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This work puts forth low-complexity Riemannian subspace descent algorithms for the minimization of functions over the symmetric positive definite (SPD) manifold. Different from the existing Riemannian gradient descent variants, the proposed approach utilizes carefully chosen subspaces that allow the update to be written as a product of the Cholesky factor of the iterate and a sparse matrix. The resulting updates avoid the costly matrix operations like matrix exponentiation and dense matrix multiplication, which are generally required in almost all other Riemannian optimization algorithms on SPD manifold. We further identify a broad class of functions, arising in diverse applications, such as kernel matrix learning, covariance estimation of Gaussian distributions, maximum likelihood parameter estimation of elliptically contoured distributions, and parameter estimation in Gaussian mixture model problems, over which the Riemannian gradients can be calculated efficiently. The proposed uni-directional and multi-directional Riemannian subspace descent variants incur per-iteration complexities of $\mathcal{O}(n)$ and $\mathcal{O}(n^2)$ respectively, as compared to the $\mathcal{O}(n^3)$ or higher complexity incurred by all existing Riemannian gradient descent variants. The superior runtime and low per-iteration complexity of the proposed algorithms is also demonstrated via numerical tests on large-scale covariance estimation problems.

We model a family of closed kinematic chains, known as Kaleidocycles, with the theory of discrete spatial curves. By leveraging the connection between the deformation of discrete curves and the semi-discrete integrable systems, we describe the motion of a Kaleidocycle by elliptic theta functions. This study showcases an interesting example in which an integrable system generates an orbit in the space of the real solutions of polynomial equations defined by geometric constraints.

A variant of the standard notion of branching bisimilarity for processes with discrete relative timing is proposed which is coarser than the standard notion. Using a version of ACP (Algebra of Communicating Processes) with abstraction for processes with discrete relative timing, it is shown that the proposed variant allows of both the functional correctness and the performance properties of the PAR (Positive Acknowledgement with Retransmission) protocol to be analyzed. In the version of ACP concerned, the difference between the standard notion of branching bisimilarity and its proposed variant is characterized by a single axiom schema.

We consider an unknown multivariate function representing a system-such as a complex numerical simulator-taking both deterministic and uncertain inputs. Our objective is to estimate the set of deterministic inputs leading to outputs whose probability (with respect to the distribution of the uncertain inputs) of belonging to a given set is less than a given threshold. This problem, which we call Quantile Set Inversion (QSI), occurs for instance in the context of robust (reliability-based) optimization problems, when looking for the set of solutions that satisfy the constraints with sufficiently large probability. To solve the QSI problem, we propose a Bayesian strategy based on Gaussian process modeling and the Stepwise Uncertainty Reduction (SUR) principle, to sequentially choose the points at which the function should be evaluated to efficiently approximate the set of interest. We illustrate the performance and interest of the proposed SUR strategy through several numerical experiments.

Data compression algorithms typically rely on identifying repeated sequences of symbols from the original data to provide a compact representation of the same information, while maintaining the ability to recover the original data from the compressed sequence. Using data transformations prior to the compression process has the potential to enhance the compression capabilities, being lossless as long as the transformation is invertible. Floating point data presents unique challenges to generate invertible transformations with high compression potential. This paper identifies key conditions for basic operations of floating point data that guarantee lossless transformations. Then, we show four methods that make use of these observations to deliver lossless compression of real datasets, where we improve compression rates up to 40 %.

A slow decaying Kolmogorov n-width of the solution manifold of a parametric partial differential equation precludes the realization of efficient linear projection-based reduced-order models. This is due to the high dimensionality of the reduced space needed to approximate with sufficient accuracy the solution manifold. To solve this problem, neural networks, in the form of different architectures, have been employed to build accurate nonlinear regressions of the solution manifolds. However, the majority of the implementations are non-intrusive black-box surrogate models, and only a part of them perform dimension reduction from the number of degrees of freedom of the discretized parametric models to a latent dimension. We present a new intrusive and explicable methodology for reduced-order modelling that employs neural networks for solution manifold approximation but that does not discard the physical and numerical models underneath in the predictive/online stage. We will focus on autoencoders used to compress further the dimensionality of linear approximants of solution manifolds, achieving in the end a nonlinear dimension reduction. After having obtained an accurate nonlinear approximant, we seek for the solutions on the latent manifold with the residual-based nonlinear least-squares Petrov-Galerkin method, opportunely hyper-reduced in order to be independent from the number of degrees of freedom. New adaptive hyper-reduction strategies are developed along with the employment of local nonlinear approximants. We test our methodology on two nonlinear time-dependent parametric benchmarks involving a supersonic flow past a NACA airfoil with changing Mach number and an incompressible turbulent flow around the Ahmed body with changing slant angle.

V. Levenshtein first proposed the sequence reconstruction problem in 2001. This problem studies the model where the same sequence from some set is transmitted over multiple channels, and the decoder receives the different outputs. Assume that the transmitted sequence is at distance $d$ from some code and there are at most $r$ errors in every channel. Then the sequence reconstruction problem is to find the minimum number of channels required to recover exactly the transmitted sequence that has to be greater than the maximum intersection between two metric balls of radius $r$, where the distance between their centers is at least $d$. In this paper, we study the sequence reconstruction problem of permutations under the Hamming distance. In this model we define a Cayley graph over the symmetric group, study its properties and find the exact value of the largest intersection of its two metric balls for $d=2r$. Moreover, we give a lower bound on the largest intersection of two metric balls for $d=2r-1$.

Using simplifying assumptions that are related to the time reversal symmetry, a 1-dimensional family of 8-stage pseudo-symplectic Runge-Kutta methods of order (4, 8), i.e., methods of order 4 that preserve symplectic structure up to order 8, is derived. An example of 7-stage method of order (4, 9) is given.

We consider a Celestial Mechanics model: the spin-orbit problem with a dissipative tidal torque, which is a singular perturbation of a conservative system. The goal of this paper is to show that it is possible to compute quasi-periodic attractors accurately and reliably for parameter values extremely close to the breakdown. Therefore, it is possible to obtain information on mathematical phenomena at breakdown. The method we use incorporates the same time numerical and rigorous improvements. Among them (i) the formalism is based on studying the time-one map of the spin-orbit problem (which reduces the dimensionality of the problem) and has mathematical advantages; (ii) very accurate integration of the ODE (high order Taylor methods implemented with extended precision) for the map at its jets; (iii) a very efficient KAM method for maps which computes the attractor and its tangent spaces ( quadratically convergent step with low storage requirements, and low operation count); (iv) the algorithms are backed by a rigorous a-posteriori KAM Theorem, which establishes that if the algorithm, produces a very approximate solution of functional equation with reasonable condition numbers. then there is a true solution nearby; and (v) the continuation algorithm is guaranteed to reach arbitrarily close to the border of existence if it is given enough computer resources. As a byproduct of the accuracy that we maintain till breakdown, we study several scale invariant observables of the tori used in the renormalization group of infinite dimensional spaces. In contrast with previously studied simple models, the behavior at breakdown of the spin-orbit problem does not satisfy standard scaling relations which implies that the spin-orbit problem is not described by a hyperbolic fixed point of a renormalization operator.

Penalized $M-$estimators for logistic regression models have been previously study for fixed dimension in order to obtain sparse statistical models and automatic variable selection. In this paper, we derive asymptotic results for penalized $M-$estimators when the dimension $p$ grows to infinity with the sample size $n$. Specifically, we obtain consistency and rates of convergence results, for some choices of the penalty function. Moreover, we prove that these estimators consistently select variables with probability tending to 1 and derive their asymptotic distribution.