- Applied and Computational Mathematics Seminar
- Monday, October 18, 2021 - 14:00 for
- Lingjiong Zhu – FSU
To treat the multiple time scales of ocean dynamics in an efficient manner, the baroclinic-barotropic splitting technique has been widely used for solving the primitive equations for ocean modeling. In this paper, we propose second and third-order multirate explicit time-stepping schemes for such split systems based on the strong stability-preserving Runge-Kutta (SSPRK) framework. Our method allows for a large time step to be used for advancing the three-dimensional (slow) baroclinic mode and a small time step for the two-dimensional (fast) barotropic mode, so that each of the two mode solves only need satisfy their respective CFL condition to maintain numerical stability. It is well known that the SSPRK method achieves high-order temporal accuracy by utilizing a convex combination of forward-Euler steps. At each time step of our method, the baroclinic velocity is first computed by using the SSPRK scheme to advance the baroclinic-barotropic system with the large time step, then the barotropic velocity is specially corrected by using the same SSPRK scheme with the small time step to advance the barotropic subsystem with a barotropic forcing interpolated based on values from the preceding baroclinic solves. Finally, the fluid thickness and the sea surface height perturbation is updated by coupling the predicted baroclinic and barotropic velocities. Two benchmark tests drawn from the ``MPAS-Ocean" platform are used to numerically demonstrate the accuracy and parallel performance of the proposed schemes.
The bluejeans link for the seminar is https://bluejeans.com/457724603/4379
Please Note: Note the hybrid mode. The speaker will be in person in Skiles 005.
In this talk, we develop algorithms for numerical computation, based on ideas from competitive games and statistical inference.
In the first part, we propose competitive gradient descent (CGD) as a natural generalization of gradient descent to saddle point problems and general sum games. Whereas gradient descent minimizes a local linear approximation at each step, CGD uses the Nash equilibrium of a local bilinear approximation. Explicitly accounting for agent-interaction significantly improves the convergence properties, as demonstrated in applications to GANs, reinforcement learning, and computer graphics.
In the second part, we show that the conditional near-independence properties of smooth Gaussian processes imply the near-sparsity of Cholesky factors of their dense covariance matrices. We use this insight to derive simple, fast solvers with state-of-the-art complexity vs. accuracy guarantees for general elliptic differential- and integral equations. Our methods come with rigorous error estimates, are easy to parallelize, and show good performance in practice.
While deep learning has been used for dynamics learning, limited physical accuracy and an inability to generalize under distributional shift limit its applicability to real world. In this talk, I will demonstrate how to incorporate symmetries into deep neural networks and significantly improve the physical consistency, sample efficiency, and generalization in learning dynamics. I will showcase the applications of these models to challenging problems such as turbulence forecasting and trajectory prediction for autonomous vehicles.
In this talk, we discuss variants of the rigid registration problem, i.e aligning objects via rigid transformation. In the simplest scenario of point-set registration where the correspondence between points are known, we investigate the robustness of registration to outliers. We also study a convex programming formulation of point-set registration with exact recovery, in the situation where both the correspondence and alignment are unknown. This talk is based on joint works with Ankur Kapoor, Cindy Orozco, and Lexing Ying.
We construct integrators to be used in Hamiltonian (or Hybrid) Monte Carlo sampling. The new integrators are easily implementable and, for a given computational budget, may deliver five times as many accepted proposals as standard leapfrog/Verlet without impairing in any way the quality of the samples. They are based on a suitable modification of the processing technique first introduced by J.C. Butcher. The idea of modified processing may also be useful for other purposes, like the construction of high-order splitting integrators with positive coefficients.
Joint work with Mari Paz Calvo, Fernando Casas, and Jesús M. Sanz-Serna
I will present MCMC algorithms as optimization over the KL-divergence in the space of probabilities. By incorporating a momentum variable, I will discuss an algorithm which performs accelerated gradient descent over the KL-divergence. Using optimization-like ideas, a suitable Lyapunov function is constructed to prove that an accelerated convergence rate is obtained. I will then discuss how MCMC algorithms compare against variational inference methods in parameterizing the gradient flows in the space of probabilities and how it applies to sequential decision making problems.
Many scientific problems involve invariant structures, and learning functions that rely on a much lower dimensional set of features than the data itself. Incorporating these invariances into a parametric model can significantly reduce the model complexity, and lead to a vast reduction in the number of labeled examples required to estimate the parameters. We display this benefit in two settings. The first setting concerns ReLU networks, and the size of networks and number of points required to learn certain functions and classification regions. Here, we assume that the target function has built in invariances, namely that it only depends on the projection onto a very low dimensional, function defined manifold (with dimension possibly significantly smaller than even the intrinsic dimension of the data). We use this manifold variant of a single or multi index model to establish network complexity and ERM rates that beat even the intrinsic dimension of the data. We should note that a corollary of this result is developing intrinsic rates for a manifold plus noise data model without needing to assume the distribution of the noise decays exponentially, and we also discuss implications in two-sample testing and statistical distances. The second setting for building invariances concerns linearized optimal transport (LOT), and using it to build supervised classifiers on distributions. Here, we construct invariances to families of group actions (e.g., shifts and scalings of a fixed distribution), and show that LOT can learn a classifier on group orbits using a simple linear separator. We demonstrate the benefit of this on MNIST by constructing robust classifiers with only a small number of labeled examples. This talk covers joint work with Timo Klock, Xiuyuan Cheng, and Caroline Moosmueller.
The problem of group synchronization asks to recover states of objects associated with group elements given possibly corrupted relative state measurements (or group ratios) between pairs of objects. This problem arises in important data-related tasks, such as structure from motion, simultaneous localization and mapping, Cryo-EM, community detection and sensor network localization. Two common groups in these problems are the rotation and symmetric groups. We propose a general framework for group synchronization with compact groups. The main part of the talk discusses a novel message passing procedure that uses cycle consistency information in order to estimate the corruption levels of group ratios. Under our mathematical model of adversarial corruption, it can be used to infer the corrupted group ratios and thus to solve the synchronization problem. We first explain why the group cycle consistency information is essential for effectively solving group synchronization problems. We then establish exact recovery and linear convergence guarantees for the proposed message passing procedure under a deterministic setting with adversarial corruption. We also establish the stability of the proposed procedure to sub-Gaussian noise. We further establish competitive theoretical results under a uniform corruption model. Finally, we discuss the MPLS (Message Passing Least Squares) or Minneapolis framework for solving real scenarios with high levels of corruption and noise and with nontrivial scenarios of corruption. We demonstrate state-of-the-art results for two different problems that occur in structure from motion and involve the rotation and permutation groups.
In quantum mechanics and the analysis of Markov processes, Monte Carlo methods are needed to identify low-lying eigenfunctions of dynamical generators. The standard Monte Carlo approaches for identifying eigenfunctions, however, can be inaccurate or slow to converge. What limits the efficiency of the currently available spectral estimation methods and what is needed to build more efficient methods for the future? Through numerical analysis and computational examples, we begin to answer these questions. We present the first-ever convergence proof and error bounds for the variational approach to conformational dynamics (VAC), the dominant method for estimating eigenfunctions used in biochemistry. Additionally, we analyze and optimize variational Monte Carlo (VMC), which combines Monte Carlo with neural networks to accurately identify low-lying eigenstates of quantum systems.