8:30	BREAKFAST
9:30	Sean Carney (George Mason University): Uncertainty in uncertainty and Rockafellian relaxation
	A critical aspect of PDE constrained optimization is to account for uncertainty in the underlying physical models, for example in model coefficients, boundary conditions, and initial data. Uncertainty in physical systems is modeled with random variables, however, in practice there may be some nontrivial ambiguity in the underlying probability distribution from which they are sampled. As stochastic optimal control problems are known to be ill-conditioned to perturbations in the sampling distribution, we describe an analytic framework that is better conditioned to such meta-uncertainties and conclude with numerical examples.
9:50	Benjamin Grimmer (Johns Hopkins University): Computer-assisted design of state-of-the-art gradient descent methods
	This talk will discuss a recent work series of works (by the presenter as well as by Altschuler and Parrilo) using computer assistance to design state-of-the-art stepsize policies of gradient descent. Considering a formulation of the algorithm design problem as a semidefinite program, the best worst-case guarantees follow from using nonconstant stepsize policies with frequent long steps potentially violating descent. We show utilizing such steps provably gives a big-O improvement in convergence of \(O(1/T^{1.2716})\) over the classic \(O(1/T)\). This process for designing these state-of-the-art methods, known as performance estimation, has proven successful across numerous areas of optimization over the last decade.
10:10	Shanyin Tong (Columbia University): Policy iteration method for inverse mean field games
	Mean-field games study the Nash Equilibrium in a non-cooperative game with infinitely many agents, they have a wide range of applications in engineering, economics, and finance. The coupling structure of the forward-backward equations of MFG raises specific difficulties in finding solutions to MFG, and makes it even more challenging to solve its inverse problem. In this talk, we will introduce a policy iteration method for solving inverse MFG, giving the initial data of value functions and recovering the potential or obstacle function in the environment. The method takes ideas from policy iterations for forward MFG and separates the inversion step as solving a linear inverse problem, which substantially simplifies the problems and results in an efficient algorithm. We will use several numerical examples to demonstrate the accuracy and efficiency of the proposed method, and also discuss its convergence.
11:00	Mansur Shakipov (University of Maryland, College Park): Inf-sup stability of parabolic TraceFEM
	We present an inf-sup analysis of continuous-in-time, discrete-in-space TraceFEM discretization of the (fixed) surface heat equation. The analysis relies on two components. First is the modification of the normal derivative volume stabilization, wherein in addition to stabilizing the solution one also stabilizes the time derivative with a different stabilization scaling. We refer to this scheme as total stabilization. The second component is the study of the stabilized \(L^2\)-projection. This analysis leads to convergence to minimum regularity solutions, symmetric error bound in an energy error functional, and ``textbook" condition number estimates. The analysis does not take into account the error in approximating the geometry and time discretization.
11:20	Max Heldman (Virginia Tech): Error estimates for averaging-type quadrature rules applied to reaction-drift-diffusion equations with discontinuous reaction coefficients
	We prove an abstract convergence result for a family of averaging quadrature rules on tensor products of simplical meshes. In the context of the multilinear tensor-product finite element discretization of reaction-drift-diffusion equations, our quadrature rule generalizes the mass-lump rule, retaining its most useful properties: for a nonnegative reaction coefficient, it gives an \(O(h^2)\)-accurate, nonnegative diagonalization of the reaction operator. The major advantage of our scheme in comparison with the standard mass lumping scheme is that, under mild conditions, it produces an \(O(h^2)\) consistency error even when the integrand has a non-mesh-aligned jump discontinuity. The primary application of our method is to systems of reaction-drift-diffusion equations related to particle-based stochastic reaction-diffusion simulations (PBSRD). In this context, the discrete reaction operator is required to be an \(M\)-matrix and a standard model for bimolecular reactions has a discontinuous reaction coefficient. Crucially, for this application the mesh cannot be aligned to the discontinuity. We describe the application to PBSRD systems and present convergence and superconvergence studies confirming the theory.
11:40	Rohit Khandelwal (George Mason University): Elliptic reconstruction and a posteriori error estimates for parabolic variational inequalities
	The elliptic reconstruction property, originally introduced by Makridakis and Nochetto for linear parabolic problems, is a well-known tool to derive optimal a posteriori error estimates. No such results are known for nonlinear and nonsmooth problems such as parabolic variational inequalities (VIs). In this talk, we establish the elliptic reconstruction property for parabolic VIs and derive a posteriori error estimates in \(L^{\infty}(0,T;L^{2}(\Omega))\) and \(L^{\infty}(0,T;L^{\infty}(\Omega))\), respectively. As an application, the residual-type error estimates are presented. Extension to VIs of the second kind will also be discussed.
12:00	Hamad El Kahza (University of Delaware): Krylov based adaptive rank implicit methods for nonlinear Fokker Planck models
	We propose high-order adaptive-rank implicit integrators for stiff time-dependent PDEs, leveraging extended Krylov subspaces to efficiently and adaptively populate low-rank solution bases. This allows for the accurate representation of solutions with significantly reduced computational costs. We further introduce an efficient mechanism for residual evaluation and an adaptive rank-seeking strategy that optimizes low-rank settings based on a comparison between the residual size and the local truncation errors of the time-stepping discretization. We demonstrate our approach with the challenging Lenard-Bernstein Fokker-Planck (LBFP) nonlinear equation, which describes collisional processes in a fully ionized plasma. The preservation of the equilibrium state is achieved through the Chang-Cooper discretization, and strict conservation of mass, momentum, and energy via a Locally Macroscopic Conservative (LoMaC) procedure. The development of implicit adaptive-rank integrators, demonstrated here up to third-order temporal accuracy via diagonally implicit Runge-Kutta schemes, showcases superior performance in terms of accuracy, computational efficiency, equilibrium preservation, and conservation of macroscopic moments. This study offers a starting point for developing scalable, efficient, and accurate methods for high-dimensional time-dependent problems.
3:00	Steven Rodriguez (U.S. Naval Research Laboratory): Developments in meshless model reduction
	The present work addresses limitations of traditional dimensionality reduction applied to meshless nonlocal methods in multiphysics modeling and simulations. Meshless nonlocal methods (MNMs) are versatile computational frameworks that enable effective modeling and simulation of complex multiphysics phenomena, from magnetohydrodynamics in astrophysics to multiphase flows in additive manufacturing. However, MNM methods are significantly less sparse in their numerical infrastructure relative to more traditional local/mesh-based methods such as finite element, finite volume, or finite difference methods. As a result, MNM methods can be more computationally expensive than traditional approaches. Model reduction would be a prime candidate to reduce the expense of these frameworks. However, traditional model reduction is not fit for the unstructured nature of meshless methods since it relies on structured data to extract low-dimensionality. This work presents an approach to enable meshless model reduction through structured reference spaces where unstructured meshless integration points can evolve in a structured low-dimensional space. The proposed approach will be showcased on a series of fluid dynamic simulations via the smoothed-particle hydrodynamic framework.
3:20	Philip Charles (University of Maryland, College Park): Pseudo-spectral methods for the integral fractional Laplacian
	When solving hyperbolic conservation laws, satisfaction of the entropy condition ensures the selection of a physically relevant (weak) solution. Entropy stable numerical schemes satisfy a discrete version of the entropy condition. In the framework of high-order entropy stable finite difference schemes, a popular class of solvers is the TeCNO schemes, which rely on reconstruction algorithms satisfying a crucial sign-property at the cell-interfaces. However, only a few reconstructions are currently known to satisfy this property. Recently, third order sign-preserving weighted essentially non-oscillatory (WENO) schemes called SP-WENO have been proposed for use in entropy stable finite difference schemes. While the resulting schemes are provably entropy stable, they fail to acceptably capture shocks, with the numerical solutions often exhibiting large oscillations near discontinuities. In this talk, we propose an alternate deep learning-based approach to constructing WENO schemes possessing the sign-property. A neural network is used to determine the polynomial weights of the WENO scheme, while being strongly constrained to satisfy the sign-property. The network is trained on smooth and discontinuous data that represent the local solution features in conservation laws. Additional constraints are built into the network to guarantee the expected order of convergence (for smooth solutions) with mesh refinement. Several numerical results are presented to demonstrate significant improvement over the existing variants of WENO with the sign-property, specifically from the perspective of shock-capturing capabilities.
3:40	Daniel Serino (Los Alamos National Laboratory): Intelligent attractors for singularly perturbed dynamical systems
	Singularly perturbed dynamical systems, commonly known as fast-slow systems, play a crucial role in various applications such as plasma physics. They are closely related to reduced order modeling, closures, and structure-preserving numerical algorithms for multiscale modeling. A powerful and well-known tool to address these systems is the Fenichel normal form, which significantly simplifies fast dynamics near slow manifolds through a transformation. However, the Fenichel normal form is difficult to realize in conventional numerical algorithms. In this work, we explore an alternative way of realizing it through structure-preserving machine learning. Specifically, a fast-slow neural network (FSNN) is proposed for learning data-driven models of singularly perturbed dynamical systems with dissipative fast timescale dynamics. Our method enforces the existence of a trainable, attracting invariant slow manifold as a hard constraint. Closed-form representation of the slow manifold enables efficient integration on the slow time scale and significantly improves prediction accuracy beyond the training data. We demonstrate the FSNN on several examples that exhibit multiple timescales, including the Grad moment system from hydrodynamics, two-scale Lorentz96 equations for modeling atmospheric dynamics, and Abraham-Lorentz dynamics modeling radiation reaction of electrons in a magnetic field.
4:00	Zezheng Song (University of Maryland, College Park): A finite expression method for solving high-dimensional committor problems
	Transition path theory (TPT) is a mathematical framework for quantifying rare transition events between a pair of selected metastable states \(A\) and \(B\). Central to TPT is the committor function, which describes the probability to hit the metastable state \(B\) prior to \(A\) from any given starting point of the phase space. Once the committor is computed, the transition channels and the transition rate can be readily found. The committor is the solution to the backward Kolmogorov equation with appropriate boundary conditions. However, solving it is a challenging task in high dimensions due to the need to mesh a whole region of the ambient space. In this work, we explore the finite expression method (FEX, Liang and Yang (2022)) as a tool for computing the committor. FEX approximates the committor by an algebraic expression involving a fixed finite number of nonlinear functions and binary arithmetic operations. The optimal nonlinear functions, the binary operations, and the numerical coefficients in the expression template are found via reinforcement learning. The FEX-based committor solver is tested on several high-dimensional benchmark problems. It gives comparable or better results than neural network-based solvers. Most importantly, FEX is capable of correctly identifying the algebraic structure of the solution which allows one to reduce the committor problem to a low-dimensional one and find the committor with any desired accuracy.
4:50	Zachary Miksis (Temple University): A new convergent Fick-Jacobs derivation with applications in computational neurophysiology
	Modeling diffusion processes in neurons has become an essential tool for researchers developing an understanding of the impact of various clinical treatments. This can quickly become an extremely expensive computational problem when modeling a network of neurons in three dimensions. To reduce this cost, we are interested in modeling neurons as collections of one-dimensional cables by reducing the three-dimensional equations to one dimension. This dimension reduction must take into account the non-constant radius of the dendrites and axons. Jacobs proposed an intuitive and straight-forward derivation of this reduction in 1935 based on Fick's laws, but it wasn't until 1992 that Zwanzig further studied this Fick-Jacobs model and demonstrated that it was unstable for domains with a quickly changing radius. Since then, multiple corrections on the diffusion coefficient or temporal derivative have been proposed to stabilize the Fick-Jacobs model for steep domains, with limited success. We propose a new simple derivation of the Fick-Jacobs model utilizing a high-order expansion of the computed flux that, when applied in a discrete computational setting, produces second-order spatial convergence and provides a much more accurate numerical solution at a wide range of radial gradients than previous correction methods.
5:10	Rasika Mahawattege (University of Maryland Baltimore County): Boundary stabilization for a Heat-Kelvin-Voigt unstable interaction model, with control and partial observation localized on the interface only
	A prototype model for a Fluid-Structure interaction is considered. We aim to stabilize [enhance stability] of the model by having access only to a portion of the state. Toward this goal we shall construct a compensator-based Luenberger design, with the following two goals: (1) reconstruct the original system asymptotically- by tracking partial information about the full state, (2) stabilize the original unstable system by feeding an admissible control based on a system which is obtained from the compensator. The ultimate result is boundary control/stabilization of partially observed and originally unstable fluid-structure interaction with restricted information on the current state and without any knowledge of the initial condition. This prevents applicability of known methods in either open-loop or closed loop stabilization/control.
5:30	Olusola Olabanjo (Morgan State University): Finite element analysis of lung parenchyma: material and parameter optimization
	The lung parenchyma plays a crucial role in respiratory physiology and patho-physiology. Finite Element Method (FEM) offers insights into its poroelastic behavior, essential for understanding lung mechanics and diseases. Aim: This study aims to explore the effects of material selection and parameter tuning on the biomechanics of lung parenchyma using FEM. Materials and Methods: Using poroelastic and hyperelastic material models, lung parenchyma deformation was simulated under various physiological conditions. Parameters such as Young's modulus and Poisson's ratio were varied to study their impact on lung mechanics. Results: The study found that poroelastic models, which consider both solid and fluid phases, provide a superior simulation of lung mechanics compared to traditional hyperelastic models. Sensitivity analysis highlighted the significant influence of material properties and parameters on lung tissue behavior. Conclusion: Variations in material properties and parameters like Young's modulus and Poisson's ratio critically affect lung tissue's mechanical response. The study enhances understanding of lung biomechanics, offering potential improvements in clinical diagnostics and treatments.
5:50	END