arXiv subjects

Shanyin Tong

Publications and source records attributed to Shanyin Tong.

14 recordsLinked to original sources

arXiv2026

An exploration of plug-and-play flow matching for full-waveform inversion on breast imaging

Full-waveform inversion (FWI) reconstructs sound-speed fields from measured waveforms, by solving an optimization problem that minimizes the discrepancy between measured waveforms and those predicted by a numerical wave-propagation model. However, choosing suitable regularization for this optimization is challenging because it must stabilize the inversion while preserving relevant structures, prompting us to investigate whether priors learned from representative sound-speed images can improve reconstruction. In this work, we adapt plug-and-play flow matching (PnP-Flow) to FWI by alternating waveform-misfit updates with updates from a pretrained flow prior. We explore the effectiveness of PnP-Flow reconstruction by comparing it with total-variation (TV) regularization using noiseless transmission-only plane-wave data. Separate flow priors are trained on two circular-inclusion model families with constant and spatially varying backgrounds. We evaluate the method both on these in-distribution circle models and under substantial distribution shift by transferring the Gaussian-circle prior and reconstruction settings to breast phantom patches without retraining or retuning. Using the same number of waveform-misfit updates, the proposed PnP-Flow framework generally improves circle reconstruction accuracy and recovers breast lesion contrast more accurately than TV baselines.

eess.IV↗

arXiv2026

Geometry-informed importance sampling for rare events

We propose GeoIS, a new importance sampling method for estimating rare event probabilities for an underlying Gaussian distribution that exploits asymptotic theory and local event geometry. The proposal distribution is a mixture of two components. The first component draws a Gaussian along the boundary near the instanton and adds to it an exponentially distributed shift into the rare event. This component is derived from the known asymptotic approximation to the optimal zero-variance proposal distribution. The second mixture component is designed to cover missing event mass due to boundary-approximation mismatch, thereby preserving unbiasedness. We rigorously analyze this sampling strategy's performance as a function of event geometry, dimension, rarity level, and number $N$ of importance samples. Under explicit and relatively mild conditions on the former three features, we prove that the squared relative error of GeoIS is bounded by $c/N$ with high probability, for an absolute constant $c$. We implement our method numerically in several examples, including a diffusion PDE problem. The numerical experiments exhibit the qualitative dependence on geometry, dimension, rarity and sample size predicted by the theory, and show that GeoIS markedly outperforms importance sampling methods that use a fully Gaussian proposal distribution.

stat.CO↗

arXiv2026

Structure-Preserving Reduced-Order Modeling via Low-Rank Transport Signatures

Parametrized PDEs with density-valued solutions are often difficult to approximate with classical linear reduced-order models, especially in transport-dominated regimes. We introduce an optimal-transport-based reduced-order modeling that represents each density by the Kantorovich potential transporting a fixed reference density to the target density, and then maps these potentials to transport signatures using a weighted Laplacian associated with the reference measure. This embeds the density-valued solution map in a Hilbert space while preserving control of the induced transport maps and Wasserstein error. We treat the signature map as a continuous matrix indexed by parameters and space, construct a low-rank skeleton decomposition using a maximal-volume criterion, and learn the parameter-to-coefficient map with a neural network for efficient non-intrusive online evaluation. The reconstructed solution is obtained by pushing forward the reference density, so mass and positivity preservation is built into the method. We prove a mean-squared Wasserstein error bound separating low-rank approximation, discretization, sampling, and learning errors, and demonstrate the method on density-valued continuity equation and Burgers equation, where transport signatures yield substantially lower-rank structure than the original density snapshots.

math.NA↗

arXiv2026

Risk Control of Traffic Flow Through Chance Constraints and Large Deviation Approximation

We develop a rare chance-constrained optimal control framework for autonomous traffic management, with traffic dynamics governed by stochastic partial differential equations. To efficiently enforce these probabilistic safety specifications, we exploit a large deviation theory (LDT) based approximation method, which converts the original highly non-convex, sampling-heavy stochastic optimization problem into a tractable deterministic nonlinear programming problem. The proposed LDT-based reformulation exhibits superior computational scalability, as its computational cost remains approximately constant regardless of the target violation probability, as opposed to the scaling challenge of traditional sampling-based methods. The effectiveness of the proposed framework is illustrated through numerical experiments on a ring road with stochastic on-ramp inflow, and evaluated with two representative risk measures, namely, maximum traffic density and speed variation. The results show that the proposed method keeps the violation probability close to the target without parameter tuning and is more than ten times faster than a risk-averse baseline.

math.OC↗

arXiv2026

Reduced order model for parametric Boltzmann equation and its application to inverse problems

The Boltzmann equation plays an important role in modeling mesoscopic behavior in a wide range of scientific and engineering applications. However, its numerical solution is computationally expensive due to the high dimensionality of the model and the nonlinear nonlocal collision operator, especially for steady-state problems that require iterative solvers. This cost becomes prohibitive for inverse problems, where the induced optimization problem requires repeated forward solves. In this work, we propose a reduced-order model (ROM) for the parametric Boltzmann equation to address this computational challenge. The ROM constructs a low-dimensional approximation space for the parameter-induced solution manifold through a residual-based greedy strategy, and the reduced solution is then obtained via residual minimization over the reduced space, subject to mass conservation. The overall efficiency of the ROM is achieved by exploiting the quadratic structure of the collision operator and a precomputed separable approximation of the collision kernel. The resulting ROM is further applied to a thermally-driven inverse problem for reconstructing collision parameters from the observed macroscopic temperature data. This is accomplished either by directly replacing the PDE constraint with the ROM, leading to a bilevel optimization formulation, or by reformulating the task as a single-level optimization problem through the Karush--Kuhn--Tucker (KKT) conditions. Numerical experiments in both collision-dominated and transport-dominated cases are performed to demonstrate the efficiency and accuracy of the proposed ROM and its effectiveness in inverse problems. In particular, the resulting inverse problem is computationally much more tractable, achieving speedups of several orders of magnitude over that based on the full-order model while maintaining comparable accuracy.

math.NA↗

arXiv2026

Data-driven reduced-order models for the radiative transfer equation

We present a data-driven reduced-order modeling (ROM) framework for the zeroth angular moment of the solution to the radiative transfer equation (RTE) rather than the full phase-space solution. Our construction is based on the Peierls integral formulation of the angularly averaged density. For media with isotropic scattering, the density satisfies a closed second-kind Fredholm equation with a globally attenuated, weakly singular kernel. We project this equation directly. For media with anisotropic scattering, we utilize the average-fluctuation decomposition to derive a closed system for a projection-based ROM. Numerical simulations are presented to illustrate the effectiveness of the ROMs we implemented.

math.NA↗

arXiv2026

A Policy Iteration Method for Inverse Mean Field Games

We propose a policy iteration method to solve an inverse problem for a mean-field game (MFG) model, specifically to reconstruct the obstacle function in the game from the partial observation data of value functions, which represent the optimal costs for agents. The proposed approach decouples this complex inverse problem, which is an optimization problem constrained by a coupled nonlinear forward and backward PDE system in the MFG, into several iterations of solving linear PDEs and linear inverse problems. This method can also be viewed as a fixed-point iteration that simultaneously solves the MFG system and inversion. We prove its linear rate of convergence. In addition, numerical examples in 1D and 2D, along with performance comparisons to a direct least-squares method, demonstrate the superior efficiency and accuracy of the proposed method for solving inverse MFGs.

math.OC↗

arXiv2025

Extreme event probability estimation using PDE-constrained optimization and large deviation theory, with application to tsunamis

We propose and compare methods for the analysis of extreme events in complex systems governed by PDEs that involve random parameters, in situations where we are interested in quantifying the probability that a scalar function of the system's solution is above a threshold. If the threshold is large, this probability is small and its accurate estimation is challenging. To tackle this difficulty, we blend theoretical results from large deviation theory (LDT) with numerical tools from PDE-constrained optimization. Our methods first compute parameters that minimize the LDT-rate function over the set of parameters leading to extreme events, using adjoint methods to compute the gradient of this rate function. The minimizers give information about the mechanism of the extreme events as well as estimates of their probability. We then propose a series of methods to refine these estimates, either via importance sampling or geometric approximation of the extreme event sets. Results are formulated for general parameter distributions and detailed expressions are provided when Gaussian distributions. We give theoretical and numerical arguments showing that the performance of our methods is insensitive to the extremeness of the events we are interested in. We illustrate the application of our approach to quantify the probability of extreme tsunami events on shore. Tsunamis are typically caused by a sudden, unpredictable change of the ocean floor elevation during an earthquake. We model this change as a random process, which takes into account the underlying physics. We use the one-dimensional shallow water equation to model tsunamis numerically. In the context of this example, we present a comparison of our methods for extreme event probability estimation, and find which type of ocean floor elevation change leads to the largest tsunamis on shore.

math.OC↗

arXiv2025

A three-stage method for reconstructing multiple coefficients in coupled photoacoustic and diffuse optical imaging

This paper studies inverse problems in quantitative photoacoustic tomography with additional optical current data supplemented from diffuse optical tomography. We propose a three-stage image reconstruction method for the simultaneous recovery of the absorption, diffusion, and Grüneisen coefficients. We demonstrate, through numerical simulations, that: (i) when the Grüneisen coefficient is known, the addition of the optical measurements allows a more accurate reconstruction of the scattering and absorption coefficients; and (ii) when the Grüneisen coefficient is not known, the addition of optical current measurements allows us to reconstruct uniquely the Grüneisen, the scattering and absorption coefficients. Numerical simulations based on synthetic data are presented to demonstrate the effectiveness of the proposed idea.

math.NA↗

arXiv2024

Sensitivity Analysis of the Information Gain in Infinite-Dimensional Bayesian Linear Inverse Problems

We study the sensitivity of infinite-dimensional Bayesian linear inverse problems governed by partial differential equations (PDEs) with respect to modeling uncertainties. In particular, we consider derivative-based sensitivity analysis of the information gain, as measured by the Kullback-Leibler divergence from the posterior to the prior distribution. To facilitate this, we develop a fast and accurate method for computing derivatives of the information gain with respect to auxiliary model parameters. Our approach combines low-rank approximations, adjoint-based eigenvalue sensitivity analysis, and post-optimal sensitivity analysis. The proposed approach also paves way for global sensitivity analysis by computing derivative-based global sensitivity measures. We illustrate different aspects of the proposed approach using an inverse problem governed by a scalar linear elliptic PDE, and an inverse problem governed by the three-dimensional equations of linear elasticity, which is motivated by the inversion of the fault-slip field after an earthquake.

math.NA↗

arXiv2023

Scalable Methods for Computing Sharp Extreme Event Probabilities in Infinite-Dimensional Stochastic Systems

We introduce and compare computational techniques for sharp extreme event probability estimates in stochastic differential equations with small additive Gaussian noise. In particular, we focus on strategies that are scalable, i.e. their efficiency does not degrade upon temporal and possibly spatial refinement. For that purpose, we extend algorithms based on the Laplace method for estimating the probability of an extreme event to infinite dimensional path space. The method estimates the limiting exponential scaling using a single realization of the random variable, the large deviation minimizer. Finding this minimizer amounts to solving an optimization problem governed by a differential equation. The probability estimate becomes sharp when it additionally includes prefactor information, which necessitates computing the determinant of a second derivative operator to evaluate a Gaussian integral around the minimizer. We present an approach in infinite dimensions based on Fredholm determinants, and develop numerical algorithms to compute these determinants efficiently for the high-dimensional systems that arise upon discretization. We also give an interpretation of this approach using Gaussian process covariances and transition tubes. An example model problem, for which we provide an open-source python implementation, is used throughout the paper to illustrate all methods discussed. To study the performance of the methods, we consider examples of stochastic differential and stochastic partial differential equations, including the randomly forced incompressible three-dimensional Navier-Stokes equations.

stat.CO↗

arXiv2023

Estimating earthquake-induced tsunami height probabilities without sampling

Given a distribution of earthquake-induced seafloor elevations, we present a method to compute the probability of the resulting tsunamis reaching a certain size on shore. Instead of sampling, the proposed method relies on optimization to compute the most likely fault slips that result in a seafloor deformation inducing a large tsunami wave. We model tsunamis induced by bathymetry change using the shallow water equations on an idealized slice through the sea. The earthquake slip model is based on a sum of multivariate log-normal distributions, and follows the Gutenberg-Richter law for moment magnitudes 7--9. For a model problem inspired by the Tohoku-Oki 2011 earthquake and tsunami, we quantify annual probabilities of differently sized tsunami waves. Our method also identifies the most effective tsunami mechanisms. These mechanisms have smoothly varying fault slip patches that lead to an expansive but moderately large bathymetry change. The resulting tsunami waves are compressed as they approach shore and reach close-to-vertical leading wave edge close to shore.

physics.geo-ph↗

arXiv2023

Large deviation theory-based adaptive importance sampling for rare events in high dimensions

We propose a method for the accurate estimation of rare event or failure probabilities for expensive-to-evaluate numerical models in high dimensions. The proposed approach combines ideas from large deviation theory and adaptive importance sampling. The importance sampler uses a cross-entropy method to find an optimal Gaussian biasing distribution, and reuses all samples made throughout the process for both, the target probability estimation and for updating the biasing distributions. Large deviation theory is used to find a good initial biasing distribution through the solution of an optimization problem. Additionally, it is used to identify a low-dimensional subspace that is most informative of the rare event probability. This subspace is used for the cross-entropy method, which is known to lose efficiency in higher dimensions. The proposed method does not require smoothing of indicator functions nor does it involve numerical tuning parameters. We compare the method with a state-of-the-art cross-entropy-based importance sampling scheme using three examples: a high-dimensional failure probability estimation benchmark, a problem governed by a diffusion equation, and a tsunami problem governed by the time-dependent shallow water system in one spatial dimension.

stat.CO↗

arXiv2022

Optimization under rare chance constraints

Chance constraints provide a principled framework to mitigate the risk of high-impact extreme events by modifying the controllable properties of a system. The low probability and rare occurrence of such events, however, impose severe sampling and computational requirements on classical solution methods that render them impractical. This work proposes a novel sampling-free method for solving rare chance constrained optimization problems affected by uncertainties that follow general Gaussian mixture distributions. By integrating modern developments in large deviation theory with tools from convex analysis and bilevel optimization, we propose tractable formulations that can be solved by off-the-shelf solvers. Our formulations enjoy several advantages compared to classical methods: their size and complexity is independent of event rarity, they do not require linearity or convexity assumptions on system constraints, and under easily verifiable conditions, serve as safe conservative approximations or asymptotically exact reformulations of the true problem. Computational experiments on linear, nonlinear and PDE-constrained problems from applications in portfolio management, structural engineering and fluid dynamics illustrate the broad applicability of our method and its advantages over classical sampling-based approaches in terms of both accuracy and efficiency.

math.OC↗