ABSTRACT. We study a class of Hamiltonian partial differential equations introduced by Bridges (2006) in space and space-time. These systems satisfy a multisymplectic conservation law, which can be viewed as a multidimensional generalization of the symplectic properties inherent in Hamiltonian ODEs. A central question in structure-preserving discretization is determining which finite element methods maintain this property at the discrete level. To address this, we introduce a unified hybridization framework that allows for the simultaneous analysis of local conservation properties across diverse discretization families. We demonstrate that while many methods, such as certain hybridizable discontinuous Galerkin (HDG) schemes, satisfy a strong version of multisymplecticity, the standard conforming finite element exterior calculus (FEEC) methods of Arnold, Falk, and Winther (2006) satisfy this property only in a weak sense. Finally, we establish the bridge between this local multisymplectic perspective and the classical treatment of PDEs as infinite-dimensional Hamiltonian ODEs. This is a joint work with Ari Stern (WashU).

ABSTRACT. In this talk, we introduce a new framework for variational problems governed by fractional differential equations, where the fractional derivatives depend on an auxiliary function. We establish a systematic variational methodology and derive the necessary optimality conditions for functionals defined through these generalized operators.

Beginning with the classical variational setting, we obtain the associated Euler–Lagrange equation and extend the analysis to more general situations, including isoperimetric and Herglotz type problems.

ABSTRACT. In this talk, we advance fractional optimal control (OC) theory by proving a version of Pontryagin’s maximum principle and establishing sufficient optimality conditions for an OC problem. The dynamical system constraint in the OC problem under investigation is described by a generalized fractional derivative: the left-sided Caputo distributed-order fractional derivative with an arbitrary kernel. This approach provides a more versatile representation of dynamic processes, accommodating a broader range of memory effects and hereditary properties inherent in diverse physical, biological, and engineering systems.

ABSTRACT. The rapid evolution of smartphone markets, characterized by intense competition, short product life cycles, and strong network effects, demands robust analytical frameworks to support strategic decision-making. This study develops a dynamic innovation diffusion framework combined with optimal control theory to examine marketing and pricing strategies that maximize product adoption and revenue throughout the smartphone lifecycle. The proposed approach captures innovation-driven adoption, social influence effects, and market saturation dynamics typical of high-technology environments. Within this setting, an optimal control problem is formulated where marketing effort, pricing intensity, and promotional timing act as policy instruments shaping the diffusion trajectory.

Numerical simulations calibrated with representative smartphone market parameters illustrate how adaptive strategies can accelerate early adoption, delay market saturation, and enhance long-term profitability. The findings emphasize threshold effects in marketing intensity, the relevance of early promotional investment, and the sensitivity of optimal policies to social interaction strength and market potential.

ABSTRACT. In this talk, we revisit the SICA (Susceptible–Infected–Chronic–AIDS) compartmental model, commonly used to model the transmission dynamics of HIV/AIDS, exploring its formulation through ordinary, fractional, and stochastic differential equations. After, we study optimal control problems associated with SICA models, applying both direct and indirect numerical methods to evaluate control strategies under diverse epidemic scenarios.

ABSTRACT. We propose a family of high-order local discontinuous Galerkin (LDG) methods, built on a parametric representation and coupled with a semi-implicit backward Euler time discretization, for isotropic and anisotropic curve-shortening flows. The spatial LDG formulation introduces auxiliary variables and carefully designed numerical fluxes which inherit the underlying variational structure. We prove the unconditional energy dissipation for the semi-discrete scheme, and establish the well-posedness for the fully discrete scheme under mild assumptions. For Pk approximations, the LDG method achieves high-order spatial convergence; extensive numerical experiments confirm optimal (k + 1)-order accuracy when the surface energy is isotropic or weakly anisotropic. Compared to classical parametric finite element methods (PFEM), the proposed LDG schemes do not need to rely on good mesh distributions or auxiliary symmetrized surface energy matrices for strongly anisotropic surface energy cases, and remain numerically stable on severely degraded meshes that typically cause PFEMs failure. This intrinsic stability enables effective capture of complex geometric evolution and sharp corner singularities produced by strong anisotropy. The approach thus provides a flexible and reliable framework for the numerical simulation of a broader class of geometric flows.

ABSTRACT. In physics and mathematics, a large class of PDE systems can be formulated as minimizing energy functionals subject to certain constraints. Lagrange multipliers are widely used for solving these problems, which however leads to minmax optimization problems, i.e., saddle point systems. The development of fast solvers for saddle point systems, especially the nonlinear ones, is particularly difficult in the sense that (i) one has to consider the preconditioning in two directions and (ii) the preconditioners have to evolve in iteration due to the nonlinearity. In this work, we introduce an efficient transformed primal-dual (TPD) algorithm to solve the aforementioned nonlinear saddle point problems. One typical application of the present work is accelerate the JKO scheme in modeling free interface dynamics.

ABSTRACT. We propose a second-order-in-time, structure-preserving, and mesh-robust parametric finite element method for surface diffusion and volume-preserving mean curvature flow. The original evolution equations are first reformulated into new systems in which the tangential motion is governed by a harmonic map heat flow. This heat flow maps a fixed reference surface onto the evolving surface and induces tangential motion that reduces the associated harmonic energy. Consequently, in the discrete setting, the mesh quality can be maintained at a level comparable to that of the reference surface unless singularities occur. The volume-preserving property is theoretically guaranteed through a careful design of the scheme, while energy dissipation is enforced via a Lagrange multiplier. Several numerical experiments are presented to demonstrate the second-order temporal convergence and the effectiveness of the proposed method in maintaining mesh quality.

ABSTRACT. Numerical approximation of geometric evolution equations that involve the first variation of the Willmore energy is particularly challenging due to the fact that highly nonlinear and fourth order terms appear. In this talk, I will introduce a new parametric fintie element method for Willmore flow of hypersurfaces in a unified framework. The method is linear and employs a splitting of the normal and tangential velocity of the flow. The normal velocity is approximated via an evolution equation for the curvature, and follows the arbitrary Lagrangian-Eulerian approach. This enables an unconditional energy stability with respect to the discrete energy. We also incorporate the `BGN’tangential velocity through a curvature identity. This helps to preserve the mesh quality. We show various numerical examples to demonstrate the favorite properties of the method.

ABSTRACT. Classical Propagation of chaos-based particle method (PoC) was developed for solving mean-field stochastic differential equations and their associated nonlinear Fokker-Planck equations. Recently, a new particle method known as the sequential propagation of chaos (SPoC) whose particle systems replace the fully interaction in PoC by sequential interaction. However, PoC and SPoC are not feasible for high dimensional problems due to their computational complexity and heavy memory requirement. In this work, we present DeepSPoC, that combines the interacting particle system of SPoC with the deep learning approach, yarding an efficient approach for tackling high dimensional problems. In DeepSPoC, a newly developed normalizing flow model called KRnet is used to approximate the empirical measure of particles. Compared with PoC and SPoC, DeepSPoC substantially reduces the memory required, computes particle interactions more efficiently, and is much more effective at solving high-dimensional problems. We apply DeepSPoC to a wide range of different types of mean-field equations and verify its effectiveness and advantages.

ABSTRACT. We present a novel framework for uncertainty reduction in machine learning (ML) models for data-driven dynamical system prediction. While ML provides an efficient surrogate for simulating dynamical systems, predictive uncertainty persists—even with abundant data—and can accumulate over time, leading to degraded long-term performance. To address this challenge, we incorporate data assimilation techniques into the training pipeline to iteratively refine model predictions by integrating observational information.

Specifically, we employ the Ensemble Score Filter (EnSF), a generative AI–based, training-free diffusion-model approach, to solve data assimilation problems in high-dimensional, nonlinear systems. This results in a hybrid data assimilation–training framework that couples ML models with EnSF to enhance long-term predictive accuracy and stability. We demonstrate that EnSF-enhanced ML effectively reduces predictive uncertainty in the Lorenz–96 system and in simulations of the Korteweg–De Vries (KdV) equation.

ABSTRACT. The talk will present recent developments in generative modeling, compressive sensing, statistical machine learning, and data assimilation tools to design data reduction methods uniquely capable for the emerging infrastructure being built for interconnected science and analysis of computational, experimental, and observational data. A critical feature of scientific data, in contrast to other types of data, is that (possibly unknown) physical principles lie at the foundation. Hence the reduction methods in this proposal seek to optimize for known physical properties as well as discover the underlying characteristics of unknown physical properties. Importantly, while scientific research is constantly growing in size and scale, there are efforts to evolve into a new paradigm that considers more connected, collaborative, autonomous, and real-time environments -- creating exciting opportunities and challenges for data reduction which this presentation aims to address.

ABSTRACT. Autoencoders help to reduce computational complexity in machine learning and simulations of data-driven dynamical systems by encoding high-dimensional data on a low-dimensional latent space. The literature demonstrates that an exploit of geometric prior information (such as variational or Hamiltonian structure) in autoencoders and in latent spaces improves reliability and structural quality of data-driven simulations. However, the extrapolation power of data-driven dynamical systems on latent spaces remains limited.

In this talk, I will show the structural benefits of temporal-spatial variational structure for simulations and machine learning. I will show how to incorporate discrete Lagrangian stencils that are local in time and space into data-driven auto encoders and latent space models.

ABSTRACT. Developing geometric integrators for the simulation of dynamical systems is a central theme in structure-preserving numerical analysis. In this talk, we present recent advances based on symplectic groupoids and Poisson geometry that provide a geometric framework for the construction of integrators for Hamiltonian dynamics. We show how these structures naturally encode fundamental properties such as symplecticity, conservation laws, and invariants associated with Poisson systems, leading to simulation methods that remain faithful to the underlying geometry. This perspective connects naturally with ideas from geometry-aware learning.

ABSTRACT. Discretisation errors are unavoidable when numerically solving evolution equations, regardless of whether one employs conventional discretisation-based solvers or neural network–based approaches. Excessively high accuracy can lead to unnecessary computational cost, whereas insufficient accuracy may introduce systematic bias, for example, in inverse problems or data-driven modelling. Therefore, quantifying such errors beyond the scope of asymptotic error analysis has recently become an important topic. In particular, this task is nontrivial in the context of nonlinear dynamics.

In this talk, we present a new approach to this motivation. The central idea is to model the discretisation errors as random variables and to estimate their statistical properties using available computational and observational data. More specifically, we formulate a state-space model that includes additional random variables representing discretization errors, and estimate these variables using filtering techniques. The proposed framework incorporates prior information from both theoretical numerical analysis, such as the order of accuracy or structure preservation, and empirical observations. We present numerical experiments demonstrating that the method can capture the behaviour of discretisation errors in a quantitative manner.

ABSTRACT. In this we work focuses on learning non-canonical Hamiltonian dynamics from data, where long-term predictions require the preservation of structure both in the learned model and in numerical schemes. Previous research focused on either facet, respectively with a potential-based architecture and with degenerate variational integrators, but new issues arise when combining both. In experiments, the learnt model is sometimes numerically unstable due to the gauge dependency of the scheme, rendering long-time simulations impossible. Here, first, we identify this problem and secondly we propose two different training strategies to address it, either by directly learning the vector field or by learning a time-discrete dynamics through the scheme. Several numerical test cases assess the ability of the methods to learn complex physical dynamics, like the guiding center from gyrokinetic plasma physics.

ABSTRACT. Sampling probability distributions on submanifolds is a relevant task in various problems in statistics. Specialized Markov Chain Monte Carlo (MCMC) methods were originally designed to address similar challenges in statistical physics and were recently adopted in various statistical contexts, as hypotheses testing, ABC, Generalized fiducial inference. However, these MCMC methods have proven to be adaptable to a variety of sampling problems in Bayesian statistics which are not naturally defined on submanifolds and offer promising avenues for efficiently sampling probability distributions.

ABSTRACT. In this talk I will discuss different constraint mechanisms in the context of underdamped Langevin dynamics. Specifically, I will speak about the realisation of constraints by either strong confinement forces or tuning physical parameters such as friction or mass, which fall in the class of so-called soft constraint techniques and include a parameter $\eps$ giving weight to the constraint mechanism. I will present quantitative pathwise convergence results for the constrained variables as well as for the unconstrained variables to a limit dynamics. This includes a discussion about (a) the requirement that the initial conditions satisfy the constraint, (b) requirements regarding the potential, (c) further implications of the results, and (d) associated invariant distributions.

ABSTRACT. Locating index-1 saddle points of an energy landscape is a central computational problem in nonconvex optimisation, with applications ranging from rare-event sampling to transition-state search in quantum chemistry. Existing methods — nudged elastic band, dimer, eigenvector-following — are inherently local: they refine a saddle near a given initialisation. The Witten Laplacian offers a complementary, global picture: its low-lying spectrum encodes the index-1 saddles of the entire landscape simultaneously, with eigenforms that concentrate on the saddles in the semiclassical limit. To expose the information encoded in the Witten Laplacian to computational methods we discuss a probabilistic interpretation via an augmented Langevin SDE and discuss a Monte-Carlo algorithm that concentrates on index-1 saddles, with $O(d)$ cost per particle, and embarrassingly parallel execution.

ABSTRACT. In this talk, I will discuss the problem of constructing numerical solutions for kinetic Langevin dynamics with holonomic constraints and complex heat bath models, e.g., tensor-valued friction and diffusion coefficients. I will discuss particular challenges posed by such settings and introduce a class of numerical integration schemes that address these.

ABSTRACT. The SUNDIALS library of time integrators and nonlinear solvers has long provided efficient adaptive multistep and multistage time integration methods. New capabilities in SUNDIALS have been added to support multiscale and multiphysics applications, allowing them the flexibility to switch between single rate and various approaches for handling multiple rate problems. These approaches include several operator splitting as well as multirate methods. This talk will start with a brief overview of the SUNDIALS library and then describe some of the newest features. Results from combustion, climate, and quantum dynamics applications will be presented.

Prepared by LLNL under Contract DE-AC52-07NA27344. LLNL-ABS-2007999.

ABSTRACT. Differentiable programming is the underpinning technology for the AI revolution. It allows neural networks to be programmed in very high level user code while still achieving very high performance for both the evaluation of the network and, crucially, its derivatives. The Firedrake project applies exactly the same concepts to the simulation of physical phenomena modelled with partial differential equations (PDEs). By exploiting the high level mathematical abstraction offered by the finite element method, users are able to write mathematical operators for the problem they wish to solve in Python. The high performance parallel implementations of these operators are then automatically generated, and composed with the PETSc solver framework to solve the resulting PDE. However, because the symbolic differential operators are available as code, it is possible to reason symbolically about them before the numerical evaluation. In particular, the operators can be differentiated with respect to their inputs, and the resulting derivative operators composed in forward or reverse order. This creates a differentiable programming paradigm congruent with (and compatible with) machine learning frameworks such as Pytorch and JAX.

In this presentation, I will present Firedrake in the context of differentiable programming, and show how this enables productivity, capability and performance to be combined in a unique way. I will also touch on the mechanism that enables Firedrake to be coupled with Pytorch and JAX.

ABSTRACT. The deal.II library (https://www.dealii.org) provides a comprehensive set of data structures and algorithms in which users build finite element applications that solve partial differential equations. In this talk, I will provide an overview of the library's features, as well as of the development processes that over the past 28 years have led to a software with nearly 2 million lines of C++ code and a large user and developer community.

ABSTRACT. In this talk we present an overview of MFEM (https://mfem.org), a scalable library for high-order finite element discretization of PDEs on general unstructured grids that employs partial assembly and matrix-free algorithms to power a wide variety of HPC applications. Our approach to efficient operator evaluation is based on a decomposed representation of the finite element operator, that factors a bilinear form into a series of sparse and dense components corresponding to the parallelism, mesh topology, basis, geometry, and pointwise physics in the problem. This exposes several layers of parallelism, enables the use of batched dgemss and tensor contractions, and only requires quadrature point values to be assembled for computing the action. The "partial assembly" formulation is a natural fit for modern HPC hardware because it results both in less (nearly optimal) computation and less (optimal) data movement compared to assembling a global sparse matrix, therefore increasing performance and reducing time to solution. In addition to discussing MFEM's capabilities and algorithms, and demonstrate their impact in several large-scale applications from the US Department of Energy.

ABSTRACT. Accurate coarse-grained (CG) models depend on reliable estimates of the potential of mean force (PMF), but standard force matching approaches require expensive unbiased sampling from the Boltzmann distribution and often fail to accurately capture transition regions. We show that performing enhanced sampling directly in CG coordinate space, combined with unbiased recomputation of forces, speeds up exploration, improves convergence, and leads to better PMF estimation. Building on this, we introduce Coarse-Grained Boltzmann Generators, which combine flow-based generative models with importance sampling in CG space guided by a learned PMF. Our framework enables scalable and statistically exact sampling of complex molecular systems.

ABSTRACT. Metastable states and the conformational transitions in between them are key to understanding the dynamical behavior and function of large-scale molecular systems. By combining basic dimensionality reduction techniques with a state-of-the-art approximation of the Koopman operator associated with molecular dynamics simulations (MD), we show that these states and transitions can be analyzed very efficiently based on MD simulation data. To construct the Koopman approximation, we employ a kernel-based method and solve the associated matrix equations using random Fourier features, leading to accurate solutions while maintaining low computational effort. On a benchmark set of fast-folding proteins, we demonstrate that key properties such as transition timescales, free energies, secondary structure elements, and hydrogen bonding patterns can be computed with remarkable robustness across hyperparameter regimes.

ABSTRACT. Sampling metastable probability measures remains a central challenge in high-dimensional computational statistical physics. Classical Monte Carlo and Langevin-based samplers often suffer from exponentially large mixing times induced by energetic and entropic barriers. A promising strategy is to introduce non- local proposal moves defined in a reduced space of collective variables (CVs), yet existing constructions primarily address linear CVs and overdamped dynamics. In this talk, we develop a general mathematical framework for non-local Monte Carlo proposals associated with nonlinear collective-variable maps in the setting of underdamped Langevin dynamics. Starting from the geometric structure of the CV mapping, we define a lifted Markov transition kernel that performs non-local moves in CV space while preserving the target measure on the full phase space. We derive sufficient conditions for reversibility, establish detailed balance of the resulting scheme, and analyze how the coupling between momentum, nonlinearity of the CV map, and the proposal distribution affects acceptance probabilities and effective mixing rates. Our results provide a principled extension of CV-based acceleration techniques to settings of intermediate CV dimensionality, where generative models can be used to approximate proposal distributions. Numerical experiments serve only to illustrate the theory: the main emphasis will be on the structural, measure-theoretic, and algorithmic properties that guarantee correctness and improved sampling performance.

ABSTRACT. We develop an algorithm to solve high-dimensional risk-averse optimization problems governed by differential equations (ODEs and/or PDEs) under uncertainty. As an example, we focus on the so-called Conditional Value at Risk (CVaR), but the approach is equally applicable to other coherent risk measures. To avoid non-smoothness of the objective function underpinning the CVaR, we propose an adaptive strategy to select the width parameter of the smoothed CVaR to balance the smoothing and approximation errors. This enables both Newton's method for asymptotically quadratic convergence in the number of iterations, and higher-order (such as spectral tensor) approximation of the high-dimensional random fields for exponential convergence in the number of degrees of freedom.

ABSTRACT. We discuss statistical learning of nonlinear mappings between infinite-dimensional spaces. Given a nonlinear (ground truth) map between separable Hilbert spaces, we study recovery of the ground truth from input-output data consisting of randomly drawn 'design' points in the input space and measurements under white noise in the output space. To infer the ground truth within a Bayesian framework, we choose a (Gaussian) prior on operator level, show well-posedness of the posterior, and establish algebraic convergence rates for the posterior mean, the maximum a posteriori and the maximum likelihood estimator towards the ground truth in terms of the data size. As an application, our theory yields algebraic learning rates for the solution operator of the diffusion equation which we undermine with numerical results.

ABSTRACT. Kernel interpolation, particularly in the context of Gaussian process emulation, is a powerful tool for surrogate modelling, allowing complex input–output maps to be approximated from relatively few function evaluations. However, in high-dimensional settings, these methods rapidly succumb to the curse of dimensionality---where the number of required function evaluations typically grows exponentially with respect to the input dimension. Fortunately, in many practical applications, functions often exhibit anisotropy in their parameters; where the influence of input parameters on the output can vary greatly. Different formulations of such functional anisotropy are routinely exploited in the high-dimensional approximation literature to mitigate dimensional dependence in the error, enabling the practical approximation of very high dimensional functions.

In this talk, we introduce a generalisation of sparse grid methods for kernel interpolation that incorporates anisotropy through the lengthscale parameter of Matérn kernels, inspired by the empirical success of hyperparameter estimation in high-dimensional Gaussian process regression. We present error bounds demonstrating reduced dependence on the input dimension, together with numerical experiments demonstrating accurate interpolation in very high dimensions when sufficient anisotropy is present.

ABSTRACT. Subdiffusion dynamics with evolving memory arise in systems where the influence of past states varies over time, reflecting changes in internal configurations or external conditions. In this talk, models governed by Scarpi-type kernels, formulated through convolution operators in the Laplace domain, are presented. Since these kernels are generally not available in closed form in the time domain, solutions are computed via numerical inversion of the Laplace transform, while temporal discretization is carried out using convolution quadrature. In the absence of exact solutions, this inversion approach also serves as a highly accurate reference benchmark. The impact of evolving memory on decay rates and transient behavior, as well as convergence to classical fractional dynamics in the limiting case, is investigated.

ABSTRACT. Fractional differential equations of variable order allow one to capture system dynamics more effectively in situations where the memory effect is not constant.

The numerical treatment of variable-order fractional differential equations is more challenging, particularly when fractional derivatives are defined in the Laplace transform domain rather than directly in the time domain. As this approach is attracting increasing interest, it is important to investigate efficient numerical strategies.

In this talk, we focus on additional difficulties arising from variable-order transitions with low regularity. As a model problem, we consider piecewise transitions in which the order of the fractional derivative is constant at early and late times, but undergoes a linear variation over a finite time interval. Unlike the case of smooth transitions, standard numerical methods for Laplace transform inversion become unreliable in this setting, and more sophisticated approaches are required.

We discuss the origin of these difficulties and outline suitable numerical methods to address them.

ABSTRACT. We investigate a numerical method for approximating stochastic Caputo fractional differential equations driven by a multiplicative noise. The nonlinear functions $f$ and $g$ are assumed to satisfy the global Lipschitz conditions as well as the linear growth conditions. The noise is approximated by a piecewise constant function, yielding a regularized stochastic fractional differential equation. We prove that the error between the exact solution and the solution of the regularized equation converges in the $L^2((0, T) \times \Omega)$ norm with an order of $O(\Delta t^{\alpha - 1/2})$, where $\alpha \in ( 1/2, 1]$ is the order of the Caputo fractional derivative, and $\Delta t$ is the time step size. Numerical experiments are provided to confirm that the simulation results are consistent with the theoretical convergence order.

ABSTRACT. In this talk the focus lies on two space-fractional reaction-diffusion equations.

The first model is the Gray-Scott system from chemistry. This system of reaction-diffusion PDEs allows a diversity of different types of patterns: splitting pulses, periodic solutions, traveling waves, and many more. The space-fractional version changes the behaviour of the patterns significantly. An adaptive moving mesh method, based on a non-uniform L2-discretization is applied to deal with the localized moving patterns.

The second model is a stationary Gelfand-Bratu model (with fractional Laplacian) of a completely different character. This model gives rise to an S-shaped bifurcation diagram, depending on the model parameter lambda and the fractional order alfa. Here, the numerical method is based on fractional powers of a finite-difference matrix in the non-linear solver of the discretized system.

Numerical experiments illustrate the theoretical observations for both models.

ABSTRACT. Plasmas inherently exhibit multiscale dynamics, and the Vlasov–Maxwell system provides a fundamental fully kinetic description of such processes. Most existing kinetic methods rely on explicit schemes, which are easy to implement but require time steps small enough to resolve the plasma period and may suffer from numerical heating in long-time simulations. Implicit schemes offer improved stability but require the solution of costly nonlinear systems. Semi-implicit methods therefore provide an appealing compromise between efficiency and stability. In particular, ECsim introduced an energy-conserving semi-implicit particle-in-cell framework. However, the design of efficient, unconditionally stable, grid-based kinetic schemes for the Vlasov–Maxwell system with exact energy conservation remains a major challenge. In this talk, I will present an inherently noise-free, energy-conserving semi-Lagrangian (ECSL) scheme that retains the simplicity and efficiency of explicit methods while offering the stability advantages of implicit schemes. Numerical results demonstrate its accuracy, efficiency, and conservation properties, making ECSL a promising approach for multiscale plasma simulations.

ABSTRACT. We develop a new implicit geometric Particle-in-Cell method for the Vlasov–Maxwell system. The method combines Galerkin Difference spatial discretization with a semi-implicit discrete gradient time integrator based on Poisson Splitting proposed by Kormann & Sonnendrücker (J. Comput. Phys, vol. 425, pages 109890, 2021). The Galerkin Difference framework provides compact, high-order Lagrange bases that achieve higher accuracy without increasing the number of interior degrees of freedom. Geometric quantities on the grid, point values, edge integrals, face integrals, and volume integrals, are used as degrees of freedom. Duality is achieved by the use of one grid and mass matrices. The resulting implicit geometric particle algorithm is implemented on uniform Cartesian meshes in GEMPICX code and validated through multi-dimensional benchmark problems, including the Landau damping, the Weibel and two-stream instabilities.

ABSTRACT. Fluid models provide collective fluid-like description of plasma and are of interest when micro-scale kinetic effects can be neglected. Relativistic effects are important both in laser-plasma interaction and certain astrophysical environments. In this talk, we consider a relativistic cold fluid model and derive a finite-element discretization that conserves mass, energy and divergence constraints on the semi-discrete level. FFor time discretization, we derive an implicit energy-conserving average-vector field method or apply an explicit strong-stability preserving Runge-Kutta scheme. We also consider a coupling of the fluid model to relativistic particles. We perform a numerical study of the scheme which shows convergence and conservation properties of the proposed methods and apply the new scheme to a plasma wake field simulation.

ABSTRACT. This work is devoted to the numerical approximation of high-dimensional advection-diffusion equations. It is well-known that classical methods, such as the finite volume method, suffer from the curse of dimensionality, and that their time step is constrained by a stability condition. The semi-Lagrangian method is known to overcome the stability issue, while recent time-discrete neural network-based approaches overcome the curse of dimensionality. In this work, we propose a novel neural semi-Lagrangian method that combines these last two approaches. It relies on projecting the initial condition onto a finite-dimensional neural space, and then solving an optimisation problem, involving the backwards characteristic equation, at each time step. It is particularly well-suited for implementation on GPUs, as it is fully parallelisable and does not require a mesh. We provide rough error estimates, and present several high-dimensional numerical experiments to assess the performance of our approach, and compare it to other neural methods.

ABSTRACT. Implicit time-stepping methods for atmospheric and oceanic fluid dynamics lead to large, ill-conditioned linear systems that must be solved efficiently at each timestep. A particular challenge arises in domains with small aspect ratios where standard iterative solvers and preconditioners based on point smoothers degrade. In this talk, we present a robust preconditioning strategy for the fully nonlinear incompressible Boussinesq equations in a two-dimensional vertical slice, designed to address this challenge within a multigrid framework.

The velocity-buoyancy-pressure block arising from an implicit discretisation is preconditioned using a shifted auxiliary operator, which decouples the pressure from the momentum equation. This system is then solved using a multigrid method with vertex-patch line smoothers oriented along the vertical direction, exploiting the anisotropic structure of the mesh. Linearising the equation around the stage of rest leads to an extension of the Mixed Poisson equation. Focusing on the linearised equation, we present an analysis of the shift parameter selection. Numerical results on the Skamarock-Klemp test case confirm the effectiveness and robustness of the approach, with iteration counts and solve times reported for the solver. The method is implemented within the Firedrake finite element framework using Irksome package.

ABSTRACT. Implicit high-order time integrators offer high accuracy over long time steps. High-order Galerkin-in-time discretizations enable preservation of invariants and better stability. However, these methods are typically deemed prohibitively expensive, since they involve the solution of large linear systems that couple together all temporal modes or stages.

In this work, we enabled scalable solvers for Galerkin time integrators in Irksome, a time stepping library built on top of the Firedrake finite element library. We introduce a special basis for Galerkin-in-time discretizations that enables a block preconditioner that decouples the temporal modes. The temporal finite element degrees of freedom are carefully chosen to obtain an equivalent implicit Runge-Kutta collocation method (with the collocated quadrature rule). The equivalent Runge-Kutta method has a cheaper Jacobian, and admits triangular approximations that enable stage-segregated block preconditioners. We illustrate our approach with numerical examples.

ABSTRACT. Fully implicit timestepping methods have several potential advantages for atmosphere/ocean simulation. First, being unconditionally stable, they degrade more gracefully as the Courant number increases, typically requiring more solver iterations rather than suddenly blowing up. Second, particular choices of implicit timestepping methods can extend energy conservation properties of spatial discretisations to the fully discrete method. Third, these methods avoid issues related to splitting errors that can occur in some situations, and avoid the complexities of splitting methods. Fully implicit timestepping methods have had limited application in geophysical fluid dynamics due to challenges of finding suitable iterative solvers, since the coupled treatment of advection prevents the standard elimination techniques. However, overlapping Additive Schwarz methods provide a robust, scalable iterative approach for solving the monolithic coupled system for all fields and Runge-Kutta stages. In this study, we investigate this approach applied to the rotating shallow water equations, facilitated by the Irksome package, which provides automated code generation for implicit Runge-Kutta methods. We compare various schemes in terms of accuracy and efficiency using an implicit/explicit splitting method, namely the ARK2 scheme of Giraldo et al (2013), as a benchmark. This provides an initial look at whether implicit Runge-Kutta methods can be viable for atmosphere and ocean simulation.

ABSTRACT. I will present a fully discrete energy conserving discretisation for the rotating shallow water equations on the sphere, using compatible finite element methods combined with the continuous Petrov-Galerkin method, enabled by Firedrake and Irksome. This will start by discussing energy conserving time discretisations in general before specialising to the rotating SWE. I will show some reformulation tricks that allow for efficient solution of these equations. If time allows, I will also provide updates about a modified version of the scheme that provides local energy conservation.

ABSTRACT. Numerical investigations of dynamo have remained active in fluid mechanics over the past decades, presenting numerous challenges and open problems. Meanwhile, the development of structure-preserving methods and finite element exterior calculus (FEEC) inspires a revisit of numerical dynamo studies and explore existing open questions in computation. In this paper, we present a FEEC approach for dynamo problems. In particular, we investigate structure-preserving finite element schemes for solving the spectra and pseudo-spectra of advection-diffusion operators for differential forms. The scheme and its analysis are based on finite element de~Rham complexes.

ABSTRACT. I will discuss two recent approaches to constructing intrinsic finite element spaces for tensor fields, developed jointly with Evan Gawlik.

One approach, blow-up finite elements, was motivated by a vexing problem when discretizing tangent vector fields on surfaces: With a polyhedral discretization of the surface, the angles at a vertex no longer sum to 360 degrees; as a result, it is not possible to construct a vector field that is continuous within each element, tangent to the surface, and continuous across each edge (in the sense of having no jump in the components tangent to the edge and normal to the edge). Previous approaches either broke tangentiality to the surface or continuity across edges. With blow-up finite elements, we can keep both of these properties by allowing the vector fields to vary rapidly near vertices. I will define these elements for vector fields and tensor fields, discuss some preliminary numerical results, and discuss potential applications to numerical geometry and to intrinsic discretization of the surface Stokes equations for creeping flow.

The second approach was motivated by recent interest in extending finite element exterior calculus from differential forms to double forms, also known as form-valued forms, for applications such as elasticity and numerical geometry. A key property of finite element exterior calculus spaces is their invariance under affine transformations, which, in particular, means that the spaces work the same way on surfaces as they do in the plane. I will discuss our construction of affine-invariant spaces of double forms with polynomial coefficients, the surprising connections to representation theory, and ongoing work on generalizing these results to arbitrary covariant tensor fields.

ABSTRACT. This talk addresses a variational discretization of incompressible Euler flow based on the Koopman representation on half-densities. After choosing finite element spaces for scalar functions and vector fields, infinitesimal advection is represented by skew-symmetric matrices acting on discrete half-densities. The admissible advection operators form a constrained subspace which is not generally closed under commutators, so the resulting dynamics naturally lead to a nonholonomic variational problem. I will describe how a vakonomic formulation treats this constraint intrinsically, in contrast to the more common Lagrange-d’Alembert approach based on projected dynamics, and yields a Lax-type evolution with exact preservation of energy and discrete Casimir invariants.

ABSTRACT. The attenuated Westervelt equation, with the attenuation governed by a non-local in time operator, is considered. The non-locality is described by a time convolution with a singular kernel, the simplest case being that of the Riemann-Liouville fractional integral. A semi-discretisation in time based on the trapezoidal rule and A-stable convolution quadrature is analysed. The theoretical error analysis results are confirmed by numerical experiments. Joint work with Katherine Baker and Lehel Banjai.

ABSTRACT. We present a new conservative discretisation for the Kadomtsev–Petviashvili equation. We will observe that energy conservation (surprisingly) does not give us numerical stability, and we will study the additional requirements to obtain a stable solution.

ABSTRACT. The Cahn--Hilliard (CH) equation with rational-like free-energy terms poses a challenge for the design of structure-preserving time discretizations. We propose a quadratic conservation elevation (QCE) method that reformulates the free energy as a quadratic functional in an extended variable system. Moreover, the proposed method preserves the original energy dissipation law at the discrete level. We also derive the associated discrete dispersion relation and analyze its consistency with the continuous one. Numerical results verify the discrete energy behavior, second-order temporal accuracy, and the expected coarsening dynamics, and further capture missing orientations for different anisotropic energy functionals. Additional simulations from several initial conditions illustrate phase separation and anisotropic evolution.

ABSTRACT. The finite-time input-to-state stability problem is addressed for a class of hybrid systems with coexisting switching and impulsive effects evolving over distinct time mechanisms. The analysis does not impose restrictive assumptions on the individual components: neither the continuous subsystems nor the impulsive effects are required to be inherently stabilizing. A Lyapunov-based framework combined with average dwell-time techniques is developed to characterize the interplay between these two mechanisms. It is shown that when the continuous dynamics admit finite-time input-to-state stability, excessive impulsive activity may destroy stability, which can be avoided by enforcing an upper bound on switching-impulse occurrences. On the other hand, even in the absence of stability in the continuous part, suitably frequent stabilizing impulses can recover the desired FT-ISS property, provided a lower bound on their activation is guaranteed. The proposed results provide a unified perspective on different stability regimes and offer practical criteria for the stability analysis of systems with hybrid dynamics.

ABSTRACT. A modern trend in optical design is the use of inverse methods to compute the shape of an optical surface (reflector/lens) that converts a given source light distribution into a desired target distribution. The associated model consists of the principles of geometrical optics, from which we can derive a Jacobian equation for the optical map connecting source and target domain, and the conservation law of luminous flux. Combining these, we can derive (generalized) Monge-Ampère equations for several optical systems. These equations are fully nonlinear, second order elliptic PDEs. In addition, these equations have to be supplemented with the nonstandard transport boundary condition, stating that the boundary of the source domain is mapped to the boundary of the target domain. We have developed several least-squares methods for these equations. In its basic formulation, these are two-stage methods; the first stage is an iterative least-squares method to compute the optical map, and upon convergence, we compute in the second stage the shape of the optical surface, also in a least-squares sense. The computation of the optical map is based on the minimization of several functionals and requires the constrained minimization of the residual of the Jacobian equation, a projection of the optical map on the target boundary and the solution of two Poisson equations, successively. Finally, the optical surface is computed from another Poisson equation. Our methods converge fast and can handle complicated target distributions. We demonstrate the performance for several illustrative examples.

ABSTRACT. The Keller-Segel system models biological aggregation driven by chemotaxis, such as bacterial pattern formation. Simulating its complex 3D dynamics, especially near blow-up singularities where density diverges, is computationally prohibitive with existing methods. To overcome this, we developed the SIPF-PIC framework, a novel hybrid method combining Lagrangian particle tracking with spectral field solvers using optimized interpolation. Crucially, SIPF-PIC reduces the cost per timestep from $\mathcal{O}(PH^3)$ to $\mathcal{O}(P + H^3 \log H)$ while maintaining convergence, where $P$ is the particle count and $H$ the grid resolution per dimension. This enables large-scale simulations capturing elusive blow-up structures like evolving ring singularities, beyond simple point collapse. To the best of our knowledge, this work represents the first numerical approach that can resolve complex 3D blowup dynamics beyond single-point collapse, including the formation of ring-shaped singular structures.

ABSTRACT. Olive Quick Decline Syndrome (OQDS), caused by the bacterium Xylella fastidiosa, is a vector-borne disease affecting olive groves across Europe, severely impacting Southern Italy. We consider an ODE model for the olive tree canopies, the vector insects, and the weeds. Under the assumption that the insects reproduce significantly faster than the trees and weeds, the model equations are singularly perturbed and can then be analysed via the dynamical systems-based geometric singular perturbation theory. This allows us to reduce the long term dynamics of the model solutions to the dynamics of a globally attracting slow manifold. We extend the model by introducing a Heaviside cut-off to the spread of the disease cross terms and regularise by applying the desingularisation technique, know as "blow-up".

ABSTRACT. The recurrence of atrial fibrillation (AF) following catheter ablation is a common complication in patients with persistent atrial fibrillation (psAF), increasing the risk of stroke and heart failure thereafter. Given the multifactorial nature of post-ablation AF, clinical predictions of successful ablation often suffer from poor accuracy and lack robustness. This paper proposes a multimodal prediction model for post-ablation AF, which extracts complex features from multidimensional data, including electrocardiogram (ECG) images, cellular characteristics, intraoperative and demographic information of patients. Specifically, a dual-module structure is proposed for ECG processing. It consists of an image module that extracts spatial features and a temporal module that captures sequential features, effectively capturing the spatiotemporal dynamics of ECG. A clinical-intraoperative data integration module is developed to combat the complex nature of cellular, demographic, and intraoperative data structures in clinical settings by leveraging sparse and dense feature integration, enabling effective representation and processing. Finally, a feature fusion module is introduced, composed of a dynamic weight mechanism and a multimodal Transformer model, enhancing feature interaction and facilitating effective information synchrony between different modalities. The experimental results demonstrate that the proposed model achieved an accuracy of 0.9079 and an Area Under the Curve (AUC) of 0.8690. These findings highlight significant effectiveness in post-ablation AF prediction, offering a comprehensive prediction framework that supports early intervention for patients with psAF at risk for AF recurrence.

ABSTRACT. Organic solar cells represent a lightweight, flexible, and low-cost alternative to traditional silicon-based cells, yet they currently struggle with stability and conversion efficiency. We present a model describing the formation of acceptor and donor regions during the production of organic solar cells and evaluate how the resulting morphology dictates device efficiency. The formation of acceptor and donor regions is based on spinodal decomposition, all while a solvent is allowed to evaporate. The derived model couples the donor-acceptor phase separation process with hydrodynamic and evaporative effects. The resulting electrical performance significantly depends on the device morphology since its layout influences exciton- and charge dynamics. Excitons require the additional energy offset provided by the donor-acceptor interface to separate into charges, while charges require energetic favorable pathways to the respective electrodes. The resulting model couples charge generation by excitons with the hopping transport typical for organic materials, while taking the device morphology into account. We present a thermodynamically consistent framework describing morphology formation, its subsequent effects on the device performance, and showcase several numerical examples based on finite element simulations.

ABSTRACT. Cell image segmentation is essential for analyzing DNA damage tolerance pathways and cellular responses in health and disease. However, traditional cell segmentation tools, such as MicrobeTracker, are computationally intensive, require substantial manual input, and perform poorly on complex or densely packed cells. In this work, we propose a hierarchical image segmentation framework based on an iterative thresholding approach derived from the Chan–Vese model. The proposed method employs a coarse-to-fine multiscale strategy tailored for cell fluorescence microscopy images. Specifically, an initial coarse segmentation is performed to separate the background, followed by a refined segmentation stage that accurately delineates cell regions and boundaries. Our approach reduces both computational complexity and human intervention. Furthermore, we propose a localized multiscale hierarchical segmentation scheme to address challenges associated with non-uniform illumination.