Comparison of POD reduced order strategies for the nonlinear 2D Shallow Water Equations

This paper introduces tensorial calculus techniques in the framework of Proper Orthogonal Decomposition (POD) to reduce the computational complexity of the reduced nonlinear terms. The resulting method, named tensorial POD, can be applied to polynomial nonlinearities of any degree $p$. Such nonlinear terms have an on-line complexity of $\mathcal{O}(k^{p+1})$, where $k$ is the dimension of POD basis, and therefore is independent of full space dimension. However it is efficient only for quadratic nonlinear terms since for higher nonlinearities standard POD proves to be less time consuming once the POD basis dimension $k$ is increased. Numerical experiments are carried out with a two dimensional shallow water equation (SWE) test problem to compare the performance of tensorial POD, standard POD, and POD/Discrete Empirical Interpolation Method (DEIM). Numerical results show that tensorial POD decreases by $76\times$ times the computational cost of the on-line stage of standard POD for configurations using more than $300,000$ model variables. The tensorial POD SWE model was only $2-8\times$ slower than the POD/DEIM SWE model but the implementation effort is considerably increased. Tensorial calculus was again employed to construct a new algorithm allowing POD/DEIM shallow water equation model to compute its off-line stage faster than the standard and tensorial POD approaches.Comment: 23 pages, 8 figures, 5 table

Dynamic Multivariate Simplex Splines For Volume Representation And Modeling

Volume representation and modeling of heterogeneous objects acquired from real world are very challenging research tasks and playing fundamental roles in many potential applications, e.g., volume reconstruction, volume simulation and volume registration. In order to accurately and efficiently represent and model the real-world objects, this dissertation proposes an integrated computational framework based on dynamic multivariate simplex splines (DMSS) that can greatly improve the accuracy and efficacy of modeling and simulation of heterogenous objects. The framework can not only reconstruct with high accuracy geometric, material, and other quantities associated with heterogeneous real-world models, but also simulate the complicated dynamics precisely by tightly coupling these physical properties into simulation. The integration of geometric modeling and material modeling is the key to the success of representation and modeling of real-world objects. The proposed framework has been successfully applied to multiple research areas, such as volume reconstruction and visualization, nonrigid volume registration, and physically based modeling and simulation

Optimization viewpoint on Kalman smoothing, with applications to robust and sparse estimation

In this paper, we present the optimization formulation of the Kalman filtering and smoothing problems, and use this perspective to develop a variety of extensions and applications. We first formulate classic Kalman smoothing as a least squares problem, highlight special structure, and show that the classic filtering and smoothing algorithms are equivalent to a particular algorithm for solving this problem. Once this equivalence is established, we present extensions of Kalman smoothing to systems with nonlinear process and measurement models, systems with linear and nonlinear inequality constraints, systems with outliers in the measurements or sudden changes in the state, and systems where the sparsity of the state sequence must be accounted for. All extensions preserve the computational efficiency of the classic algorithms, and most of the extensions are illustrated with numerical examples, which are part of an open source Kalman smoothing Matlab/Octave package.Comment: 46 pages, 11 figure

Total variation on a tree

We consider the problem of minimizing the continuous valued total variation subject to different unary terms on trees and propose fast direct algorithms based on dynamic programming to solve these problems. We treat both the convex and the non-convex case and derive worst case complexities that are equal or better than existing methods. We show applications to total variation based 2D image processing and computer vision problems based on a Lagrangian decomposition approach. The resulting algorithms are very efficient, offer a high degree of parallelism and come along with memory requirements which are only in the order of the number of image pixels.Comment: accepted to SIAM Journal on Imaging Sciences (SIIMS

Efficient Nonlinear Solvers for Nodal High-Order Finite Elements in 3D

Conventional high-order finite element methods are rarely used for industrial problems because the Jacobian rapidly loses sparsity as the order is increased, leading to unaffordable solve times and memory requirements. This effect typically limits order to at most quadratic, despite the favorable accuracy and stability properties offered by quadratic and higher order discretizations. We present a method in which the action of the Jacobian is applied matrix-free exploiting a tensor product basis on hexahedral elements, while much sparser matrices based on Q 1 sub-elements on the nodes of the high-order basis are assembled for preconditioning. With this "dual-order” scheme, storage is independent of spectral order and a natural taping scheme is available to update a full-accuracy matrix-free Jacobian during residual evaluation. Matrix-free Jacobian application circumvents the memory bandwidth bottleneck typical of sparse matrix operations, providing several times greater floating point performance and better use of multiple cores with shared memory bus. Computational results for the p-Laplacian and Stokes problem, using block preconditioners and AMG, demonstrate mesh-independent convergence rates and weak (bounded) dependence on order, even for highly deformed meshes and nonlinear systems with several orders of magnitude dynamic range in coefficients. For spectral orders around 5, the dual-order scheme requires half the memory and similar time to assembled quadratic (Q 2) elements, making it very affordable for general us

Diffeomorphic Transformations for Time Series Analysis: An Efficient Approach to Nonlinear Warping

The proliferation and ubiquity of temporal data across many disciplines has sparked interest for similarity, classification and clustering methods specifically designed to handle time series data. A core issue when dealing with time series is determining their pairwise similarity, i.e., the degree to which a given time series resembles another. Traditional distance measures such as the Euclidean are not well-suited due to the time-dependent nature of the data. Elastic metrics such as dynamic time warping (DTW) offer a promising approach, but are limited by their computational complexity, non-differentiability and sensitivity to noise and outliers. This thesis proposes novel elastic alignment methods that use parametric \& diffeomorphic warping transformations as a means of overcoming the shortcomings of DTW-based metrics. The proposed method is differentiable \& invertible, well-suited for deep learning architectures, robust to noise and outliers, computationally efficient, and is expressive and flexible enough to capture complex patterns. Furthermore, a closed-form solution was developed for the gradient of these diffeomorphic transformations, which allows an efficient search in the parameter space, leading to better solutions at convergence. Leveraging the benefits of these closed-form diffeomorphic transformations, this thesis proposes a suite of advancements that include: (a) an enhanced temporal transformer network for time series alignment and averaging, (b) a deep-learning based time series classification model to simultaneously align and classify signals with high accuracy, (c) an incremental time series clustering algorithm that is warping-invariant, scalable and can operate under limited computational and time resources, and finally, (d) a normalizing flow model that enhances the flexibility of affine transformations in coupling and autoregressive layers.Comment: PhD Thesis, defended at the University of Navarra on July 17, 2023. 277 pages, 8 chapters, 1 appendi