Thursday at 1:50-2:45 PM (unless otherwise noted)
Hybrid format: either virtually or in Math Tower Room 154 (See below for details)
For questions, contact Dr. Yulong Xing or Dr. Dongbin Xiu, Email: xing dot 205@osu.edu or xiu dot 16@osu.edu
Dahlquist, Liniger, and Nevanlinna proposed a two-step time-stepping scheme for systems of ordinary differential equations (ODEs) in 1983. The little-explored variable time-stepping scheme has advantages in numerical simulations for its fine properties such as unconditional G-stability and second-order accuracy. We simplify its implementation through time filters (pre-filter and post-filter) on a certain first-order implicit method. The adaptivity algorithm for this variable time-stepping scheme, highly reducing computation cost as well as keeping time accuracy, has been applied to systems of ODEs and flow models. Moreover we have applied this method to Navier Stokes equations for stability and error analysis.
For many decades, physics-based PDEs have been commonly employed for modeling complex system responses, then traditional numerical methods were employed to solve the PDEs and provide predictions. However, when governing laws are unknown or when high degrees of heterogeneity present, these classical models may become inaccurate. In this talk we propose to use data-driven modeling which directly utilizes high-fidelity simulation and experimental measurements to learn the hidden physics and provide further predictions. In particular, we develop PDE-inspired neural operator architectures, to learn the mapping between loading conditions and the corresponding system responses. By parameterizing the increment between layers as an integral operator, our neural operator can be seen as the analog of a time-dependent autonomous nonlocal equation, which captures the long-range dependencies in the feature space and is guaranteed to be resolution-independent. Moreover, when applying to (hidden) PDE solving tasks, our neural operator provides a universal approximation to a fixed point iterative procedure, and partial physical knowledge can be incorporated to further improve the model’s generalizability and transferability. As an application, we learn the material models directly from digital image correlation (DIC) displacement tracking measurements on a porcine tricuspid valve leaflet tissue, and show that the learnt model substantially outperforms conventional constitutive models.
Recent years have witnessed tremendous progress in developing and analyzing quantum computing algorithms for quantum dynamics simulation of bounded operators (Hamiltonian simulation). However, many scientific and engineering problems require the efficient treatment of unbounded operators, which frequently arise due to the discretization of differential operators. Such applications include molecular dynamics, electronic structure theory, quantum control and quantum machine learning. We will introduce some recent advances in quantum algorithms for efficient unbounded Hamiltonian simulation, including Trotter type splitting and the quantum highly oscillatory protocol (qHOP) in the interaction picture. The latter yields a surprising superconvergence result for regular potentials. (The talk does not assume a priori knowledge on quantum computing.)
Cells make fate decisions in response to dynamic environments, and multicellular structures emerge from multiscale interplays among cells and genes in space and time. The recent single-cell genomics technology provides an unprecedented opportunity to profile cells. However, those measurements are taken as static snapshots of many individual cells that often lose spatial information. How to obtain temporal relationships among cells from such measurements? How to recover spatial interactions among cells, such as cell-cell communication? In this talk I will present our newly developed computational tools that dissect transition properties of cells and infer cell-cell communication based on nonspatial single-cell genomics data. In addition, I will present methods to derive multicellular spatiotemporal pattern from spatial transcriptomics datasets. Through applications of those methods to systems in development and regeneration, we show the discovery power of such methods and identify areas for further development for spatiotemporal reconstruction of single-cell genomics data.
High order methods are known to be unstable when applied to nonlinear conservation laws whose solutions exhibit shocks and turbulence. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of numerical resolution or solution regularization and shock capturing. In this talk, we will review different approaches for constructing robust entropy stable discontinuous Galerkin methods and discuss the impact of different discretization choices on robustness for under-resolved compressible flows.
Many engineering and scientific problems can be described by equations of fluid dynamics, namely, systems of time dependent nonlinear hyperbolic PDEs. The mathematical description of these processes as well as the numerical discretisation of the resulting PDEs will depend on the level of detail required to study them. In this talk I will focus on two directions towards higher fidelity numerical simulations: first, I present a novel structure-preserving arbitrarily high-order method that solves the nonlinear ideal magneto-hydrodynamics equations. Secondly, I will focus on our work using Machine Learning (ML) methods with the aim to improve or speed up numerical simulations, through the development of parameter-free routines as part of a numerical solver or surrogate models, with the goal of creating hybrid simulation pipelines that can improve over time.
In this talk, we consider the problem of minimizing multi-modal loss functions with a large number of local optima. Since the local gradient points to the direction of the steepest slope in an infinitesimal neighborhood, an optimizer guided by the local gradient is often trapped in a local minimum. To address this issue, we develop a novel nonlocal gradient to skip small local minima by capturing major structures of the loss's landscape in black-box optimization. The nonlocal gradient is defined by a directional Gaussian smoothing (DGS) approach. The key idea is to conducts 1D long-range exploration with a large smoothing radius along orthogonal directions, each of which defines a nonlocal directional derivative as a 1D integral. Such long-range exploration enables the nonlocal gradient to skip small local minima. We use the Gauss-Hermite quadrature rule to approximate the d 1D integrals to obtain an accurate estimator. We also provide theoretical analysis on the convergence of the method on nonconvex landscape. In this work, we investigate the scenario where the objective function is composed of a convex function, perturbed by a highly oscillating, deterministic noise. We provide a convergence theory under which the iterates converge to a tightened neighborhood of the solution, whose size is characterized by the noise frequency. Furthermore, if the noise level decays to zero when approaching global minimum, we prove that the DGS optimization converges to the exact global minimum with linear rates, similarly to standard gradient-based method in optimizing convex functions. We complement our theoretical analysis with numerical experiments to illustrate the performance of this approach.
Bayesian sampling and neural networks are seemingly two different machine learning areas, but they both deal with many particle systems. In sampling, one evolves a large number of samples (particles) to match a target distribution function, and in optimizing over-parameterized neural networks, one can view neurons particles that feed each other information in the DNN flow. These perspectives allow us to employ mean-field theory, a powerful tool that translates dynamics of many particle system into a partial differential equation (PDE), so rich PDE analysis techniques can be used to understand both the convergence of sampling methods and the zero-loss property of over-parameterization of ResNets. We showcase the use of mean-field theory in these two machine learning areas.
In scientific and engineering applications, people usually encounter time-dependent systems with a decaying total energy. A natural question to ask is that when using a time integration method to approximate such systems, whether and how the energy-decay law will be preserved at the discrete level. In this talk, we will focus on linear autonomous systems and present a general framework on deriving discrete energy identities for both explicit and implicit Runge-Kutta (RK) method. The established identities provide a precise characterization on energy generation or dissipation in the RK discretization, thereby leading to weak and strong stability criteria of RK methods. Furthermore, we also present a general energy identity for all the diagonal Padé approximations, based on an analytical Cholesky type decomposition of a class of symmetric matrices.
Year 2019-2020 Seminar
Does this page seem familiar?