Related papers: Numerical Stability of Lanczos Methods
We show that by using higher order precision arithmetic, i.e., using floating point types with more significant bits than standard double precision numbers, one may accurately compute eigenvalues for non-normal matrices arising in…
Backward error (BE) analysis emerges as a powerful tool for assessing the backward stability and strong backward stability of numerical algorithms. In this paper, we explore structured BEs for a class of double saddle point problems…
A new procedure is constructed by means of APS in APLAN language. The procedure solves the initial-value problem for linear differential equations of order $k$ with polynomial coefficients and regular singularity in the initialization point…
In variable or graph selection problems, finding a right-sized model or controlling the number of false positives is notoriously difficult. Recently, a meta-algorithm called Stability Selection was proposed that can provide reliable…
We consider a family of conforming space-time finite element discretizations for the wave equation based on splines of maximal regularity in time. Traditional techniques may require a CFL condition to guarantee stability. Recent works by O.…
We consider the problem of designing uniformly stable first-order optimization algorithms for empirical risk minimization. Uniform stability is often used to obtain generalization error bounds for optimization algorithms, and we are…
Sensitivity to noise makes most of the current quantum computing schemes prone to error and nonscalable, allowing only for small proof-of-principle devices. Topologically-protected quantum computing aims at solving this problem by encoding…
We investigate errors in tangents and adjoints of implicit functions resulting from errors in the primal solution due to approximations computed by a numerical solver. Adjoints of systems of linear equations turn out to be unconditionally…
A matrix framework is presented for the solution of ODEs, including initial-, boundary and inner-value problems. The framework enables the solution of the ODEs for arbitrary nodes. There are four key issues involved in the formulation of…
In this article, we provide stability estimates for the finite element discretization of a class of inverse parameter problems of the form $-\nabla\cdot(\mu S) = \g f$ in a domain $\Omega$ of $\R^d$. Here $\mu$ is the unknown parameter to…
Iterative methods with certified convergence for the computation of Gauss--Jacobi quadratures are described. The methods do not require a priori estimations of the nodes to guarantee its fourth-order convergence. They are shown to be…
In this paper we construct a third order method for solving additively split autonomous stiff systems of ordinary differential equations. The constructed additive method is L-stable with respect to the implicit part and allows to use an…
Orthogonality constraints are ubiquitous in robust and probabilistic machine learning. Unfortunately, current optimizers are computationally expensive and do not scale to problems with hundreds or thousands of constraints. One notable…
In this paper, we focus on the analysis of discrete versions of the Calderon problem with partial boundary data in dimension d >= 3. In particular, we establish logarithmic stability estimates for the discrete Calderon problem on an…
A desirable property of control systems is to be robust to inputs, that is small perturbations of the inputs of a system will cause only small perturbations on its outputs. But it is not clear whether this property is maintained at the…
In this work we explore the fidelity of numerical approximations to the analytic spectra of hyperbolic partial differential equation systems with variable coefficients. We are particularly interested in the ability of discrete methods to…
Stability margins for linear time-varying (LTV) and switched-linear systems are traditionally computed via quadratic Lyapunov functions, and these functions certify the stability of the system under study. In this work, we show how the more…
Mathematically rigorous inversion method is developed to recover compactly supported potentials from the fixed-energy scattering data in three dimensions. Error estimates are given for the solution. An algorithm for inversion of noisy…
Algorithms of control of differential equations solutions are under investigation in the article. Idealized and real modifications of the algorithms are distinguished. An equation, which can be the base equation for investigation of the…
Stochastic optimization has found wide applications in minimizing objective functions in machine learning, which motivates a lot of theoretical studies to understand its practical success. Most of existing studies focus on the convergence…