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Goal-Oriented Real-Time Bayesian Inference for Linear Autonomous Dynamical Systems With Application to Digital Twins for Tsunami Early Warning

This paper presents a real-time, goal-oriented Bayesian framework that overcomes the computational challenges of high-dimensional inverse problems in autonomous dynamical systems by decomposing the process into offline and online phases, enabling the construction of digital twins for tsunami early warning that infer earthquake-induced seafloor motion from seafloor pressure data and forecast tsunami propagation with rigorous uncertainty quantification.

Original authors: Stefan Henneking, Sreeram Venkat, Omar Ghattas

Published 2026-01-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Stefan Henneking, Sreeram Venkat, Omar Ghattas

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out exactly what happened in a dark room just by listening to the echoes of a single clap. Now, imagine that room is the entire ocean floor, the "clap" is a massive earthquake, and the "echoes" are pressure waves traveling through the water. This is the challenge scientists face when trying to predict tsunamis: they need to reconstruct the invisible, chaotic movement of the ocean floor from sparse data collected by a few sensors, and then instantly predict how big the resulting waves will be when they hit the shore.

This paper presents a "digital twin"—a perfect, virtual copy of the real world—that can solve this puzzle in real-time (seconds) rather than taking days or weeks. Here is how they did it, explained through simple analogies.

The Problem: The "Impossible" Puzzle

Usually, trying to figure out the shape of an object based on a few scattered measurements is like trying to guess the exact pattern of a giant, intricate rug by looking at only a few threads.

The Scale: The ocean floor is huge. To model it accurately, you need to track billions of tiny points (parameters).
The Noise: The data from sensors is messy and incomplete.
The Speed: Tsunamis travel fast. By the time a supercomputer finishes calculating the answer, the wave might have already hit the coast.
The Trap: Traditional methods try to solve this by running the physics equations over and over again. For a problem this big, that would take hundreds of days. It's like trying to find a needle in a haystack by checking every single piece of hay one by one.

The Solution: The "Digital Twin" Framework

The authors built a system that splits the work into two distinct phases: Offline (doing the heavy lifting beforehand) and Online (the split-second reaction when an earthquake actually happens).

Phase 1: The "Offline" Library (Doing the Homework)

Before any earthquake happens, the team uses a massive supercomputer to build a "library" of how the ocean behaves.

The Analogy: Imagine a chef who wants to serve a complex dish instantly. Instead of chopping vegetables and cooking from scratch every time a customer orders, the chef pre-chops, pre-cooks, and pre-mixes everything into a "ready-to-serve" kit.
The Science: They use the fact that the physics of sound and water waves are "time-invariant" (the rules don't change over time). They run the complex physics simulations just enough times to create a massive, pre-computed map. This map tells them exactly how a signal from any sensor would look if the ocean floor moved in a specific way.
The Result: They store this map on a hard drive. It's huge, but it's ready to go.

Phase 2: The "Online" Magic (The Real-Time Reaction)

When an earthquake strikes, the system doesn't run the physics equations again. Instead, it uses the pre-made library.

The Analogy: When the customer orders, the chef doesn't cook; they just grab the pre-made kit and assemble it. It takes seconds.
The Science: The system takes the new sensor data (the "echoes") and uses a special mathematical trick (Fast Fourier Transforms on powerful graphics cards) to look up the answer in the pre-computed library.
The Speed: Because they aren't re-solving the physics, they can infer the ocean floor movement and predict the tsunami wave height in fractions of a second.

The "Goal-Oriented" Twist

Usually, scientists try to figure out everything about the ocean floor first, then predict the wave. But the authors realized they don't need to know every detail of the ocean floor to know how big the wave will be.

The Analogy: If you want to know if a car will crash into a wall, you don't need to know the exact color of the paint or the brand of the tires. You just need to know the speed and the angle.
The Science: They created a direct shortcut from the sensor data straight to the wave prediction. They built a specific "data-to-wave" map. This means they can predict the tsunami height and its uncertainty (how confident they are) without even explicitly calculating the full ocean floor movement first.

The Results: What Did They Prove?

They tested this on a realistic 3D model of a tsunami scenario:

The Scale: The model had 132 million parameters (points of data) to track.
The Speed: The entire process—from receiving noisy sensor data to predicting the wave height with uncertainty estimates—took less than a second (about 58 milliseconds).
The Accuracy: Even though the data was noisy and the problem was mathematically "ill-posed" (meaning there are many possible answers), the system accurately predicted the smooth, large-scale movements of the ocean floor that actually cause the tsunami. It correctly identified that the "rough" details (high-frequency noise) didn't matter much for the final wave prediction.

Why This Matters

This framework proves that we can build a "digital twin" of a subduction zone (like the Cascadia zone in the Pacific Northwest) that is:

Exact: It uses the full, high-fidelity physics equations, not simplified guesses.
Fast: It works in real-time, fast enough to warn people before a tsunami hits.
Uncertainty-Aware: It doesn't just give a number; it tells us how confident it is in that number, which is crucial for making life-or-death evacuation decisions.

In short, they turned a problem that used to take months of supercomputer time into a task a laptop can do in a heartbeat, simply by doing the heavy math beforehand and using a clever "lookup" system when the emergency happens.

Technical Summary: Goal-Oriented Real-Time Bayesian Inference for Linear Autonomous Dynamical Systems

Problem Statement
The paper addresses the challenge of solving very large-scale linear inverse problems governed by high-fidelity partial differential equation (PDE) models within a Bayesian framework, specifically for real-time applications. The target application is a digital twin for tsunami early warning in subduction zones, such as the Cascadia Subduction Zone (CSZ). The specific task involves inferring earthquake-induced spatiotemporal seafloor motion (the parameter $m$ ) from sparse seafloor pressure observations (the data $d$ ) and subsequently predicting tsunami wave heights (quantities of interest, or QoIs, denoted $q$ ) in vulnerable coastal regions.

The problem is characterized by extreme dimensionality: a discretized acoustic–gravity PDE model for the CSZ yields approximately $O(10^9)$ inversion parameters. Traditional Bayesian inference methods, which rely on sampling or Laplace approximations of the posterior, are computationally intractable for such scales. Furthermore, standard surrogate modeling techniques (e.g., Gaussian processes, neural operators) are ineffective because the parameter-to-observable map for wave propagation problems does not admit a low-dimensional subspace representation; the singular values of the map decay slowly due to the high-frequency nature of the governing hyperbolic PDEs. Additionally, the need for real-time execution (within seconds) precludes the use of conventional iterative solvers that require repeated forward and adjoint PDE solves during the inference phase.

Methodology
The authors propose a framework that exploits the autonomous structure of the forward dynamical system (specifically, a linear time-invariant or LTI system) to decompose the solution into an offline precomputation phase and an online real-time phase. The core mathematical insight is that the parameter-to-observable (p2o) and parameter-to-QoI (p2q) maps for such systems exhibit a block Toeplitz structure due to time-shift invariance.

The methodology proceeds through four distinct phases:

Phase 1 (Offline): Precomputation of Maps.
Instead of solving the forward problem for every parameter, the framework computes the p2o map ( $F$ ) and p2q map ( $F_q$ ) by solving the adjoint PDE equations. Due to the block Toeplitz structure, only one adjoint solve per sensor location (for $F$ ) and per QoI location (for $F_q$ ) is required. These solutions are shifted in time to construct the full maps. This step is computationally expensive but performed only once.
Phase 2 (Offline): Compact Representation of the Posterior.
The posterior covariance of the parameters, $\Gamma_{post}(m) = (F^* \Gamma_{noise}^{-1} F + \Gamma_{prior}^{-1})^{-1}$ , is too large to store directly ( $O(10^9 \times 10^9)$ ). The authors employ the Sherman–Morrison–Woodbury formula to shift the inversion from the high-dimensional parameter space to the lower-dimensional data space ( $O(N_d N_t \times N_d N_t)$ ). This requires precomputing a matrix $K = \Gamma_{noise} + F \Gamma_{prior} F^*$ and a matrix $G^* = \Gamma_{prior} F^*$ . Crucially, if the prior covariance is time-invariant, $G^*$ also inherits the block Toeplitz structure, allowing for efficient storage and computation.
Phase 3 (Offline): Goal-Oriented Precomputation.
To enable real-time prediction of QoIs without explicitly inferring the full parameter field, the framework precomputes a data-to-QoI (d2q) map ( $Q$ ). This map directly relates observed data to the QoI posterior mean and covariance, bypassing the intermediate step of parameter inference during the online phase.
Phase 4 (Online): Real-Time Inference and Prediction.
Given new observational data, the system performs:
- Parameter Inference: Computes the Maximum A Posteriori (MAP) estimate of the seafloor motion using the precomputed matrices and fast matrix-vector multiplications (matvecs).
- QoI Prediction: Directly computes the predicted tsunami wave heights and their uncertainties using the precomputed d2q map.
The computational bottleneck in the online phase is eliminated by using a multi-GPU-accelerated Fast Fourier Transform (FFT) algorithm. This algorithm exploits the block Toeplitz structure to perform matvecs with $F$ , $F^*$ , and their adjoints in $O(N \log N)$ time, avoiding any PDE solves during the online phase.

Key Contributions

Exact Real-Time Inference: The paper presents a framework that solves the discretized inverse problem exactly (up to rounding error) without approximations or surrogate models, achieving real-time performance for problems with $O(10^8)$ to $O(10^9)$ parameters.
Offline-Online Decomposition: By leveraging the autonomous nature of the system, the computationally intensive PDE solves are moved entirely to an offline phase. The online phase relies solely on linear algebra operations on precomputed matrices.
FFT-Based Acceleration: The authors introduce a method to compute Hessian matvecs using FFT-based algorithms on multi-GPU clusters, achieving speedups of approximately $44,000\times$ to $160,000\times$ compared to conventional matrix-free methods involving forward/adjoint PDE solves.
Goal-Oriented Efficiency: The framework explicitly targets the prediction of QoIs, allowing for the direct mapping of data to QoIs (via the d2q map), which drastically reduces storage and compute requirements for the online prediction step.
Handling High-Rank Hessians: The method successfully addresses inverse problems where the Hessian has a slow decay of singular values (high rank), a scenario where standard low-rank approximation techniques fail.

Results
The authors demonstrate the framework on a 3D test configuration of the acoustic–gravity wave equations representing a subduction zone scenario:

Scale: The test problem involves approximately 132 million parameters ( $1.3 \times 10^8$ ), 24,500 data points, and 800 QoIs.
Performance:
- Offline: Precomputing the necessary maps and matrices required solving 65 adjoint PDEs, taking approximately 8.6 hours on a supercomputer cluster.
- Online: The full end-to-end process of inferring the seafloor motion and predicting tsunami wave heights (including uncertainty quantification) was completed in fractions of a second (specifically, ~57 ms for inference and ~1 ms for prediction on a laptop for the QoI step).
Accuracy: For synthetic data with 6% additive noise, the relative error in the inferred seafloor motion was approximately 9.5%, while the relative error in the predicted tsunami wave heights (QoIs) was only ~1.6%. The results indicate that while high-frequency components of the seafloor motion are poorly resolved (due to sparse data), the low-frequency components driving tsunami formation are accurately captured, leading to highly accurate QoI predictions.
Comparison: Compared to a conventional conjugate gradient approach, which would require days to solve the same problem, the proposed method offers a speedup of roughly 400 million-fold for the online stage.

Significance and Claims
The paper claims to provide a fundamental advancement in the construction of digital twins for critical infrastructure and disaster management. By enabling goal-oriented Bayesian inference that is both exact (faithful to the high-fidelity PDE model) and real-time, the framework makes it feasible to construct digital twins for systems previously considered intractable due to scale and computational constraints.

The authors emphasize that their approach does not rely on data-driven surrogates or low-rank approximations, which are often insufficient for wave propagation problems. Instead, they exploit the specific mathematical structure of autonomous dynamical systems to achieve efficiency. The ultimate goal is to apply this framework to the Cascadia Subduction Zone to provide early warnings for tsunamis generated by megathrust earthquakes, potentially saving lives by reducing the time required for inference and prediction from hours to seconds. The paper notes that while the current test case is smaller than the full CSZ scale, the methodology is designed to scale to the $O(10^9)$ parameter regime required for the full digital twin.