiv:2503.03909 and arXiv:2509.18554

Research Updates:

•  Pablo Cortés Castillo

•  Nick Fisher 

•  Gabriel Pinochet Soto

•  Mason Spears

•  Austen Nelson

Title: A Real-Time Goal-Oriented Bayesian Inversion-Based Digital Twin for Tsunami Early Warning Applied to the Cascadia Subduction Zone

Abstract: We present a digital twin (DT) for tsunami early warning in the Cascadia subduction zone (CSZ). This DT assimilates pressure data from seafloor sensors into an acoustic-gravity wave equation model, solves an inverse problem to infer spatiotemporal seafloor deformation, and forward predicts tsunami wave heights. The entire end-to-end data-to-inference-to-prediction computation is carried out in real time through a Bayesian framework that rigorously accounts for uncertainties. Creating such a DT is challenging due to the enormous size and complexity of both the forward and inverse problems. For example, a discretization of the spatiotemporal seafloor velocity in the CSZ – the parameter field to be inferred – gives rise to a system with one billion parameters. Using conventional state-of-the-art methods, computing the posterior mean alone would require more than 50 years on 512 GPUs. We exploit the shift invariance of the parameter-to-observable map and devise novel parallel algorithms for fast offline-online decomposition. The offline component requires just one adjoint wave propagation per sensor; the PDE solver is implemented with MFEM and exhibits excellent scalability to 43,520 GPUs on LLNL’s El Capitan system. Fast Hessian applications are enabled by an FFT-based algorithm for the resulting block Toeplitz matrices. Using this framework, the Bayesian inverse solution and wave height forecasts are computed in 0.2 seconds, representing a ten-billion-fold speedup over state-of-the-art methods.

This work has been selected as a finalist for the 2025 ACM Gordon Bell Prize and is joint work with Sreeram Venkat, Milinda Fernando, and Omar Ghattas (UT Austin), John Camier, Veselin Dobrev, and Tzanio Kolev (Lawrence Livermore National Laboratory), and Alice-Agnes Gabriel (UC San Diego), described in https://arxiv.org/abs/2504.16344 and https://arxiv.org/abs/2501.14911.

Research Updates:

• Michael  Neunteufel

• Emily Bogle

• 
  
• 
  

Title: 

Abstract: 

Title: 

Abstract: 

Research updates: