| 9:00 | BREAKFAST |
|---|---|
| 9:30 | Alex Kaltenbach (Technical University of Berlin): A Prox-Based Semi-Smooth Newton Method for Convex Variational Problems |
| In this talk, we present a prox-based semi-smooth Newton method, which applies to a broad class of convex variational problems with non-differentiable energy densities, including TV-minimisation, p-Dirichlet type energies, elasto-plastic torsion, and obstacle-type problems. The key idea is to reformulate the subdifferential optimality relations using proximity operators, yielding a non-linear operator equation with a Newton-differentiable structure. Under mild local strong convexity assumptions on the energy densities, we establish local super-linear convergence via invertibility and uniform boundedness of Newton derivatives. The framework recovers established semi-smooth Newton schemes for classical obstacle-type problems and satisfies a primal-dual invariance. Numerical examples illustrate the efficiency of the method for representative model problems. | |
| 9:50 | Michael Ackermann (Virginia Tech): A refined adaptive nonlinear least-squares method for the rational approximation problem |
| In modern computational sciences, rational approximation has been gaining greater attention due to its rapid rate of approximation to non differentiable functions and functions with branch points. The adaptive Antoulas Anderson (AAA) algorithm has played a large role in generating this interest due to its ability to rapidly produce rational approximations accurate to a user defined tolerance. In short, AAA alternates between greedy interpolation and least-squares fitting. AAA's ability to produce rational approximations extremely quickly is derived from solving a linearized rational least-squares problem instead of the nonlinear rational least-squares problem. In some cases, this linearized least-squares problem introduces significant errors and causes AAA to produce a rational approximation with degree much larger than required to meet the specified tolerance. In this presentation, we present an algorithm (NL-AAA) derived from AAA which solves the rational least-squares problem instead of the linearized version. The minimizer is computed via a sequence of linear least-squares problems. We provide an analysis of the linearized least-squares problem which sheds light on some cases where the method fails. Finally, we demonstrate our algorithm on several numerical examples including applications to data driven modeling of dynamical systems, an application where low-degree rational approximations are crucial. | |
| 10:10 | Sylvia Amihere (University of Maryland Baltimore County): Embedded Strong Stability Preserving Runge Kutta Methods |
| Strong Stability Preserving (SSP) Runge-Kutta methods are time integration schemes designed to preserve monotonicity and nonlinear stability properties of spatial discretization methods when solving time-dependent differential equations. In recent times, new classes of problems have emerged which can be modeled by time-dependent equations that involve both stiff and non-stiff terms, where the non-stiff terms are solved explicitly and stiff terms are treated implicitly. This has motivated the development of Implicit-Explicit (ImEx) SSP methods designed for fixed time stepping. However, this may not be ideal for problems with rapidly changing dynamics. To address this limitation, we introduce a new class of ImEx SSP methods that employ adaptive time stepping to adjust step sizes while preserving stability properties. Numerical experiments are conducted to analyze the efficiency and robustness of these new methods. | |
| 11:00 | Yaw Owusu-Agyemang (George Mason University): Electromagnetic Source Cloaking Via Optimal Control |
| Maxwell’s equations are an important modeling component in a variety of modern engineering applications, ranging from nanoscale optical devices to magnetic confinement fusion. Despite the significant advances in the mathematical analysis and numerical methods for the solution of Maxwell’s equations, little progress has been made in our understanding of optimal control and optimal design problems governed by Maxwell’s equations. In this work we study the mathematical and computational challenges associated with such optimization problems. We establish the well-posedness of the forward problem and optimization problems under minimal regularity. Next, we derive the first order necessary and sufficient optimality conditions. The discretization of electric and magnetic fields are carried out using Nedelec and Raviart-Thomas finite elements. We establish the convergence of this discretization to the continuous problems. Finally, the efficacy of the theoretical framework is illustrated via numerical simulations done in MrHyDE, an advanced electromagnetic software and ROL (the Rapid Optimization Library), both developed at Sandia National Laboratory. | |
| 11:20 | Charles Parker (U.S Naval Research Lab): Structure-Preserving Finite Elements and Augmented Lagrangian Methods |
| Problems with physical constraints, such as the incompressibility constraint for mass conservation in fluids or Gauss's laws for electric and magnetic fields, often result in generalized saddle point systems. So-called structure-preserving finite elements respect the constraints pointwise, resulting in more physically accurate solutions that are typically robust with respect to some problem parameters. However, constructing these finite elements may involve complicated spaces for the Lagrange multiplier variables. Augmented Lagrangian methods (ALMs) provide one process to compute the solution without the need for an explicit basis for the Lagrange multiplier space. In this talk, we present new convergence estimates for a standard ALM method, sometimes called the iterated penalty method, applied to structure-preserving discretizations of linear saddle point systems. As a by-product, we also obtain new stability results for perturbed saddle point systems. Numerical examples drawn from incompressible flow, Helmholtz-Hodge decompositions, and \(C^1\)-continuous splines demonstrate the theory. | |
| 11:40 | Umarkhon Rakhimov (George Mason University): Non-overlapping domain decomposition methods for elliptic optimal control problems |
| Large-scale PDE-constrained optimization problems arise in applications such as optimal heat distribution, image processing, and shape optimization. Their discretization by, e.g., the finite element method leads to large linear systems whose solution demands substantial memory and computational resources, motivating the use of parallel algorithms on modern high-performance architectures. Domain decomposition methods are a natural framework for parallelism: the computational domain is partitioned into subdomains, and the global problem is solved by coupling local subdomain solves on separate processors through a linear system for the unknowns shared along subdomain boundaries. This interface system becomes increasingly ill-conditioned as the mesh is refined or the number of subdomains increases. Balancing Domain Decomposition by Constraints (BDDC) preconditioners address this by enforcing continuity at a small subset of interface unknowns, creating a global coupling mechanism that enables information to propagate across the entire decomposition. In this talk, we develop a BDDC preconditioner for elliptic optimal control problems with control and state constraints and sparsity-promoting \(L^1\)-regularization. The constraints render the optimality system nonsmooth, and a semismooth Newton method yields a sequence of symmetric but indefinite linear systems whose conditioning deteriorates as the regularization parameters and mesh size tend to zero. In contrast to classical BDDC methods for elliptics PDEs, where the preconditioner is assembled from local Neumann solves, each subdomain solve here is itself a local optimal control problem. Because these local problems inherit the saddle-point structure of the global system, the preconditioner is indefinite, rendering the preconditioned system nonsymmetric and necessitating GMRES. Our numerical experiments demonstrate robustness with respect to the regularization parameters and logarithmic growth of GMRES iterations in the ratio of subdomain diameter to mesh size. | |
| 12:00 | Mansur Shakipov (University of Maryland, College Park): Transient Surface Stokes Without Inf-Sup Condition |
| Using geometric identities, we reformulate the transient Stokes problem posed on a closed hypersurface as a system of a parabolic problem for the velocity and an elliptic problem for the pressure. We then discretize this reformulation using (lifted) Lagrange finite elements on surfaces and derive uniform stability and quasi-best approximation results that do not appear possible for bounded Euclidean domains. | |
| 3:00 | Muhammad Jalil Ahmad
(University of Maryland Baltimore County): A Quantum Optimization Framework for Parameter Estimation in Epidemiological and Chaotic Systems |
| Parameter estimation is a fundamental challenge in the calibration of ordinary differential equation (ODE) models, where repeated numerical integration can lead to high computational cost. In this work, we investigate whether near-term quantum algorithms can be leveraged to assist parameter estimation in nonlinear dynamical systems. Focusing first on the susceptible–infected–recovered (SIR) model, we develop a hybrid classical–quantum framework that reformulates a data-assimilation–based parameter estimation problem as a combinatorial optimization task. Model dynamics and data assimilation are enforced entirely on the classical side, while the resulting parameter estimation objective is discretized and approximated by a Quadratic Unconstrained Binary Optimization (QUBO) surrogate. This surrogate is mapped to an Ising Hamiltonian and minimized using the Quantum Approximate Optimization Algorithm (QAOA). We further apply this approach to the Lorenz system in the chaotic regime. In this setting, the method is used to recover classical system parameters from partial state observations, illustrating its ability to handle chaotic dynamical systems. Numerical experiments with synthetic data show that the proposed approach accurately recovers parameters and reproduces observed dynamics while requiring substantially fewer ODE solves than classical iterative methods. The framework avoids quantum state tomography and is compatible with near-term quantum hardware, illustrating a viable pathway for integrating quantum optimization into data-driven parameter estimation for epidemiological and chaotic dynamical systems. | |
| 3:20 | Mahathi Vempati
(University of Maryland, College Park): Quantum Algorithms for PDEs in Heterogeneous Media: The Diffusion Eigenvalue Problem |
| Partial differential equations (PDEs) arise in many real-world applications, where they often have discontinuous coefficients and geometries, presenting significant challenges for classical numerical methods. In this work, we investigate one such PDE in discontinuous media: the diffusion k-eigenvalue problem, which is relevant to nuclear reactor design. To solve it, we consider the standard method of uniform finite elements, and show that quantum algorithms implementing this method can provide significant polynomial end-to-end speedups over their classical counterparts. This speedup leverages recent advances in quantum linear systems---"fast inversion" and "quantum preconditioning"---and uses Hamiltonian simulation as a subroutine. Our results suggest that quantum algorithms may provide speedups for heterogeneous PDEs, though the extent of this advantage over the fastest classical algorithm depends on the effectiveness of other classical approaches such as nonuniform or adaptive meshing for a given problem instance. | |
| 3:40 | Gianluca Fabiani (Johns Hopkins University): Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach |
|
We address a numerical framework for the stability and bifurcation analysis of nonlinear partial differential equations (PDEs) in which the solution is sought in the function space spanned by physics-informed random projection neural networks (PI-RPNNs), and discretized via a collocation approach. These are single-hidden-layer networks with randomly sampled and fixed a priori hidden-layer weights; only the linear output layer weights are optimized, reducing training to a single least-squares solve. This linear output structure enables the direct and explicit formulation of the eigenvalue problem governing the linear stability of stationary solutions. This takes a generalized eigenvalue form, which naturally separates the physical domain interior dynamics from the algebraic constraints imposed by boundary conditions, at no additional training cost and without requiring additional PDE solves. However, the random basis collocation matrix is inherently numerically rank-deficient, rendering naive eigenvalue computation unreliable and contaminating the true eigenvalue spectrum with spurious near-zero modes. To overcome this limitation, we introduce a matrix-free shift-invert Krylov-Arnoldi method that operates directly in weight space, avoiding explicit inversion of the numerically rank-deficient collocation matrix. This enables the reliable computation of several leading eigenpairs of the PDE Jacobian, whose eigenvalue spectrum determines linear stability. We further prove that the PI-RPNN-based generalized eigenvalue problem is almost surely regular, guaranteeing solvability with standard eigensolvers, and that the singular values of the random projection collocation matrix decay exponentially for analytic activation functions. Reference: Fabiani, G., Kavousanakis, M. E., Siettos, C., & Kevrekidis, I. G. (2026). Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach. arXiv preprint arXiv:2603.21568 | |
| 4:00 | Di Wu (University of Maryland, College Park): Beyond Expected Information Gain: Stable Bayesian Optimal Experimental Design with Integral Probability Metrics and Plug-and-Play Extensions |
| Bayesian Optimal Experimental Design (BOED) provides a rigorous framework for decision-making tasks in which data acquisition is often the critical bottleneck, especially in resource-constrained settings. Traditionally, BOED typically selects designs by maximizing expected information gain (EIG), commonly defined through the Kullback-Leibler (KL) divergence. However, classical evaluation of EIG often involves challenging nested expectations, and even advanced variational methods leave the underlying log-density-ratio objective unchanged. As a result, support mismatch, tail underestimation, and rare-event sensitivity remain intrinsic concerns for KL-based BOED. To address these fundamental bottlenecks, we introduce an IPM-based BOED framework that replaces density-based divergences with integral probability metrics (IPMs), including the Wasserstein distance, Maximum Mean Discrepancy, and Energy Distance, resulting in a highly flexible plug-and-play BOED framework. We establish theoretical guarantees showing that IPM-based utilities provide stronger geometry-aware stability under surrogate-model error and prior misspecification than classical EIG-based utilities. We also validate the proposed framework empirically, demonstrating that IPM-based designs yield highly concentrated credible sets. Furthermore, by extending the same sample-based BOED template in a plug-and-play manner to geometry-aware discrepancies beyond the IPM class, illustrated by a neural optimal transport estimator, we achieve accurate optimal designs in high-dimensional settings where conventional nested Monte Carlo estimators and advanced variational methods fail. | |
| 4:50 | Gitansh Dandona (University of Delaware): A Discontinuous Galerkin Method with Cell-Wise Hierarchical Tucker Tensor Decomposition for Multiscale BGK Dynamics |
| BGK kinetic models arise in rarefied gas dynamics and plasma transport, but their numerical simulation is severely challenged by the high dimensionality of phase space, particularly in the velocity variables. In this work, we develop a high-order discontinuous Galerkin (DG) solver for the one-dimensional, two-velocity (1D2V) BGK equation that combines cell-wise tensor compression with robust time integration and shock stabilization. Within each spatial cell, the velocity dependence of the distribution function is represented in a low-rank Hierarchical Tucker (HT) tensor format, enabling efficient compression and adaptive control of tensor ranks. This cell-localized tensor structure significantly reduces memory and computational cost while preserving high-order accuracy. Time integration is performed using implicit–explicit (IMEX) Runge–Kutta schemes to efficiently handle the stiff relaxation regime associated with small Knudsen numbers. | |
| 5:10 | Nour Al Hassanieh (Flatiron Institute): Truncated kernel windowed Fourier projection: a fast algorithm for the 3D free-space wave equation |
| We present a spectrally accurate fast algorithm for evaluating the solution to the scalar wave equation in free space driven by a large collection of point sources in a bounded domain. With \(M\) sources temporally discretized by \(N_t\) time steps of size \(\Delta t\), a naive potential evaluation at $M$ targets on the same time grid requires \(O(M^2N_t)\) work. Our scheme requires \(O((M + N^3\log N)N_t)\) work, where \(N\) scales as \(O(1/\Delta t)\), i.e., the maximum signal frequency. This is achieved by using the recently-proposed windowed Fourier projection (WFP) method to split the potential into a local part, evaluated directly, plus a smooth history part approximated by an \(N^3\)-point equispaced discretization of the Fourier transform, where each Fourier coefficient obeys a simple recursion relation. The growing oscillations in the spectral representation (which would be present with a naive use of the Fourier transform) are controlled by spatially truncating the hyperbolic Green’s function itself. Thus, the method avoids the need for absorbing boundary conditions. We demonstrate the performance of our algorithm with up to a million sources and targets at 6-digit accuracy. We believe it can serve as a key component in addressing time-domain wave equation scattering problems. | |
| 5:30 | Kwassi Joseph Dzahini (Argonne National Laboratory): Noise-Aware Scalable Stochastic Trust-Region via Adaptive Random Subspaces |
| We introduce ANASTAARS, a noise-aware stochastic trust-region algorithm with adaptive random subspace strategies that is effective from low to potentially large dimensions. The method achieves scalability by optimizing random models restricted to low-dimensional affine subspaces defined via Johnson--Lindenstrauss transforms, thereby sharply reducing function evaluation cost per iteration. Unlike earlier random subspace approaches with fixed dimensions and fully rebuilt poised sets, ANASTAARS adaptively selects the subspace dimension and updates the model by adding only a few new interpolation points while reusing past evaluations in a way that preserves poisedness, yielding substantially more efficient progress. To handle noisy objectives, it incorporates a noise-aware trust-region mechanism based on local estimates of the noise level, and its effectiveness is demonstrated on problems arising from the quantum approximate optimization algorithm (QAOA) framework. | |
| 5:50 | END |