Jiahua Jiang

Wuwei Ren, Siyuan Shen, Linlin Li et al.

Light scattering imposes a major obstacle for imaging objects seated deeply in turbid media, such as biological tissues and foggy air. Diffuse optical tomography (DOT) tackles scattering by volumetrically recovering the optical absorbance and has shown significance in medical imaging, remote sensing and autonomous driving. A conventional DOT reconstruction paradigm necessitates discretizing the object volume into voxels at a pre-determined resolution for modelling diffuse light propagation and the resulting spatial resolution of the reconstruction is generally limited. We propose NeuDOT, a novel DOT scheme based on neural fields (NF) to continuously encode the optical absorbance within the volume and subsequently bridge the gap between model accuracy and high resolution. Comprehensive experiments demonstrate that NeuDOT achieves submillimetre lateral resolution and resolves complex 3D objects at 14 mm-depth, outperforming the state-of-the-art methods. NeuDOT is a non-invasive, high-resolution and computationally efficient tomographic method, and unlocks further applications of NF involving light scattering.

Bo Dong, Jiahua Jiang, Yanlai Chen

We develop and analyze the first hybridizable discontinuous Galerkin (HDG) method for solving fifth-order Korteweg-de Vries (KdV) type equations. We show that the semi-discrete scheme is stable with proper choices of the stabilization functions in the numerical traces. For the linearized fifth-order equations, we prove that the approximations to the exact solution and its four spatial derivatives as well as its time derivative all have optimal convergence rates. The numerical experiments, demonstrating optimal convergence rates for both the linear and nonlinear equations, validate our theoretical findings.

Jiahua Jiang, Yanlai Chen, Akil Narayan

The Reduced Basis Method (RBM) is a popular certified model reduction approach for solving parametrized partial differential equations. One critical stage of the \textit{offline} portion of the algorithm is a greedy algorithm, requiring maximization of an error estimate over parameter space. In practice this maximization is usually performed by replacing the parameter domain continuum with a discrete "training" set. When the dimension of parameter space is large, it is necessary to significantly increase the size of this training set in order to effectively search parameter space. Large training sets diminish the attractiveness of RBM algorithms since this proportionally increases the cost of the offline {phase}. In this work we propose novel strategies for offline RBM algorithms that mitigate the computational difficulty of maximizing error estimates over a training set. The main idea is to identify a subset of the training set, a "surrogate parameter domain" (SPD), on which to perform greedy algorithms. The SPD's we construct are much smaller in size than the full training set, yet our examples suggest that they are accurate enough to represent the solution manifold of interest at the current offline RBM iteration. We propose two algorithms to construct the SPD: Our first algorithm, the Successive Maximization Method (SMM) method, is inspired by inverse transform sampling for non-standard univariate probability distributions. The second constructs an SPD by identifying pivots in the Cholesky Decomposition of an approximate error correlation matrix. We demonstrate the algorithm through numerical experiments, showing that the algorithm is capable of accelerating offline RBM procedures without degrading accuracy, assuming that the solution manifold has low Kolmogorov width.

Jiahua Jiang, Carmeliza Navasca

We address the optimal control problems arising from partial differential equations with large discrete dimensional control systems. To obtain reduced order models, we find basis elements from the canonical polyadic (CP) decomposition. Tensor datasets are from snapshots of the large models. Our method to reduce the control system is to use dimensionality reduction approaches through sparse optimization and flexible hybrid methods is to obtain low rank CP tensor basis elements. The reduced optimal control problem leads to reduced state-dependent Riccati Equations which can be solved efficiently.

Shihan Zhao, Jianru Zhang, Yanan Wu et al.

Fluorescence Molecular Tomography (FMT) is a promising technique for non-invasive 3D visualization of fluorescent probes, but its reconstruction remains challenging due to the inherent ill-posedness and reliance on inaccurate or often-unknown tissue optical properties. While deep learning methods have shown promise, their supervised nature limits generalization beyond training data. To address these problems, we propose $μ$NeuFMT, a self-supervised FMT reconstruction framework that integrates implicit neural-based scene representation with explicit physical modeling of photon propagation. Its key innovation lies in jointly optimize both the fluorescence distribution and the optical properties ($μ$) during reconstruction, eliminating the need for precise prior knowledge of tissue optics or pre-conditioned training data. We demonstrate that $μ$NeuFMT robustly recovers accurate fluorophore distributions and optical coefficients even with severely erroneous initial values (0.5$\times$ to 2$\times$ of ground truth). Extensive numerical, phantom, and in vivo validations show that $μ$NeuFMT outperforms conventional and supervised deep learning approaches across diverse heterogeneous scenarios. Our work establishes a new paradigm for robust and accurate FMT reconstruction, paving the way for more reliable molecular imaging in complex clinically related scenarios, such as fluorescence guided surgery.

Ruchi Guo, Jiahua Jiang, Bangti Jin et al.

Fluorescence Molecular Tomography (FMT) is a widely used non-invasive optical imaging technology in biomedical research. It usually faces significant accuracy challenges in depth reconstruction, and conventional iterative methods struggle with poor $z$-resolution even with advanced regularization. Supervised learning approaches can improve recovery accuracy but rely on large, high-quality paired training dataset that is often impractical to acquire in practice. This naturally raises the question of how learning-based approaches can be effectively combined with iterative schemes to yield more accurate and stable algorithms. In this work, we present a novel warm-basis iterative projection method (WB-IPM) and establish its theoretical underpinnings. The method is able to achieve significantly more accurate reconstructions than the learning-based and iterative-based methods. In addition, it allows a weaker loss function depending solely on the directional component of the difference between ground truth and neural network output, thereby substantially reducing the training effort. These features are justified by our error analysis as well as simulated and real-data experiments.

Yanlai Chen, Jiahua Jiang, Akil Narayan

The Reduced Basis Method (RBM) is a rigorous model reduction approach for solving parametrized partial differential equations. It identifies a low-dimensional subspace for approximation of the parametric solution manifold that is embedded in high-dimensional space. A reduced order model is subsequently constructed in this subspace. RBM relies on residual-based error indicators or {\em a posteriori} error bounds to guide construction of the reduced solution subspace, to serve as a stopping criteria, and to certify the resulting surrogate solutions. Unfortunately, it is well-known that the standard algorithm for residual norm computation suffers from premature stagnation at the level of the square root of machine precision. In this paper, we develop two alternatives to the standard offline phase of reduced basis algorithms. First, we design a robust strategy for computation of residual error indicators that allows RBM algorithms to enrich the solution subspace with accuracy beyond root machine precision. Secondly, we propose a new error indicator based on the Lebesgue function in interpolation theory. This error indicator does not require computation of residual norms, and instead only requires the ability to compute the RBM solution. This residual-free indicator is rigorous in that it bounds the error committed by the RBM approximation, but up to an uncomputable multiplicative constant. Because of this, the residual-free indicator is effective in choosing snapshots during the offline RBM phase, but cannot currently be used to certify error that the approximation commits. However, it circumvents the need for \textit{a posteriori} analysis of numerical methods, and therefore can be effective on problems where such a rigorous estimate is hard to derive.

Jiahua Jiang, Yanlai Chen, Akil Narayan

The non-intrusive generalized Polynomial Chaos (gPC) method is a popular computational approach for solving partial differential equations (PDEs) with random inputs. The main hurdle preventing its efficient direct application for high-dimensional input parameters is that the size of many parametric sampling meshes grows exponentially in the number of inputs (the "curse of dimensionality"). In this paper, we design a weighted version of the reduced basis method (RBM) for use in the non-intrusive gPC framework. We construct an RBM surrogate that can rigorously achieve a user-prescribed error tolerance, and ultimately is used to more efficiently compute a gPC approximation non-intrusively. The algorithm is capable of speeding up traditional non-intrusive gPC methods by orders of magnitude without degrading accuracy, assuming that the solution manifold has low Kolmogorov width. Numerical experiments on our test problems show that the relative efficiency improves as the parametric dimension increases, demonstrating the potential of the method in delaying the curse of dimensionality. Theoretical results as well as numerical evidence justify these findings.