Abstract : The simulation of complex physical systems for prediction and control requires robust, efficient mathematical modeling and numerical algorithms, as the problem size and time scales of interest in many applications are typically beyond which can be simulated directly. In recent years, a wealth of new techniques and algorithms have been developed to help reduce problem size/dimension and accelerate the accurate simulation of various classes of physical observables while quantifying the uncertainty of the resulting predictions made. Examples of such techniques include Coarse-Grained Molecular Dynamics, Nonlocal Theories of Mechanics, Time Accelerated Dynamics, Hyperdynamics, Space-Time Homogenization, and so on. These techniques and numerical algorithms have made successful applications in a diverse range of models.
In view of the wide range of applicability of these algorithms and the ideas which lie behind them, this minisymposium seeks to bring together both theoreticians and practitioners who study and use numerical simulations for a range of practical scientific problems, aiming to facilitate discussion and two-way dissemination of ideas across disciplinary and topical boundaries.
[02156] Ahyper-reduced MAC scheme for the parametric Stokes and Navier-Stokes equations
Format : Talk at Waseda University
Author(s) :
Lijie Ji (Shanghai Jiao Tong University)
Abstract : The classical reduced basis method is popular due to an offline-online decomposition and a mathematically rigorous a posteriorerror estimator which guides a greedy algorithm offline. For nonlinear and nonaffine problems, hyper reduction techniques have been introduced to make this decomposition efficient. However, they may be tricky to implement and often degrade the offline and online computational efficiency. In this talk, I will introduce an adaptive enrichment strategy for R2-ROC rendering it capable of handling parametric fluid flow problems. Tests on lid-driven cavity and flow past a backward-facing step problems demonstrate its high efficiency, stability and accuracy.
[02166] Hybrid Projection Methods for Solution Decomposition in Large-scale Bayesian Inverse Problems
Format : Talk at Waseda University
Author(s) :
Jiahua Jiang (University of Birmingham)
Julianne Chung (Emory University)
Arvind Krishna Saibaba (North Carolina State University)
Scot Miller (John Hopkins )
Abstract : We develop hybrid projection methods for computing solutions to large-scale inverse problems, where the solution represents a sum of different stochastic components. Such scenarios arise in many imaging applications (e.g., anomaly detection in atmospheric emissions tomography) where the reconstructed solution can be represented as a combination of two or more components and each component contains different smoothness or stochastic properties. In a deterministic inversion or inverse modeling framework, these assumptions correspond to different regularization terms for each solution in the sum. Although various prior assumptions can be included in our framework, we focus on the scenario where the solution is a sum of a sparse solution and a smooth solution. For computing solution estimates, we develop hybrid projection methods for solution decomposition that are based on a combined flexible and generalized Golub-Kahan processes. This approach integrates techniques from the generalized Golub-Kahan bidiagonalization and the flexible Krylov methods. The benefits of the proposed methods are that the decomposition of the solution can be done iteratively, and the regularization terms and regularization parameters are adaptively chosen at each iteration. Numerical results from photoacoustic tomography and atmospheric inverse modeling demonstrate the potential for these methods to be used for anomaly detection.
[02484] A reduced basis method for the parametrized Monge-Ampere equation
Format : Talk at Waseda University
Author(s) :
Shijin Hou (University of Science and Technology of China)
Abstract : In this talk, we first introduce a highly efficient solver for the parameterized optimal mass transport problem by adapting the reduced residual reduced over-collocation approach to the parameterized Monge-Amp$\grave{\rm e}$re equation. This new reduced basis technique allows us to handle the strong and unique nonlinearity pertaining to the Monge-Amp$\grave{\rm e}$re equation achieving online efficiency. After that, several numerical tests will be presented to demonstrate the accuracy and high efficiency of our reduced solver.
[04068] Novel Reduced Basis Method for Radiative Transfer Equation
Format : Talk at Waseda University
Author(s) :
ZHICHAO PENG (Michigan State University)
Yanlai Chen (University of Massachusetts Dartmout)
Yingda Cheng (Michigan State University)
Fengyan Li (Rensselaer Polytechnic Institute)
Abstract : One prominent computational challenge to simulate radiative transfer (RTE), a fundamental kinetic description of energy or particle transport through mediums affected by scattering and absorption processes, comes from the high dimensionality of the phase space. Leveraging the existence of a low-rank structure in the solution manifold induced by the angular variable in the scattering dominating regime, reduced order models are designed and tested here for the linear RTE model based on reduced basis methods.
[03752] Large Deviations for Model Coarse Graining
Format : Talk at Waseda University
Author(s) :
Tobias Grafke (Warwick Mathematics Institute)
Abstract : Systems with time-scale separation allow effective model reduction via averaging and homogenization, where average effects of fluctuating degrees of freedom are as reduced dynamics. In the language of probability theory, this averaging corresponds to a law-of-large numbers, making it natural to ask about expected fluctuations and large deviations. In this talk, I will introduce developments regarding large deviations in the presence of time-scale separation for the computation of rare event probabilities in reduced models.
[03592] Mean curvature flow as the limit of a spin system
Format : Talk at Waseda University
Author(s) :
Patrick van Meurs (Kanazawa University)
Abstract : I will present the derivation of a continuum mean-curvature flow as a certain hydrodynamic scaling limit of a stochastic particle system (Glauber-Kawasaki) on the discrete torus in d dimensions. The particles can jump to neighboring vacant sites and there is a creation and annihilation mechanism. Our work combines techniques from probability theory (in particular the relative entropy method), numerical analysis and PDE theory.
Collaborators: T. Funaki, S. Sethuraman, K. Tsunoda.
[04679] Model reduction methods for non-reversible multiscale dynamics: a comparison
Format : Talk at Waseda University
Author(s) :
Lara Neureither (BTU Cottbus)
Abstract : In this talk we will compare existing coarse graining methods such as averaging, effective dynamics as well as the normal form approach among others for multiscale dynamics given by a non-reversible Ornstein-Uhlenbeck process driven by degenerate noise. We will address the following questions: which of the methods yields the best approximation to the original dynamics? What causes the differences in the approaches, if there are any?
[04002] Model Reduction using the Koopman Operator
Format : Talk at Waseda University
Author(s) :
Xiu Yang (Lehigh University)
Bian Li (Lehigh University)
Yi-An Ma (University of California at San Diego)
J. Nathan Kutz (University of Washington)
Abstract : We propose the adaptive spectral Koopman (ASK) method to solve nonlinear autonomous dynamical systems. ASK leverages the spectral method and the Koopman operator to obtain the solution. Specifically, this solution is represented by Koopman operator’s eigenfunctions, eigenvalues, and Koopman modes. Numerical experiments demonstrate high accuracy of ASK for solving both ordinary and partial differential equations. Using ASK as a surrogate model, we can design novel efficient uncertainty quantification methods.
[04693] Machine-learning-based spectral methods for partial differential equations
Format : Talk at Waseda University
Author(s) :
Panos Stinis (Pacific Northwest National Laboratory)
Saad Qadeer (Pacific Northwest National Laboratory)
Brek Meuris (University of Washington)
Abstract : We use deep neural operators to identify custom-made basis functions for constructing spectral methods for partial differential equations. The custom-made basis functions are studied both for their approximation capability and used to expand the solution of linear and nonlinear time-dependent PDEs. The proposed approach advances the state of the art and versatility of spectral methods and, more generally, promotes the synergy between traditional scientific computing and machine learning.
[03852] Optimal control for fractional order equations
Format : Talk at Waseda University
Author(s) :
Christian Glusa (Sandia National Laboratories)
Abstract : We consider adjoint-based optimization for control problems involving fractional-order state equations, applied to the inference of kernel parameters. We will discuss optimality conditions, error estimates and techniques to efficiently explore the parameter space and approximate gradients.
[04163] Multi-Resolution and FVM inspired Neural Network (MuRFiV-Net) for PDE prediction
Format : Talk at Waseda University
Author(s) :
Xin-Yang Liu (University of Notre Dame)
Jian-Xun Wang (University of Notre Dame)
Abstract : Predicting physical processes requires modeling complex spatiotemporal dynamics. Traditional numerical methods are expensive in many-query tasks, while data-driven neural networks face issues of high training costs and poor generalizability. Physics-informed deep learning (PiDL) combines numerical methods and deep learning, offering a promising approach to overcome these limitations. This work proposes MuRFiV-Net, a novel PiDL architecture based on a multi-resolution mesh and finite volume method. The merit of MuRFiV-Net is demonstrated on several PDE-governed dynamic systems.
