We propose a novel methodology for forecasting spatio-temporal data using supervised semi-nonnegative matrix factorization (SSNMF) with frequency regularization. Matrix factorization is employed to decompose spatio-temporal data into spatial and temporal components. To improve clarity in the temporal patterns, we introduce a nonnegativity constraint on the time domain along with regularization in the frequency domain. Specifically, regularization in the frequency domain involves selecting features in the frequency space, making an interpretation in the frequency domain more convenient. We propose two methods in the frequency domain: soft and hard regularizations, and provide convergence guarantees to first-order stationary points of the corresponding constrained optimization problem. While our primary motivation stems from geophysical data analysis based on GRACE (Gravity Recovery and Climate Experiment) data, our methodology has the potential for wider application. Consequently, when applying our methodology to GRACE data, we find that the results with the proposed methodology are comparable to previous research in the field of geophysical sciences but offer clearer interpretability.
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By using the stochastic particle method, the truncated Euler-Maruyama (TEM) method is proposed for numerically solving McKean-Vlasov stochastic differential equations (MV-SDEs), possibly with both drift and diffusion coefficients having super-linear growth in the state variable. Firstly, the result of the propagation of chaos in the L^q (q\geq 2) sense is obtained under general assumptions. Then, the standard 1/2-order strong convergence rate in the L^q sense of the proposed method corresponding to the particle system is derived by utilizing the stopping time analysis technique. Furthermore, long-time dynamical properties of MV-SDEs, including the moment boundedness, stability, and the existence and uniqueness of the invariant probability measure, can be numerically realized by the TEM method. Additionally, it is proven that the numerical invariant measure converges to the underlying one of MV-SDEs in the L^2-Wasserstein metric. Finally, the conclusions obtained in this paper are verified through examples and numerical simulations.
This paper proposes several approaches as baselines to compute a shared active subspace for multivariate vector-valued functions. The goal is to minimize the deviation between the function evaluations on the original space and those on the reconstructed one. This is done either by manipulating the gradients or the symmetric positive (semi-)definite (SPD) matrices computed from the gradients of each component function so as to get a single structure common to all component functions. These approaches can be applied to any data irrespective of the underlying distribution unlike the existing vector-valued approach that is constrained to a normal distribution. We test the effectiveness of these methods on five optimization problems. The experiments show that, in general, the SPD-level methods are superior to the gradient-level ones, and are close to the vector-valued approach in the case of a normal distribution. Interestingly, in most cases it suffices to take the sum of the SPD matrices to identify the best shared active subspace.
We introduce two iterative methods, GPBiLQ and GPQMR, for solving unsymmetric partitioned linear systems. The basic mechanism underlying GPBiLQ and GPQMR is a novel simultaneous tridiagonalization via biorthogonality that allows for short-recurrence iterative schemes. Similar to the biconjugate gradient method, it is possible to develop another method, GPBiCG, whose iterate (if it exists) can be obtained inexpensively from the GPBiLQ iterate. Whereas the iterate of GPBiCG may not exist, the iterates of GPBiLQ and GPQMR are always well defined as long as the biorthogonal tridiagonal reduction process does not break down. We discuss connections between the proposed methods and some existing methods, and give numerical experiments to illustrate the performance of the proposed methods.
We propose a finite difference scheme for the numerical solution of a two-dimensional singularly perturbed convection-diffusion partial differential equation whose solution features interacting boundary and interior layers, the latter due to discontinuities in source term. The problem is posed on the unit square. The second derivative is multiplied by a singular perturbation parameter, $\epsilon$, while the nature of the first derivative term is such that flow is aligned with a boundary. These two facts mean that solutions tend to exhibit layers of both exponential and characteristic type. We solve the problem using a finite difference method, specially adapted to the discontinuities, and applied on a piecewise-uniform (Shishkin). We prove that that the computed solution converges to the true one at a rate that is independent of the perturbation parameter, and is nearly first-order. We present numerical results that verify that these results are sharp.
We introduce in this paper the numerical analysis of high order both in time and space Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order $q=5$. The development and analysis of the methods are performed in the framework of time evolving finite elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical Analysis \textbf{41}, 1696-1845 (2021). The error estimates show through their dependence on the parameters of the equation the existence of different regimes in the behavior of the numerical solution; namely, in the diffusive regime, that is, when the diffusion parameter $\mu$ is large, the error is $O(h^{k+1}+\Delta t^{q})$, whereas in the advective regime, $\mu \ll 1$, the convergence is $O(\min (h^{k},\frac{h^{k+1} }{\Delta t})+\Delta t^{q})$. It is worth remarking that the error constant does not have exponential $\mu ^{-1}$ dependence.
We introduce a novel sampler called the energy based diffusion generator for generating samples from arbitrary target distributions. The sampling model employs a structure similar to a variational autoencoder, utilizing a decoder to transform latent variables from a simple distribution into random variables approximating the target distribution, and we design an encoder based on the diffusion model. Leveraging the powerful modeling capacity of the diffusion model for complex distributions, we can obtain an accurate variational estimate of the Kullback-Leibler divergence between the distributions of the generated samples and the target. Moreover, we propose a decoder based on generalized Hamiltonian dynamics to further enhance sampling performance. Through empirical evaluation, we demonstrate the effectiveness of our method across various complex distribution functions, showcasing its superiority compared to existing methods.
This paper presents a concrete and a symbolic rewriting logic semantics for parametric time Petri nets with inhibitor arcs (PITPNs), a flexible model of timed systems where parameters are allowed in firing bounds. We prove that our semantics is bisimilar to the "standard" semantics of PITPNs. This allows us to use the rewriting logic tool Maude, combined with SMT solving, to provide sound and complete formal analyses for PITPNs. We develop and implement a new general folding approach for symbolic reachability, so that Maude-with-SMT reachability analysis terminates whenever the parametric state-class graph of the PITPN is finite. Our work opens up the possibility of using the many formal analysis capabilities of Maude -- including full LTL model checking, analysis with user-defined analysis strategies, and even statistical model checking -- for such nets. We illustrate this by explaining how almost all formal analysis and parameter synthesis methods supported by the state-of-the-art PITPN tool Romeo can be performed using Maude with SMT. In addition, we also support analysis and parameter synthesis from parametric initial markings, as well as full LTL model checking and analysis with user-defined execution strategies. Experiments show that our methods outperform Romeo in many cases.
We discuss Cartan-Schouten metrics (Riemannian or pseudo-Riemannian metrics that are parallel with respect to the Cartan-Schouten canonical connection) on perfect Lie groups. Applications are foreseen in Information Geometry. Throughout this work, the tangent bundle TG and the cotangent bundle T*G of a Lie group G, are always endowed with their Lie group structures induced by the right trivialization. We show that TG and T*G are isomorphic if G possesses a biinvariant Riemannian or pseudo-Riemannian metric. We also show that, if on a perfect Lie group, there exists a Cartan-Schouten metric, then it must be biinvariant. We compute all such metrics on the cotangent bundles of simple Lie groups. We further show the following. Endowed with their canonical Lie group structures, the set of unit dual quaternions is isomorphic to TSU(2), the set of unit dual split quaternions is isomorphic to T*SL(2,R). The group SE(3) of special rigid displacements of the Euclidean 3-space is isomorphic to T*SO(3). The group SE(2,1) of special rigid displacements of the Minkowski 3-space is isomorphic to T*SO(2,1). Some results on SE(3) by N. Miolane and X. Pennec, and M. Zefran, V. Kumar and C. Croke, are generalized to SE(2,1) and to T*G, for any simple Lie group G.
The so-called independent low-rank matrix analysis (ILRMA) has demonstrated a great potential for dealing with the problem of determined blind source separation (BSS) for audio and speech signals. This method assumes that the spectra from different frequency bands are independent and the spectral coefficients in any frequency band are Gaussian distributed. The Itakura-Saito divergence is then employed to estimate the source model related parameters. In reality, however, the spectral coefficients from different frequency bands may be dependent, which is not considered in the existing ILRMA algorithm. This paper presents an improved version of ILRMA, which considers the dependency between the spectral coefficients from different frequency bands. The Sinkhorn divergence is then exploited to optimize the source model parameters. As a result of using the cross-band information, the BSS performance is improved. But the number of parameters to be estimated also increases significantly, and so is the computational complexity. To reduce the algorithm complexity, we apply the Kronecker product to decompose the modeling matrix into the product of a number of matrices of much smaller dimensionality. An efficient algorithm is then developed to implement the Sinkhorn divergence based BSS algorithm and the complexity is reduced by an order of magnitude.
A well-balanced second-order finite volume scheme is proposed and analyzed for a 2 X 2 system of non-linear partial differential equations which describes the dynamics of growing sandpiles created by a vertical source on a flat, bounded rectangular table in multiple dimensions. To derive a second-order scheme, we combine a MUSCL type spatial reconstruction with strong stability preserving Runge-Kutta time stepping method. The resulting scheme is ensured to be well-balanced through a modified limiting approach that allows the scheme to reduce to well-balanced first-order scheme near the steady state while maintaining the second-order accuracy away from it. The well-balanced property of the scheme is proven analytically in one dimension and demonstrated numerically in two dimensions. Additionally, numerical experiments reveal that the second-order scheme reduces finite time oscillations, takes fewer time iterations for achieving the steady state and gives sharper resolutions of the physical structure of the sandpile, as compared to the existing first-order schemes of the literature.
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