Midwest Numerical Analysis Day — Program
Dates: April 11–12, 2026 • Jordan Hall of Science, University of Notre Dame
Schedule at a glance
Day 1 — April 11
| Time | Event | Location |
|---|---|---|
| 7:30 AM | Registration table opens | Lobby |
| 8:00–8:10 AM | Opening remarks — Steve Corcelli (College of Science Dean) | Room 105 |
| 8:10–9:00 AM |
Plenary Lecture #1 — Mark Ainsworth, Brown University Galerkin Neural Network Approximation of Variational Problems with Error Control |
Room 105 |
| 9:00–9:30 AM | Coffee break | Galleria |
| 9:30–11:10 AM | Parallel sessions (A–D) | Rooms 101, 105, 310, 322 |
| 11:10–11:40 AM | Coffee break | Galleria |
| 11:40 AM–12:30 PM |
Plenary Lecture #2 — Hongkai Zhao, Duke University Mathematical and Computational Understanding of Neural Networks: From Representation to Learning and From Shallow to Deep, and Beyond |
Room 105 |
| 12:30–12:40 PM | Group photo (before lunch) | Outside venue |
| 12:40–2:30 PM | Lunch break | Reading Room |
| 2:30–4:10 PM | Parallel sessions (E–H) | Rooms 101, 105, 310, 322 |
| 4:10–5:00 PM | Poster session & coffee break | Galleria |
| 5:00–5:45 PM | Panel discussion | Room 105 |
| 6:00–8:00 PM | Conference dinner | Reading Room |
Day 2 — April 12
| 7:30 AM | Registration table opens | Lobby |
| 8:10–9:00 AM | Plenary Lecture #3 — Susanne C. Brenner, Louisiana State University Finite Element Methods for Least-Squares Problems |
Room 105 |
| 9:00–9:30 AM | Coffee break | Galleria |
| 9:30–11:10 AM | Parallel sessions (I–K) | Rooms 101, 105, 310 |
| 11:10–11:40 AM | Coffee break | Galleria |
| 11:40 AM–12:30 PM | Plenary Lecture #4 — William Layton, University of Pittsburgh The quest for time accuracy in CFD |
Room 105 |
| 12:30 PM | Concluding remarks | Lobby / Exit |
Plenary Abstracts
Mark Ainsworth (Brown University)
Galerkin Neural Network Approximation of Variational Problems with Error Control
Abstract
Recent years have seen an unprecedented surge of interest in applying neural networks to a very wide range of areas including non-scientific applications and artificial intelligence. In principle, neural networks offer benefits for scientific applications including the approximate solution of differential equations. However, if such methods are to be taken as a serious computational tool then it is important that they are set on a firm theoretical foundation and ideally include quantitative measures on the reliability of the results that are obtained.
We present an overview of some of our recent work in this direction on what we call Extended Galerkin Neural Networks (xGNN) where we aim to provide a variational framework for approximating general boundary value problems (BVPs) with error control. The main contributions of this work are (1) a rigorous theory guiding the construction of weighted least squares variational formulations suitable for use in neural network approximation of general BVPs and (2) an “extended” feedforward network architecture which incorporates and is even capable of learning singular solution structures, thus offering the potential to greatly improve the efficiency for singular solutions.
This is joint work with Justin Dong, Lawrence Livermore National Laboratory, USA.
Hongkai Zhao (Duke University)
Mathematical and Computational Understanding of Neural Networks: From Representation to Learning and From Shallow to Deep, and Beyond
Abstract
In this talk I will present some understanding of a few basic mathematical and computational questions for neural networks, as a particular form of nonlinear representation, and show how the network structure, activation function, and parameter initialization can affect its approximation properties and the learning process. In particular, we propose structured and balanced multi-component and multi-layer neural networks (MMNN) using sine as the activation function with an initialization scaling strategy. At the end, I will discuss a few issues and challenges when using neural networks to solve partial differential equations.
Susanne C. Brenner (Louisiana State University)
Finite Element Methods for Least-Squares Problems
Abstract
Finite dimensional linear and nonlinear least-squares problems appear in data fitting and the solution of nonlinear equations. In this talk I will present some recent results for the infinite dimensional analogs of such problems. They include (i) a general framework for solving distributed elliptic optimal control problems with pointwise state constraints by finite element methods originally designed for fourth order elliptic boundary value problems, (ii) a multiscale finite element method for solving distributed elliptic optimal control problems with rough coefficients and pointwise control constraints, and (iii) a convexity enforcing nonlinear least-squares finite element method for solving the Monge-Ampere equation.
William Layton (University of Pittsburgh)
The quest for time accuracy in CFD
Abstract
Advancements in algorithms and computational resources have made it possible to solve reliably (stable) steady state flows and even many evolutionary flows at statistical equilibrium. Time accuracy is still elusive for essentially time dependent problems. This is due to many factors including primitive time discretizations, turbulence models over dissipation and not incorporating intermittence, complexity of ensemble simulations, legacy codes and so on. This talk will present promising paths to overcome these impediments to time accuracy.
Parallel sessions (detailed)
Session A — Fluids and Fluid-Structure Interaction (9:30–11:10, Room 101)
Session Chair: Martina Bukač
Session Chair: Martina Bukač
9:30–9:50 (A-1) Zhiwei Zhang, Illinois Institute of Technology — An Efficient CC-MSAV Scheme for Phase-Field Vesicle Dynamics in Stokes Flow
9:50–10:10 (A-2) Tamas Horvath, Oakland University — Adaptive 2+1D space-time mesh generation
10:10–10:30 (A-3) Andrew Mangini, University of Notre Dame — A Partitioned, Second-Order Method for Two-Phase Flow
10:30–10:50 (A-4) Xiaoming Zheng, Central Michigan University — Iterative Projection Method with Grad-Div stabilization for unsteady Navier–Stokes equations
10:50–11:10 (A-5) Martina Bukač, University of Notre Dame — The recursive correction method for fluid-structure interaction
View session abstracts
A-1. Zhiwei Zhang
An Efficient CC-MSAV Scheme for Phase-Field Vesicle Dynamics in Stokes Flow
An Efficient CC-MSAV Scheme for Phase-Field Vesicle Dynamics in Stokes Flow
We extend the constant-coefficient multiple scalar auxiliary variable (CC-MSAV) framework to phase-field vesicle models with hydrodynamic interactions governed by the incompressible Stokes equations. The model couples an Allen–Cahn–Cahn–Hilliard (AC–CH) phase-field subsystem describing membrane dynamics with a Stokes flow driven by phase-field forces. Using the CC-MSAV formulation, we design a decoupled scheme in which the phase-field variables and velocity are computed sequentially. The method reduces the coupled system to a sequence of linear constant-coefficient elliptic problems that can be solved efficiently. Numerical examples illustrate the performance of the proposed approach.
A-2. Tamas Horvath
Adaptive 2+1D space-time mesh generation
Adaptive 2+1D space-time mesh generation
Space-time finite element methods provide a natural framework for solving time-dependent PDEs on evolving domains. We develop conforming space-time mesh-generation strategies compatible with hybridized DG discretizations. The framework supports large deformations, adaptive spatial refinement, and local temporal refinement. Numerical experiments demonstrate robustness in fluid-structure interaction scenarios.
A-3. Andrew Mangini
A Partitioned, Second-Order Method for Two-Phase Flow
A Partitioned, Second-Order Method for Two-Phase Flow
This work develops a novel partitioned, strongly-coupled, second-order numerical method for the Cahn–Hilliard–Navier–Stokes (CHNS) equations modeling immiscible, viscous, incompressible two-phase flows. The method is based on a refactorization of Cauchy’s one-legged θ-like method, yielding a predictor–corrector scheme consisting of a Backward Euler step followed by a Forward Euler step. In the Backward Euler stage, the Cahn–Hilliard and Navier–Stokes systems are decoupled, linearized, and solved iteratively; the Forward Euler stage extrapolates all unknowns. We prove stability under parameter-dependent conditions and demonstrate second-order temporal accuracy for θ = 0.5. Numerical experiments include rising and merging bubbles and phase-separation coarsening.
A-4. Xiaoming Zheng
Iterative Projection Method with Grad-Div stabilization for unsteady Navier–Stokes equations
Iterative Projection Method with Grad-Div stabilization for unsteady Navier–Stokes equations
This work continues our previous study on the iterative projection method (Advances in Computational Mathematics, 2025) by incorporating grad–div stabilization. We show that the L2 error bound for velocity is independent of viscosity and establish convergence and stability for inf-sup stable finite element pairs. Numerical experiments validate the theory.
A-5. Martina Bukač
The recursive correction method for fluid-structure interaction
The recursive correction method for fluid-structure interaction
We present a novel partitioned numerical method for simulating the interaction between a viscous, incompressible fluid and a thick, elastic structure. The method integrates a recently developed recursive correction approach with a Robin-based partitioning strategy and the Refactorized Midpoint method. Uniquely, this scheme achieves a second-order accuracy using only two subiterations per time step. We show that the method is unconditionally stable and evaluate its performance on benchmark problems. Numerical results demonstrate that the proposed scheme matches the accuracy of a second-order strongly-coupled method while maintaining a computational cost comparable to that of a single loosely-coupled solve, offering a highly efficient solution strategy for challenging fluid-structure interaction problems.
Session B — Analysis, Optimization, and Quantum Dynamics (9:30–11:10, Room 105)
Session Chair: Jiguang Sun
Session Chair: Jiguang Sun
9:30–9:50 (B-1) Yuan Gao, Purdue University — Convergence of monotone scheme for HJE and large deviation principle
9:50–10:10 (B-2) Tianyun Tang, The University of Chicago — Bregman ADMM for Bethe variational problem
10:10–10:30 (B-3) Shixu Meng, University of Texas at Dallas — Exploring Low-Rank Structures in Inverse Scattering
10:30–10:50 (B-4) Ke Wang, University of Michigan — Beyond Lindblad Dynamics: Rigorous Guarantees for Thermal and Ground State Preservation under System Bath Interactions
10:50–11:10 (B-5) Jiguang Sun, Michigan Technological University — Computation of Scattering Resonances
View session abstracts
B-1. Yuan Gao
Convergence of monotone scheme for HJE and large deviation principle
Convergence of monotone scheme for HJE and large deviation principle
We establish a rigorous large deviation principle (LDP) for stochastic chemical reaction networks by connecting monotone scheme convergence theory to the viscosity solution theory of first-order Hamilton–Jacobi equations (HJE). The key observation is that Varadhan’s discrete nonlinear semigroup can be recast as a monotone scheme approximating a limiting first-order HJE, and convergence of the scheme — via a Barles–Souganidis-type argument — yields the LDP directly. A central technical contribution is the treatment of the state constraint boundary condition arising from the confinement of species concentrations to a bounded domain: we show the scheme respects this constraint and that the limiting HJE is well-posed in the constrained sense. This framework provides both a rigorous analytical foundation for LDPs in reaction kinetics and a principled basis for numerical approximation of the rate function.
B-2. Tianyun Tang
Bregman ADMM for Bethe variational problem
Bregman ADMM for Bethe variational problem
In this work, we propose a novel Bregman ADMM with nonlinear dual update to solve the Bethe variational problem (BVP), a key optimization formulation in graphical models and statistical physics. Our algorithm provides rigorous convergence guarantees, even if the objective function of BVP is non-convex and non-Lipschitz continuous on the boundary. A central result of our analysis is proving that the entries in local minima of BVP are strictly positive, effectively resolving non-smoothness issues caused by zero entries. Beyond theoretical guarantees, the algorithm possesses high level of separability and parallelizability to achieve highly efficient subproblem computation. Our Bregman ADMM can be easily extended to solve the quantum Bethe variational problem. Numerical experiments are conducted to validate the effectiveness and robustness of the proposed method.
B-3. Shixu Meng
Exploring Low-Rank Structures in Inverse Scattering
Exploring Low-Rank Structures in Inverse Scattering
The inverse scattering problem plays an important role in a wide range of physical and engineering applications where hidden or internal features need to be analyzed. This talk will introduce a low-rank structure based on the disk prolate spheroidal wave functions, which are eigenfunctions of both the Born forward operator and a Sturm–Liouville differential operator. The potential of this low-rank structure will be demonstrated through regularization techniques, explicit a priori estimates, and its application using nonlinear techniques.
B-4. Ke Wang
Beyond Lindblad Dynamics: Rigorous Guarantees for Thermal and Ground State Preservation under System Bath Interactions
Beyond Lindblad Dynamics: Rigorous Guarantees for Thermal and Ground State Preservation under System Bath Interactions
We establish new theoretical results demonstrating the efficiency and robustness of system–bath interaction models for quantum thermal and ground state preparation. Unlike prior analyses, which typically rely on the Lindblad limit and require vanishing coupling strengths o(1), we rigorously show that efficient state preparation remains possible far beyond this regime, even when the cumulative coupling strength is Θ(1). We first prove that even with constant cumulative coupling strength, the induced quantum channel still approximately fixes the target state. For thermal state preparation, we then develop a general perturbative framework that yields end-to-end complexity bounds outside weak coupling, and in particular proves that the mixing time scales as the inverse square of the coupling strength. This framework extends to broad Hamiltonians for which KMS detailed balance Lindbladians are known to mix. These bounds substantially improve upon prior results, and numerical simulations further confirm the robustness of the system–bath interaction framework across both weak and strong coupling regimes.
B-5. Jiguang Sun
Computation of Scattering Resonances
Computation of Scattering Resonances
We review the abstract approximation theory for eigenvalue problems of holomorphic Fredholm operator-valued functions. The associated error estimates rely on the notion of regular convergence for the discrete approximation operators. We consider two nonlinear eigenvalue problems: the computation of band-structure diagrams for dispersive photonic crystals, and scattering poles for obstacles. For both problems, we develop finite element schemes and present numerical examples that demonstrate the effectiveness of the framework. Error estimates are obtained using Karma’s theory.
Session C — Scientific Machine Learning and Data-Driven Methods (9:30–11:10, Room 310)
Session Chair: Yue Zhao
Session Chair: Yue Zhao
9:30–9:50 (C-1) Ziyue Chen, Iowa State University — Bound-Preserving Cell-Average-Based Neural Network Method for Linear Hyperbolic and Parabolic Equations
9:50–10:10 (C-2) Harihara Maharna, University of Notre Dame — A Neural-Network-Based Lagrangian Method for Generalized Diffusion with Adaptive Refinement
10:10–10:30 (C-3) Sayantan Sarkar, SUNY Buffalo — From Video to Equations: Interpretable PDE Discovery via Sparse Learning
10:30–10:50 (C-4) Siyao Yang, University of Chicago — Tensor Density Estimator by Convolution-Deconvolution
10:50–11:10 (C-5) Yue Zhao, Michigan State University — Structure-Preserving Construction of Collision Operators for Kinetic Equations from Molecular Dynamics
View session abstracts
C-1. Ziyue Chen
Bound-Preserving Cell-Average-Based Neural Network Method for Linear Hyperbolic and Parabolic Equations
Bound-Preserving Cell-Average-Based Neural Network Method for Linear Hyperbolic and Parabolic Equations
In this paper, we develop a bound-preserving cell-average-based neural network (CANN) method for solving time-dependent linear partial differential equations. Biases are removed from the feedforward network and most entries of the weight matrices are enforced to be nonnegative. With these constraints imposed on the weight matrices, the piecewise constant solution of the bound-preserving CANN method can be proved to be bounded at all time levels. The L∞ sense stability can effectively remove previously observed blow-ups encountered by the original CANN method. For the advection equation, the well-trained bound-preserving CANN method can be generalized to solve a large group of initial value problems. Numerical approximations remain bounded at any instance, even after long time simulations. For the linear parabolic equation, the ultra-generalization capability of the method is maintained once the network solver is trained over a representative solution trajectory of data.
C-2. Harihara Maharna
A Neural-Network-Based Lagrangian Method for Generalized Diffusion with Adaptive Refinement
A Neural-Network-Based Lagrangian Method for Generalized Diffusion with Adaptive Refinement
Generalized diffusion models, like the Cahn–Hilliard equation, present computational challenges due to multiscale dynamics and steep interfacial gradients. Traditional methods often struggle with long-term energy stability and memory costs in high dimensions. To address this, we present the Energetic Variational Neural Network (EVNN) approach, a structure-preserving Lagrangian algorithm. Using neural networks for mesh-free spatial discretization, EVNN is derived directly from the energy-dissipation law via the Energetic Variational Approach (EnVarA). This guarantees monotonic free energy decay, preventing unphysical states and ensuring stability. To accurately capture steep gradients, we introduce an adaptive sampling method that dynamically refines the domain. Additionally, solving for flow map increments significantly improves memory efficiency. Ultimately, this mesh-free framework enables the accurate computation of complex gradient flows across high dimensions. Numerical experiments are provided to demonstrate the exceptional accuracy, computational efficiency, and energy stability of the proposed EVNN scheme.
C-3. Sayantan Sarkar
From Video to Equations: Interpretable PDE Discovery via Sparse Learning
From Video to Equations: Interpretable PDE Discovery via Sparse Learning
Recent advances in scientific machine learning have enabled high-fidelity prediction of complex spatiotemporal systems, yet extracting governing equations from data remains challenging. In this talk, we propose a sparse identification framework for learning partial differential equations directly from video observations. The method constructs a physics-informed library of spatial differential operators and employs sparse regression to recover interpretable dynamical models. Unlike purely neural approaches, our formulation emphasizes mechanistic transparency, numerical stability, and consistency with PDE structure. We discuss implementation challenges, including noise, numerical differentiation, and limited sampling, and present results for diffusion and transport processes. This approach bridges classical PDE modeling and modern data-driven techniques.
C-4. Siyao Yang
Tensor Density Estimator by Convolution-Deconvolution
Tensor Density Estimator by Convolution-Deconvolution
In this talk, we propose a natural approach to density estimation, entirely based on linear algebra. Our goal is to achieve computational complexity that is linear in dimensionality. This is accomplished by viewing the density estimation process as applying a low-dimensional orthogonal projector in function space onto the noisy empirical density. Due to the dimension reduction, the exponentially large variance in the empirical density is suppressed. Additionally, we leverage tensor-train algebra to perform this projection with linear computational complexity in terms of both dimensionality and sample size. This approach also yields a final density estimator in the form of a tensor train. Furthermore, we theoretically show that, in addition to achieving linear scaling in computational complexity, the estimation error also enjoys linear scaling under certain assumptions. Numerical experiments demonstrate that our estimator outperforms deep learning methods on several tasks, including Boltzmann sampling and generative modeling for real-world datasets.
C-5. Yue Zhao
Structure-Preserving Construction of Collision Operators for Kinetic Equations from Molecular Dynamics
Structure-Preserving Construction of Collision Operators for Kinetic Equations from Molecular Dynamics
We present a data-driven, multiscale framework for constructing generalized collisional kinetic models directly from the micro-scale molecular dynamics. The learned operator takes an anisotropic and non-stationary form, capturing heterogeneous collisional energy transfer arising from the many-body interactions, which is overbroadly overlooked by conventional collision operators beyond the weakly coupled regime. The constructed model strictly preserves frame-indifference, conservation laws, and physical constraints such as the H-theorem. To enable efficient numerical evaluation, we develop a fast spectral separation method that represents the kernel as a low-rank tensor product of univariate basis functions. This formulation admits an O(N log N) algorithm and structure-preserving discretization. Numerical results demonstrate that the proposed model accurately predicts transport coefficients and 1D3V plasma kinetics across a wide range of plasma conditions, where the conventional Landau model shows limitations.
Session D — Waves, Scattering, and Integral Equations (9:30–11:10, Room 322)
Session Chair: Songting Luo
Session Chair: Songting Luo
9:30–9:50 (D-1) Jeremy Hoskins, The University of Chicago — Fast algorithms for flexural gravity waves
9:50–10:10 (D-2) Cat Su Tran, Kansas State University — Numerical solution to the inverse electromagnetic scattering problem for periodic chiral media
10:10–10:30 (D-3) Tristan Goodwill, University of Chicago — A complex scaling method for junctions of semi-infinite interfaces
10:30–10:50 (D-4) Jacob Linden, University of Chicago — Acoustic Boundary Layers: A Boundary Integral Formulation
10:50–11:10 (D-5) Songting Luo, Iowa State University — A parallelizable, high-order exponential integrator for the wave equation
View session abstracts
D-1. Jeremy Hoskins
Fast algorithms for flexural gravity waves
Fast algorithms for flexural gravity waves
Flexural waves, the propagation of waves in thin elastic sheets, arise in a number of contexts, particularly in the study of ice shelves. In the frequency domain, they are commonly modeled as a fourth order PDE in two dimensions with clamped, free, or supported plate boundary conditions. We review existing approaches and propose novel representations reducing the problems to second-kind integral equations. These are amenable to high-order discretizations and fast algorithms. Numerical examples illustrate their effectiveness and generalizations to other wave phenomena.
D-2. Cat Su Tran
Numerical solution to the inverse electromagnetic scattering problem for periodic chiral media
Numerical solution to the inverse electromagnetic scattering problem for periodic chiral media
We study inverse scattering problems for Maxwell’s equation in periodic chiral media, focusing on recovering the location and shape of the scatterer. We propose an imaging function based on boundary measurements that is efficient and robust to noise. Theoretical analysis and numerical experiments validate the approach.
D-3. Tristan Goodwill
A complex scaling method for junctions of semi-infinite interfaces
A complex scaling method for junctions of semi-infinite interfaces
Scattering problems involving unbounded interfaces occur frequently in physics and engineering settings. Due to this prevalence, there exist many numerical methods for solving such problems. Unfortunately, the complicated behavior of solutions in the vicinity of infinite interfaces can make it challenging to deriving explicit error bounds for these methods. Many of these methods also require a large computational domain and so require a large number of discretization points to accurately solve the problem. In this talk, I present a class of decomposable scattering problems. For this class of problems, the PDE domain can be decomposed into a collection of simple subdomains. The fundamental solutions for these simple regions can then be used to reduce the scattering problem into an integral equation on the interfaces between these subdomains. These integral equations can be analytically continued into the complex plane, where they can be safely truncated. I demonstrate this procedure for a junction of two dielectric waveguides. For this problem, I show that the fundamental solutions and integral equation densities decay exponentially in the complex plane, and so the analytically continued integral equation can be truncated with provable exponential accuracy. I will also demonstrate how this method can be applied to a number of other scattering problems, including junctions of periodic gratings and corrugated waveguides.
D-4. Jacob Linden
Acoustic Boundary Layers: A Boundary Integral Formulation
Acoustic Boundary Layers: A Boundary Integral Formulation
We present boundary integral techniques for the Helmholtz equation with visco-thermal boundary conditions, modeling losses in acoustic devices. Using cancellations, image methods, and analytic preconditioners, we derive well-conditioned integral equations that enable fast and accurate solutions.
D-5. Songting Luo
A parallelizable, high-order exponential integrator for the wave equation
A parallelizable, high-order exponential integrator for the wave equation
We present a parallelizable high-order split-exponential integrator for the wave equation. The spatial operator is split into a nilpotent component containing the Laplacian and a scalar component given by a 2×2 matrix. Each exponential can be computed exactly, enabling stable and efficient high-order schemes. Combined with Gauss–Lobatto quadrature, the method reduces evaluations and allows straightforward parallelization. Numerical experiments and analysis demonstrate the effectiveness of the approach.
Session E — Kinetic, Plasma, and Transport Models (2:30–4:10, Room 101)
Session Chair: James Rossmanith
Session Chair: James Rossmanith
2:30–2:50 (E-1) Dauda Gambo, Iowa State University — Energy conserving semi-Lagrangian discontinuous Galerkin methods for multi-species Vlasov-Ampère models of plasma
2:50–3:10 (E-2) Shiping Zhou, Michigan State University — A deterministic particle method for the relativistic Landau equation
3:10–3:30 (E-3) Huan Lei, Michigan State University — From Micro-physics to Stable Macro-models: Variational Learning for Non-Newtonian Fluids
3:30–3:50 (E-4) Geshuo Wang, University of Washington — Dynamical Tensor Train Approximation for Kinetic Equations
3:50–4:10 (E-5) James Rossmanith, Iowa State University — High-Order Micro-Macro Decomposition Schemes for Kinetic Plasma Models
View session abstracts
E-1. Dauda Gambo
Energy conserving semi-Lagrangian discontinuous Galerkin methods for multi-species Vlasov-Ampère models of plasma
Energy conserving semi-Lagrangian discontinuous Galerkin methods for multi-species Vlasov-Ampère models of plasma
Long-time kinetic simulations of multi-species plasmas are challenged by severe ion-electron scale separation and by the need to preserve invariants under under-resolved Debye length and plasma time scales. We present a high-order, energy conserving semi-Lagrangian discontinuous Galerkin (SLDG) methods for the 1D1V multispecies Vlasov-Ampère system. The scheme advances the split phase space transport with conservative SLDG updates, eliminating the CFL constraint while preserving each species particle number at the fully discrete level. To control particle-field energy exchange, we introduce an exact, local field-current evolution for the acceleration stage by freezing the appropriate density moment; the coupled update reduces to a harmonic oscillator subsystem that provides a closed-form, time-dependent electric field for characteristic tracing. A local energy projection correction then compensates for the kinetic energy defect, restoring discrete conservation of the total energy. Accuracy and robustness are demonstrated through manufactured solution tests together with standard plasma benchmark problems.
E-2. Shiping Zhou
A deterministic particle method for the relativistic Landau equation
A deterministic particle method for the relativistic Landau equation
We propose a deterministic particle method for the spatially homogeneous relativistic Landau equation, based on a gradient-flow variational formulation with a regularized collision operator. This method preserves conservation laws and entropy decay, and achieves second-order accuracy. Comparison against theoretical equilibria, as well as the transition to the non-relativistic Landau case, confirms its accuracy and consistency, especially for low-speed particles, highlighting its potential for high-order time integration schemes in plasma simulations.
E-3. Huan Lei
From Micro-physics to Stable Macro-models: Variational Learning for Non-Newtonian Fluids
From Micro-physics to Stable Macro-models: Variational Learning for Non-Newtonian Fluids
We introduce a general approach for learning stable, interpretable macroscale PDEs by constructing the energy variational structure directly from microscale physical laws. In this framework, we introduce a set of micro-macro encoders to model the unresolved micro-physics as generalized field variables, along with an extendable energy functional and variational form that strictly preserve the conservation laws and entropy production. We illustrate this approach through the non-Newtonian hydrodynamics of polymeric fluids, a canonical multiscale problem where conventional empirical closures often fail. The resulting model naturally inherits microscale structural-dependent, nonlinear interactions that challenge empirical closures. More importantly, the variational informed formulation guarantees frame-indifference objectivity, free energy decay, and positivity-preserving dynamics; various pre-existing energy-stable numerical schemes can be used to establish long-time simulations. In contrast, the direct PDE form-based learning leads to models that may fit training data but fail beyond it due to instability and loss of physical fidelity.
E-4. Geshuo Wang
Dynamical Tensor Train Approximation for Kinetic Equations
Dynamical Tensor Train Approximation for Kinetic Equations
The numerical solution of kinetic equations is challenging due to the high dimensionality of the underlying phase space. In this paper, we develop a dynamical low-rank method based on the projector-splitting integrator in tensor-train (TT) format. The key idea is to discretize the three-dimensional velocity variable using tensor trains while treating the spatial variable as a parameter, thereby exploiting the low-rank structure of the distribution function in velocity space. In contrast to the standard step-and-truncate approach, this method updates the tensor cores through a sweeping procedure, allowing the use of relatively small TT-ranks and leading to substantial reductions in memory usage and computational cost. We demonstrate the effectiveness of the proposed approach on several representative kinetic equations.
E-5. James Rossmanith
High-Order Micro-Macro Decomposition Schemes for Kinetic Plasma Models
High-Order Micro-Macro Decomposition Schemes for Kinetic Plasma Models
Kinetic models are widely used to simulate the dynamics of plasmas and rarefied gases across many application areas, including astrophysics, magnetically confined fusion, high-altitude aircraft, various spacecraft propulsion mechanisms, and microfluidic devices. Kinetic models arise from a statistical-mechanics approximation of the underlying particle motion and typically take the form of nonlinear integro-differential equations defined on a high-dimensional phase space. The challenge of reducing this phase space to a lower-dimensional space is called the moment-closure problem. In this talk, I will describe an approach known as micro-macro decomposition that allows us to efficiently split the kinetic equations into their low-order statistics and the remaining high-order corrections. A high-order discontinuous Galerkin finite element method will be developed to numerically solve the resulting micro-macro system. The resulting scheme is verified on several standard test cases.
Session F — Neural Networks and PDEs (2:30–4:10, Room 105)
Session Chair: Qingguo Hong
Session Chair: Qingguo Hong
2:30–2:50 (F-1) Chuqi Chen, University of Michigan — Learn Sharp Interface Solution by Homotopy Dynamics
2:50–3:10 (F-2) Cesar Herrera, Purdue University — Efficient Neural Network Methods for Numerical PDEs: Singularly Perturbed Problems
3:10–3:30 (F-3) Nisha Chandramoorthy, The University of Chicago — Toward generative modeling for physical systems
3:30–3:50 (F-4) Qingguo Hong, Missouri University of Science and Technology — Greedy algorithms for neural networks approximations for indefinite problems
View session abstracts
F-1. Chuqi Chen
Learn Sharp Interface Solution by Homotopy Dynamics
Learn Sharp Interface Solution by Homotopy Dynamics
Solving partial differential equations (PDEs) using neural networks has become a central focus in scientific machine learning. Training neural networks for singularly perturbed problems is particularly challenging due to certain parameters in the PDEs that introduce near-singularities in the loss function. In this study, we overcome this challenge by introducing a novel method based on homotopy dynamics to effectively manipulate these parameters. From a theoretical perspective, we analyze the effects of these parameters on training difficulty and establish convergence of the proposed method. Experimentally, we demonstrate significant acceleration in convergence and improved accuracy.
F-2. Cesar Herrera
Efficient Neural Network Methods for Numerical PDEs: Singularly Perturbed Problems
Efficient Neural Network Methods for Numerical PDEs: Singularly Perturbed Problems
Approximating PDE solutions using piecewise linear functions is central to finite element methods. Shallow ReLU neural networks produce similar structures but with adaptive mesh points as parameters. This enables efficient handling of discontinuities and unknown interfaces. However, optimizing these parameters introduces computational challenges, which we analyze and address.
F-3. Nisha Chandramoorthy
Toward generative modeling for physical systems
Toward generative modeling for physical systems
Generative models produce samples from target distributions and can be interpreted as random dynamical systems. We investigate when such models generate physically meaningful samples and explore invertibility in noisy transformations. This work connects score-based methods and flow-matching frameworks with physical system modeling.
F-4. Qingguo Hong
Greedy algorithms for neural networks approximations for indefinite problems
Greedy algorithms for neural networks approximations for indefinite problems
We present error analysis and implementation of the Orthogonal Greedy Algorithm (OGA) for neural network approximation of indefinite elliptic problems. Numerical experiments validate the approach and highlight improved performance over conventional methods in non-coercive settings.
Session G — Numerical Methods and Nonlinear Problems (2:30–4:10, Room 310)
Session Chair: Xinyu Zhao
Session Chair: Xinyu Zhao
2:30–2:50 (G-1) Chushan Wang, University of Chicago — Optimal error bounds on the exponential integrator for dispersive equations with highly concentrated potential
2:50–3:10 (G-2) Huy Pham, Texas Tech University — Exponential Nyström integrators for stiff and highly oscillatory differential equations
3:10–3:30 (G-3) Galiya Myrzabayeva, Iowa State University — Limiting Strategies for High-Order Discontinuous Galerkin Methods Based on Entropy Dissipation
3:30–3:50 (G-4) Zhonggang Zeng, Northeastern Illinois University — Singularities of univariate equations
3:50–4:10 (G-5) Xinyu Zhao, New Jersey Institute of Technology — Systematic search for singularities in periodic 3D Euler flows
View session abstracts
G-1. Chushan Wang
Optimal error bounds on the exponential integrator for dispersive equations with highly concentrated potential
Optimal error bounds on the exponential integrator for dispersive equations with highly concentrated potential
We study a one-dimensional linear dispersive equation of order κ ≥ 2 with a highly concentrated potential of wavelength ε with 0 < ε ≪ 1, featuring a competition between weak dispersion of strength εα (0 ≤ α ≤ κ) and the localization induced by the concentrated potential. In particular, the second-order case arises from the equation of statistical moments of the solution to the Itô-Schrödinger equation in the scintillation scaling. We apply a standard first-order exponential integrator to solve the equation numerically, and establish an error bound of order O(τ εβ) (up to some logarithmic factors) where β = min{1 + (κ−1)α/κ, 2(1−α/κ)} ≥ 0. It should be noted that (i) the error bound is not only uniform in ε but improves as ε → 0 and (ii) there is no step size restriction on τ in terms of ε; hence, the error bound is valid even when τ ≫ ε. Extensive numerical results confirm the error estimates and suggest that the rate is optimal in both τ and ε.
G-2. Huy Pham
Exponential Nyström integrators for stiff and highly oscillatory differential equations
Exponential Nyström integrators for stiff and highly oscillatory differential equations
Second-order differential equations arise naturally in many physical and biological applications, including molecular dynamics, string vibration, and wave propagations. For the time integration of such systems, classical Runge-Kutta-Nyström integrators and their extended variants are widely used. While effective for small-scale or non-stiff problems, these methods often suffer from stability restrictions and inefficiency when applied to large, stiff, or highly oscillatory systems. To address these limitations, we develop and analyze a new class of time integration methods, termed exponential Nyström (expN) methods. These methods permit significantly larger time steps without sacrificing accuracy. Within the framework of strongly continuous semigroups on a Banach space, we prove convergence results up to fifth-order accuracy, with error bounds independent of the stiffness or high frequencies of the system. Our numerical experiments demonstrate that the proposed expN methods outperform existing Nyström-type integrators in both efficiency and accuracy.
G-3. Galiya Myrzabayeva
Limiting Strategies for High-Order Discontinuous Galerkin Methods Based on Entropy Dissipation
Limiting Strategies for High-Order Discontinuous Galerkin Methods Based on Entropy Dissipation
Subcell limiting has been extensively developed in a series of works by Dumbser and collaborators. The approach advances an unlimited high-order DG solution, applies local admissibility tests, and recomputes the solution only in troubled cells. In these cells, the DG solution is redistributed into subcell averages, evolved using a robust TVD or monotone finite-volume scheme, and then projected back to the DG space. Recent work by Chan (2025) introduces an entropy-correction artificial viscosity based on local violations of a cell entropy inequality and entropy dissipation, yielding nonzero viscosity near discontinuities. We combine these ideas by using Chan’s entropy-correction coefficient solely as a smoothness indicator to trigger an a posteriori subcell fallback. Cells with negligible entropy violation retain the DG solution, while cells with significant violation are recomputed on a finite-volume subgrid before projection back to DG. This strategy preserves high-order accuracy in smooth regions while providing robust resolution near discontinuities.
G-4. Zhonggang Zeng
Singularities of univariate equations
Singularities of univariate equations
Singular solutions inevitably arise in almost all types of equations. Solving such singular equations has been a challenge in scientific computing due to unbounded sensitivity and an insurmountable barrier that is known as attainable accuracy. This study investigates singularities in one of the most fundamental problems: solving a univariate equation for a zero of the underlying function under perturbations. Contrary to common assumptions, simple zeros of a function can still be singular and multiple zeros may not be. Numerical algorithms fail at singular multiple zeros because the conventional zero-finding model is ill-posed. However, the problem can be regularized by a least square model whose solution uniquely exists, enjoys Lipschitz continuity with respect to data, and accurately approximates the intended multiple zero in numerical computation even if the equation is perturbed. A simple iterative method is constructed for solving the least squares model with fast local convergence.
G-5. Xinyu Zhao
Systematic search for singularities in periodic 3D Euler flows
Systematic search for singularities in periodic 3D Euler flows
It remains one of the central questions in mathematical fluid mechanics whether solutions of the three-dimensional incompressible Euler equations can develop finite-time singularities from smooth initial conditions, i.e., whether certain norms of the solutions blow up in finite time. In this talk, I will present a numerical approach to this problem where we develop a PDE optimization method to systematically search for initial data that may lead to a potential singularity. The behavior of the obtained extreme flow, which features two colliding distorted vortex rings, suggests a finite-time singularity formation. This is based on a joint work with Bartosz Protas.
Session H — Finite Elements, DG, and Applied Scientific Computing (2:30–4:10, Room 322)
Session Chair: Dexuan Xie
Session Chair: Dexuan Xie
2:30–2:50 (H-1) Calvin Reedy, Washington University in St. Louis — HDG Methods in Finite Element Exterior Calculus
2:50–3:10 (H-2) Ziyao Xu, Binghamton University — A Conservative and Positivity-Preserving Discontinuous Galerkin Method for the Population Balance Equation
3:10–3:30 (H-3) Xu Zhang, Oklahoma State University — Frenet Immersed Finite Element Spaces on Triangular Meshes
3:30–3:50 (H-4) Connor Parrow, University of Notre Dame — A Fully Partitioned Robin-Robin Method for Fluid-Poroelastic Structure Interaction: Stability and Improved Convergence
3:50–4:10 (H-5) Dexuan Xie, University of Wisconsin–Milwaukee — Nonlocal Nonuniform Size Modified Poisson–Boltzmann Model and Efficient Finite Element Solver for Protein Electrostatics and Ion Distributions
View session abstracts
H-1. Calvin Reedy
HDG Methods in Finite Element Exterior Calculus
HDG Methods in Finite Element Exterior Calculus
The study of finite element exterior calculus has been primarily focused on analyzing conforming methods using the commuting projections developed by Arnold, Falk, and Winther. More recently, hybridization of both conforming and non-conforming methods for the Hodge–Laplace and Hodge–Dirac problems has been studied in works by Awanou et al. (2023) and Stern and Zampa (2025), but thus far error analysis has only been developed for conforming methods. In this talk, I will present the background and motivation for studying hybridizable non-conforming methods, in particular equal-order HDG methods, and outline an analysis inspired by Castillo et al. (2000) that shows optimal convergence for these methods under suitable assumptions on the domain.
H-2. Ziyao Xu
A Conservative and Positivity-Preserving Discontinuous Galerkin Method for the Population Balance Equation
A Conservative and Positivity-Preserving Discontinuous Galerkin Method for the Population Balance Equation
We develop a conservative, positivity-preserving discontinuous Galerkin (DG) method for the population balance equation (PBE), which models the distribution of particle numbers across particle sizes due to growth, nucleation, aggregation, and breakage. To ensure number conservation in growth and mass conservation in aggregation and breakage, we design a DG scheme that applies standard treatment for growth and nucleation, and introduces a novel discretization for aggregation and breakage. The birth and death terms are discretized in a symmetric double-integral form, evaluated using a common refinement of the integration domain and carefully selected quadrature rules. Beyond conservation, we focus on preserving the positivity of the number density in aggregation-breakage. Since local mass corresponds to the first moment, the classical Zhang–Shu limiter, which preserves the zeroth moment (cell average), is not directly applicable. We address this by proving the positivity of the first moment on each cell and constructing a moment-conserving limiter that enforces nonnegativity across the domain.
H-3. Xu Zhang
Frenet Immersed Finite Element Spaces on Triangular Meshes
Frenet Immersed Finite Element Spaces on Triangular Meshes
In this talk, we introduce geometry-conforming immersed finite element (GC-IFE) spaces on triangular meshes for elliptic interface problems. The proposed spaces are constructed using a Frenet–Serret mapping that transforms the interface curve into a straight line, allowing the interface jump conditions to be imposed exactly. Building on earlier work for rectangular meshes, we extend the framework to triangular meshes and present several construction strategies for high-order Frenet-based IFE spaces. Numerical experiments demonstrate the optimal approximation properties of the resulting spaces. We further utilize these GC-IFE spaces into interior penalty discontinuous Galerkin formulations for elliptic interface problems and observe optimal convergence rates in both the H1 and L2 norms.
H-4. Connor Parrow
A Fully Partitioned Robin-Robin Method for Fluid-Poroelastic Structure Interaction: Stability and Improved Convergence
A Fully Partitioned Robin-Robin Method for Fluid-Poroelastic Structure Interaction: Stability and Improved Convergence
Efficient and accurate numerical solvers for fluid-poroelastic structure interaction problems are critical to a variety of applications in geomechanics, biomedical engineering, and subsurface flow modeling. This work introduces and analyzes a fully partitioned method based on Robin-Robin coupling conditions for these problems. We propose to solve the fluid subproblem separately from the Biot subproblem, and to further decompose the Biot subproblem into a mechanics subproblem and a Darcy subproblem. We first consider a numerical method based on the Backward Euler time discretization, and show that it is conditionally stable, and sub-optimally convergent. To improve the rate of convergence, we introduce a second numerical method based on Cauchy’s theta-like scheme combined with Robin-Robin coupling conditions. We further investigate the convergence rates and the impact of the Robin-Robin coupling parameter, L, using numerical examples.
H-5. Dexuan Xie
Nonlocal Nonuniform Size Modified Poisson–Boltzmann Model and Efficient Finite Element Solver for Protein Electrostatics and Ion Distributions
Nonlocal Nonuniform Size Modified Poisson–Boltzmann Model and Efficient Finite Element Solver for Protein Electrostatics and Ion Distributions
This talk presents a new nonlocal, nonuniform size-modified Poisson–Boltzmann (NNSMPB) model for predicting electrostatic potentials and ionic concentration functions. The model couples a nonlocal interface boundary-value problem with nonlinear algebraic equations and is challenging to solve using finite element techniques due to solution singularities induced by atomic charges, discontinuous solute–solvent interface conditions, and nonlocal convolution terms. These challenges are addressed, resulting in an efficient NNSMPB finite element iterative algorithm and a corresponding software package capable of handling three-dimensional protein structures and ionic solutions containing multiple species with distinct ion sizes. Numerical experiments demonstrate the convergence of the iterative algorithm and the high performance of the software.
Session I — Computational Methods and Applications (9:30–11:10, Room 101)
Session Chair: Longfei Gao
Session Chair: Longfei Gao
9:30–9:50 (I-1) Sanchita Chakraborty, University of Notre Dame — Orientation-Sensitive MFPT on Fly Muscle Cell Geometries: Green’s Functions and Elliptical-Nucleus Optimization
9:50–10:10 (I-2) Yuguan Wang, University of Chicago — Fast multipole method with complex coordinates
10:10–10:30 (I-3) Joanna Held, Iowa State University — Micro-macro decomposition for modeling the kinetic Vlasov system
10:30–10:50 (I-4) Hunter La Croix, University of Notre Dame — A Lightning Solver for the solution of planar diffusion equations
10:50–11:10 (I-5) Longfei Gao, Argonne National Laboratory — Floating point arithmetic and system validation testing
View session abstracts
I-1. Sanchita Chakraborty
Orientation-Sensitive MFPT on Fly Muscle Cell Geometries: Green’s Functions and Elliptical-Nucleus Optimization
Orientation-Sensitive MFPT on Fly Muscle Cell Geometries: Green’s Functions and Elliptical-Nucleus Optimization
Diffusive first-passage processes are central to cellular transport, including signaling to nuclei in multinucleated muscle fibers, yet classical narrow capture theory typically models targets as isotropic disks. Motivated by microscopy data from fly muscle cells showing strongly anisotropic nuclear geometries, this talk develops a data-driven framework for mean first passage time (MFPT) problems with small absorbing elliptical targets in bounded two-dimensional domains. Using matched asymptotic analysis as the target size ε → 0+, I derive a two-term MFPT expansion that shows target orientation enters at O(ε2) through polarization tensors and derivatives of the regular part of the Neumann Green’s function. These predictions are validated against finite-element simulations in canonical geometries. I extend the theory to many elliptical targets, deriving leading-order linear systems for capture strengths with orientation-dependent corrections. To handle irregular cell geometries from fly muscle datasets, I compute Neumann Green’s functions using boundary integral methods (chunkIE). The resulting asymptotic surrogate enables fast optimization of nuclear positions, sizes, and orientations on experimental geometries, linking cell shape, nuclear anisotropy, and transport efficiency.
I-2. Yuguan Wang
Fast multipole method with complex coordinates
Fast multipole method with complex coordinates
In this work we present a variant of the fast multipole method (FMM) for efficiently evaluating standard layer potentials on geometries with complex coordinates in two and three dimensions. The complex scaled boundary integral method for the efficient solution of scattering problems on unbounded domains results in complex point locations upon discretization. Classical real-coordinate FMMs are no longer applicable, hindering the use of this approach for large-scale problems. Here we develop the complex-coordinate FMM based on the analytic continuation of certain special function identities used in the construction of the classical FMM. To achieve the same linear time complexity as the classical FMM, we construct a hierarchical tree based solely on the real parts of the complex point locations, and derive convergence rates for truncated expansions when the imaginary parts of the locations are a Lipschitz function of the corresponding real parts. We demonstrate the efficiency of our approach through several numerical examples and illustrate its application for solving large-scale time-harmonic water wave problems and Helmholtz transmission problems.
I-3. Joanna Held
Micro-macro decomposition for modeling the kinetic Vlasov system
Micro-macro decomposition for modeling the kinetic Vlasov system
Kinetic theory is a classical description of the thermodynamic behavior of gases. In this model, the movement and collisions of particles govern the large-scale behavior of a system. An important kinetic equation which describes the dynamics of particles in plasma is the Vlasov equation. Computationally, this is challenging to model because its dynamics are separated into collisional regimes. In a highly collisional zone, called the fluid regime, the distribution is driven toward equilibrium and can be modeled by fluid models such as Euler equations, Navier-Stokes, and magnetohydrodynamics equations. In regions with low interparticle collisions, called the kinetic regime, differences in length scales require the full kinetic equation be solved. To handle these two regions, we investigate using a micro-macro formulation to decompose the distribution function into the sum of its equilibrium and perturbations away from this equilibrium. Using this new formulation, we obtain coupled evolution equations which are equivalent to the original equation. We are interested in applying this method to Vlasov equations paired with different collision operators to describe plasma dynamics.
I-4. Hunter La Croix
A Lightning Solver for the solution of planar diffusion equations
A Lightning Solver for the solution of planar diffusion equations
This talk will describe a rapid, accurate and easy to implement numerical solution of the planar heat equation based on a lightning solver, a recent development in the numerical solution of linear PDEs which expresses solutions using sums of polynomials and rational functions, or more generally as sums of fundamental solutions. This method has demonstrated an ability to accurately represent solutions of PDEs with solution singularities. The method solves elliptic PDEs by forming series solutions whose coefficients are determined by an overdetermined linear system at a collocation grid. Our approach utilizes the Laplace transform to obtain a modified Helmholtz equation, solve it via the lightning method, and then numerical inversion of the Laplace transform to yield a solution to the diffusion problem. Our validation of the method against existing results and multiple challenging test problems, shows the method is robust across a wide range of time ranges and geometric configurations.
I-5. Longfei Gao
Floating point arithmetic and system validation testing
Floating point arithmetic and system validation testing
In this talk, I will present the challenges of deploying and supporting modern large scale computer systems such as Aurora at Argonne. In particular, I will focus on the need for well designed validation tests and how knowledge in floating point arithmetic, numerical linear algebra, and numerical methods in general can help in this effort. I will give an example using compensated summation algorithms and demonstrate how they can be applied to design tests that can detect small inadvertent errors caused by hardware defects or software bugs. Detailed error analysis involving floating point arithmetic will be provided, illustrating how analysis and practice may connect.
Session J — Computational Methods for PDEs and Related Problems (9:30–11:10, Room 105)
Session Chair: Xiangxiong Zhang
Session Chair: Xiangxiong Zhang
9:30–9:50 (J-1) Jielin Yang, University of Notre Dame — A new h-adaptive DG method with oscillation-elimination for Euler system
9:50–10:10 (J-2) Jue Yan, Iowa State University — Conservative cell-average-based neural network method for nonlinear conservation laws
10:10–10:30 (J-3) Chen Liu, University of Arkansas — Efficient optimization-based invariant-domain-preserving limiters in solving gas dynamics equations
10:30–10:50 (J-4) Zhuoran Wang, University of Kansas — A regularization approach to parameter-free block preconditioners for singular and nearly singular Stokes and poroelasticity problems
10:50–11:10 (J-5) Xiangxiong Zhang, Purdue University — Computing Gross–Pitaevskii Ground States by Wasserstein Gradient Flow on Diffeomorphism Space
View session abstracts
J-1. Jielin Yang
A new h-adaptive DG method with oscillation-elimination for Euler system
A new h-adaptive DG method with oscillation-elimination for Euler system
We will present a new h-adaptive discontinuous Galerkin method for compressible Euler equations. The scheme is both entropy-preserving and positive-preserving. In addition, we introduce an oscillation-elimination mechanism to damp numerical oscillations near shocks. Numerical examples will be presented to demonstrate the effectiveness of the proposed method.
J-2. Jue Yan
Conservative cell-average-based neural network method for nonlinear conservation laws
Conservative cell-average-based neural network method for nonlinear conservation laws
In this talk, we present the recently developed Cell-Average-Based Neural Network (CANN) for time-dependent PDEs. Spatial and temporal discretization in conventional methods is replaced by training a simple feedforward network to obtain an explicit one-step finite volume method. The well-trained network parameters function as the scheme coefficients. The network method has a minimum number of unknowns, which can be quickly trained on a small training dataset. Unlike conventional numerical methods, the CANN approach is not limited by the CFL conditions, enabling the use of significantly larger time steps. This leads to a highly efficient computational method for solving PDEs. The conservative version of the CANN method for nonlinear conservation laws will be discussed. The method, trained on the smooth-solution data from one initial-value problem, is verified to solve any initial-value problem involving shocks and rarefaction waves. The network can accurately learn a numerical flux such as the Lax–Friedrichs flux, which guarantees convergence to physically relevant entropy solutions. The extension of the method to non-uniform meshes will be considered. A bound-preserving version of the network method is presented for linear advection equations.
J-3. Chen Liu
Efficient optimization-based invariant-domain-preserving limiters in solving gas dynamics equations
Efficient optimization-based invariant-domain-preserving limiters in solving gas dynamics equations
The PDEs raised in modeling compressible flow, such as the compressible Euler and Navier–Stokes equations, are fundamental in gas dynamics with numerous applications. In this talk, we introduce effective splitting methods for implementing optimization-based limiters to enforce invariant domains in high-order numerical schemes. Key ingredients include an efficient explicit projection onto invariant sets and the use of Douglas–Rachford and Davis–Yin splitting methods. The approach can be applied broadly to construct high-order accurate, globally conservative, and invariant-domain-preserving schemes for compressible flows. Numerical tests validate robustness and performance of both L2– and L1-based limiters.
J-4. Zhuoran Wang
A regularization approach to parameter-free block preconditioners for singular and nearly singular Stokes and poroelasticity problems
A regularization approach to parameter-free block preconditioners for singular and nearly singular Stokes and poroelasticity problems
While Schur complement preconditioning has been widely studied for saddle point systems, challenges remain when dealing with singular and nearly singular systems that arise from Stokes flows and nearly incompressible poroelasticity flows. In this talk we will present a new approach for developing effective and parameter-free block Schur complement preconditioners for those saddle point systems. The key idea of this approach is to regularize original systems with inherent identities and construct preconditioners based on the regularized systems. It will be shown that these preconditioners are straightforward to construct and implement. Moreover, bounds on the eigenvalues of the preconditioned systems will be derived. The convergence of MINRES and GMRES applied to those systems will be analyzed and shown to be independent of physical parameters and mesh size. Numerical results in 2D and 3D will be presented.
J-5. Xiangxiong Zhang
Computing Gross–Pitaevskii Ground States by Wasserstein Gradient Flow on Diffeomorphism Space
Computing Gross–Pitaevskii Ground States by Wasserstein Gradient Flow on Diffeomorphism Space
We compute the ground state of the Gross–Pitaevskii equation (GPE) via Wasserstein gradient descent on diffeomorphism space. Writing ρ = u2 via the Madelung transform, we represent the density as the push-forward of a reference measure through a parameterized transport map realized by a Neural ODE. The method is mesh-free, preserves the unit-mass constraint, and exhibits strong performance in dimensions 1–3.
Session K — Time Integration, Integral Equations, and Applied Computation (9:30–11:10, Room 310)
Session Chair: Lei Wang
Session Chair: Lei Wang
09:30–09:50 (K-1) Van Hoang Nguyen, Texas Tech University — Two-derivative exponential Runge–Kutta methods
09:50–10:10 (K-2) Peter Nekrasov, University of Chicago — Efficient Representations for Elastodynamics: Integral Equations and Applications
10:10–10:30 (K-3) Baoli Hao, Illinois Institute of Technology — A Finite Element Framework for Crime Hotspot Simulation: From Agent-Based Models to Police Dynamics in Realistic Urban Geometries
10:30–10:50 (K-4) Rongbiao Wang, University of Chicago — A Ten-fold Way of Matrix Decompositions
10:50–11:10 (K-5) Lei Wang, University of Wisconsin–Milwaukee — Numerical experiments using the barycentric Lagrange treecode to compute correlated random displacements for Brownian dynamics simulations
View session abstracts
K-1. Van Hoang Nguyen
Two-derivative exponential Runge–Kutta methods
Two-derivative exponential Runge–Kutta methods
In this work, we propose a new class of time integration methods, referred to as two-derivative exponential Runge–Kutta (TDexpRK) methods for stiff semilinear parabolic PDEs. Specifically, we construct TDexpRK integrators that inherit the favorable properties of both two-derivative Runge–Kutta (TDRK) methods and explicit exponential Runge–Kutta (ExpRK) methods. In particular, TDexpRK methods treat the stiff linear part exactly via exponential operator, while handling the nonlinear term with a two-derivative correction weighted by exponential φ-functions of the linear operator. The structure of our TDexpRK schemes enables a local error expansion involving only four stiff order conditions for methods up to fifth order, which is significantly fewer than the sixteen conditions required for ExpRK integrators. Based on this analysis, we rigorously prove convergence up to fifth-order accuracy, with an error bound that remains uniform with respect to the stiffness of the linear operator. As a result, we obtain high-order, explicit, stiffly accurate TDexpRK schemes that exhibit unconditional linear stability and require only a few stages per step. Numerical experiments on PDEs in one and two spatial dimensions confirm the superior accuracy and efficiency of the proposed methods compared with existing exponential Runge–Kutta/Rosenbrock schemes from the literature.
K-2. Peter Nekrasov
Efficient Representations for Elastodynamics: Integral Equations and Applications
Efficient Representations for Elastodynamics: Integral Equations and Applications
Elastic waves are involved in a number of important phenomena, from the detection of earthquakes to the bending and flexing of floating ice sheets. These effects are frequently modeled using a time-harmonic wave equation with zero traction boundary conditions. In this talk, we show that it is possible to reduce the problem to an integral equation defined solely on the boundary of the domain. This integral representation uses charge strings to cancel singularities in the integral operators, leading to a Fredholm second kind integral equation. This formulation is readily amenable to existing fast algorithms, and we demonstrate its effectiveness by computing vibrational modes for some relevant problems of interest.
K-3. Baoli Hao
A Finite Element Framework for Crime Hotspot Simulation: From Agent-Based Models to Police Dynamics in Realistic Urban Geometries
A Finite Element Framework for Crime Hotspot Simulation: From Agent-Based Models to Police Dynamics in Realistic Urban Geometries
We present a finite element framework for a nonlinear coupled PDE system modeling residential burglary, derived from a probabilistic agent-based model via a mean-field limit. Unlike prior spectral approaches, our framework enforces natural Neumann boundary conditions, enabling simulations on realistic urban domains such as the city of Chicago. An iterative partitioned algorithm decouples the system at each time step into two sparse linear solves, achieving iteration counts between 1 and 6 across a wide range of parameters and mesh sizes. We further extend the model to incorporate dynamic police intervention, coupling criminal density to a time-lagged crime intensity map governing police movement. Linear stability analysis characterizes hotspot formation conditions and reveals oscillatory instability regimes. Numerical experiments validate both solvers against agent-based simulations, and all code is released open-source.
K-4. Rongbiao Wang
A Ten-fold Way of Matrix Decompositions
A Ten-fold Way of Matrix Decompositions
Matrix decomposition is a fundamental technique in matrix analysis from both theoretical and numerical perspectives. In this work, we provide a systematic treatment of matrix decompositions through the lens of Lie theory. This geometric framework theoretically organizes matrix factorizations into a tenfold classification that directly corresponds to the classes of compact symmetric spaces, topological insulators, and Gaussian ensembles. Furthermore, it provides a rigorous geometric interpretation of the variational methods used to compute these decompositions—generalizing concepts like the Rayleigh quotient and certain variational quantum algorithms. Through this unified framework, we extend classical results previously known only for specific factorizations to the entirety of the ten classes, including the existence of generalized Toda flows and double-bracket isospectral flows.
K-5. Lei Wang
Numerical experiments using the barycentric Lagrange treecode to compute correlated random displacements for Brownian dynamics simulations
Numerical experiments using the barycentric Lagrange treecode to compute correlated random displacements for Brownian dynamics simulations
Brownian dynamics simulations require correlated random displacements g = D1/2z, where D is the Rotne–Prager–Yamakawa diffusion tensor. The O(N2) cost of direct summation is prohibitive for large systems. We employ a barycentric Lagrange treecode to reduce the cost to O(N log N) while controlling approximation error. Numerical experiments compare performance and accuracy.
Poster Session (Day 1, 4:15–5:00 PM)
P-1 Abdul Quayam Khan, Universidade de Trás-os-Montes e Alto Douro — Distribute order fractional order PDEs
P-2 Soham Sarkar, The University of Texas at Dallas — Stability of pulses in a lumped model of a laser with a slow saturable absorber
P-3 Sayantan Sarkar, State University of New York at Buffalo — A Family of Second Order, Linear, Unconditionally Stable Methods for the Cahn-Hilliard-Navier-Stokes Equations
P-4 Angela Wang, University of Chicago — Inverse scattering for time-harmonic fractional waves via continuation in frequency
View poster abstracts
P-1. Abdul Quayam Khan
Distribute order fractional order PDEs
Distribute order fractional order PDEs
We provide an a posteriori error analysis for finite difference schemes on non-uniform meshes, for distributed-order differential equations. This analysis will allow the construction of mesh adaptive algorithms, for a posteriori adapted meshes. Although such schemes based on a posteriori error estimates already exist in the literature for single-term Caputo fractional differential equations, to the best of the author’s knowledge these cannot be found for distributed-order equations. Some numerical experiments and results will provided.
P-2. Soham Sarkar
Stability of pulses in a lumped model of a laser with a slow saturable absorber
Stability of pulses in a lumped model of a laser with a slow saturable absorber
We develop a computational model to analyze the stability of periodically stationary pulses in a lumped fiber laser model with slow saturable absorber. Pulse stability is determined by computing spectrum of the linearized roundtrip (monodromy) operator. We present a formula for its essential spectrum and describe a numerical method for computing discrete eigenvalues. Unlike previous models assuming instantaneous absorber response, our work incorporates a more realistic slow-response saturable absorber, extending prior work by Shinglot.
P-3. Sayantan Sarkar
A Family of Second Order, Linear, Unconditionally Stable Methods for the Cahn-Hilliard-Navier-Stokes Equations
A Family of Second Order, Linear, Unconditionally Stable Methods for the Cahn-Hilliard-Navier-Stokes Equations
Modeling complex interfacial dynamics in matched-density two-phase flows requires numerical methods that are both highly efficient and thermodynamically stable over extended integration periods. In this poster, we present a family of second-order, linear, unconditionally stable implicit-explicit (IMEX) finite element methods for the Cahn-Hilliard-Navier-Stokes (CHNS) equations, supported by various numerical simulations.
P-4. Angela Wang
Inverse scattering for time-harmonic fractional waves via continuation in frequency
Inverse scattering for time-harmonic fractional waves via continuation in frequency
We present a computational method for inverse scattering of time-harmonic fractional waves in inhomogeneous media. Such media are modeled by fractional Helmholtz equations with spatially varying potential. We evaluate the forward map using the adjoint Lippmann–Schwinger volume integral equation, discretized as a second-kind system and accelerated with FFTs. We then formulate the inverse problem as an optimization problem, and address its challenges via continuation in frequency, using reconstructions at low frequency to initialize successively higher-frequency problems. The performance of our approach is illustrated through several representative numerical experiments. The effects of potential amplitude, noise, and regularization are discussed, as well as comparisons to the standard Helmholtz scattering problem. We propose a regularization scheme to enforce frequency-consistent spectral discriminations, which improves the rate of convergence and stabilizes reconstructions.