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This paper is concerned with recovering the solution of a final value problem associated with a parabolic equation involving a non linear source and a non-local term, which to the best of our knowledge has not been studied earlier. It is shown that the considered problem is ill-posed, and thus, some regularization method has to be employed in order to obtain stable approximations. In this regard, we obtain regularized approximations by solving some non linear integral equations which is derived by considering a truncated version of the Fourier expansion of the sought solution. Under different Gevrey smoothness assumptions on the exact solution, we provide parameter choice strategies and obtain the error estimates. A key tool in deriving such estimates is a version of Gr{\"o}nwalls' inequality for iterated integrals, which perhaps, is proposed and analysed for the first time.

Introducing a coupling framework reminiscent of FETI methods, but here on abstract form, we establish conditions for stability and minimal requirements for well-posedness on the continuous level, as well as conditions on local solvers for the approximation of subproblems. We then discuss stability of the resulting Lagrange multiplier methods and show stability under a mesh conditions between the local discretizations and the mortar space. If this condition is not satisfied we show how a stabilization, acting only on the multiplier can be used to achieve stability. The design of preconditioners of the Schur complement system is discussed in the unstabilized case. Finally we discuss some applications that enter the framework.

The arrival of AI techniques in computations, with the potential for hallucinations and non-robustness, has made trustworthiness of algorithms a focal point. However, trustworthiness of the many classical approaches are not well understood. This is the case for feature selection, a classical problem in the sciences, statistics, machine learning etc. Here, the LASSO optimisation problem is standard. Despite its widespread use, it has not been established when the output of algorithms attempting to compute support sets of minimisers of LASSO in order to do feature selection can be trusted. In this paper we establish how no (randomised) algorithm that works on all inputs can determine the correct support sets (with probability $> 1/2$) of minimisers of LASSO when reading approximate input, regardless of precision and computing power. However, we define a LASSO condition number and design an efficient algorithm for computing these support sets provided the input data is well-posed (has finite condition number) in time polynomial in the dimensions and logarithm of the condition number. For ill-posed inputs the algorithm runs forever, hence, it will never produce a wrong answer. Furthermore, the algorithm computes an upper bound for the condition number when this is finite. Finally, for any algorithm defined on an open set containing a point with infinite condition number, there is an input for which the algorithm will either run forever or produce a wrong answer. Our impossibility results stem from generalised hardness of approximation -- within the Solvability Complexity Index (SCI) hierarchy framework -- that generalises the classical phenomenon of hardness of approximation.

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 $O(n)$ and $O(n^2)$ respectively, as compared to the $O(n^3)$ 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 and matrix square root problems. MATLAB code implementation is publicly available on GitHub : //github.com/yogeshd-iitk/subspace_descent_over_SPD_manifold

In this article, we study some anisotropic singular perturbations for a class of linear elliptic problems. A uniform estimates for conforming $Q_1$ finite element method are derived, and some other results of convergence and regularity for the continuous problem are proved.

This paper is dedicated to the numerical solution of a fourth-order singular perturbation problem using the interior penalty virtual element method (IPVEM) proposed in [42]. The study introduces modifications to the jumps and averages in the penalty term, as well as presents an automated mesh-dependent selection of the penalty parameter. Drawing inspiration from the modified Morley finite element methods, we leverage the conforming interpolation technique to handle the lower part of the bilinear form. Through our analysis, we establish optimal convergence in the energy norm and provide a rigorous proof of uniform convergence concerning the perturbation parameter in the lowest-order case.

In this paper we present a mathematical and numerical analysis of an eigenvalue problem associated to the elasticity-Stokes equations stated in two and three dimensions. Both problems are related through the Herrmann pressure. Employing the Babu\v ska--Brezzi theory, it is proved that the resulting continuous and discrete variational formulations are well-posed. In particular, the finite element method is based on general inf-sup stables pairs for the Stokes system, such that, Taylor--Hood finite elements. By using a general approximation theory for compact operators, we obtain optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Under mild assumptions, we have that these estimates hold with constants independent of the Lam\'e coefficient $\lambda$. In addition, we carry out the reliability and efficiency analysis of a residual-based a posteriori error estimator for the spectral problem. We report a series of numerical tests in order to assess the performance of the method and its behavior when the nearly incompressible case of elasticity is considered.

The aim of this paper is to give a systematic mathematical interpretation of the diffusion problem on which Graph Neural Networks (GNNs) models are based. The starting point of our approach is a dissipative functional leading to dynamical equations which allows us to study the symmetries of the model. We discuss the conserved charges and provide a charge-preserving numerical method for solving the dynamical equations. In any dynamical system and also in GRAph Neural Diffusion (GRAND), knowing the charge values and their conservation along the evolution flow could provide a way to understand how GNNs and other networks work with their learning capabilities.

In this paper, we introduce a new simple approach to developing and establishing the convergence of splitting methods for a large class of stochastic differential equations (SDEs), including additive, diagonal and scalar noise types. The central idea is to view the splitting method as a replacement of the driving signal of an SDE, namely Brownian motion and time, with a piecewise linear path that yields a sequence of ODEs $-$ which can be discretised to produce a numerical scheme. This new way of understanding splitting methods is inspired by, but does not use, rough path theory. We show that when the driving piecewise linear path matches certain iterated stochastic integrals of Brownian motion, then a high order splitting method can be obtained. We propose a general proof methodology for establishing the strong convergence of these approximations that is akin to the general framework of Milstein and Tretyakov. That is, once local error estimates are obtained for the splitting method, then a global rate of convergence follows. This approach can then be readily applied in future research on SDE splitting methods. By incorporating recently developed approximations for iterated integrals of Brownian motion into these piecewise linear paths, we propose several high order splitting methods for SDEs satisfying a certain commutativity condition. In our experiments, which include the Cox-Ingersoll-Ross model and additive noise SDEs (noisy anharmonic oscillator, stochastic FitzHugh-Nagumo model, underdamped Langevin dynamics), the new splitting methods exhibit convergence rates of $O(h^{3/2})$ and outperform schemes previously proposed in the literature.

This paper proposes a specialized autonomous driving system that takes into account the unique constraints and characteristics of automotive systems, aiming for innovative advancements in autonomous driving technology. The proposed system systematically analyzes the intricate data flow in autonomous driving and provides functionality to dynamically adjust various factors that influence deep learning models. Additionally, for algorithms that do not rely on deep learning models, the system analyzes the flow to determine resource allocation priorities. In essence, the system optimizes data flow and schedules efficiently to ensure real-time performance and safety. The proposed system was implemented in actual autonomous vehicles and experimentally validated across various driving scenarios. The experimental results provide evidence of the system's stable inference and effective control of autonomous vehicles, marking a significant turning point in the development of autonomous driving systems.