Publication
MATHICSE Technical Report: Stability properties of a projector-splitting scheme for the dynamical low rank approximation of random parabolic equations
Related concepts (23)
Chicken (Scheme implementation)
Chicken (stylized as CHICKEN) is a programming language, specifically a compiler and interpreter which implement a dialect of the programming language Scheme, and which compiles Scheme source code to standard C. It is mostly R5RS compliant and offers many extensions to the standard. The newer R7RS standard is supported through an extension library. Chicken is free and open-source software available under a BSD license. It is implemented mostly in Scheme, with some parts in C for performance or to make embedding into C programs easier.
Stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.
Scheme (programming language)
Scheme is a dialect of the Lisp family of programming languages. Scheme was created during the 1970s at the MIT Computer Science and Artificial Intelligence Laboratory (MIT AI Lab) and released by its developers, Guy L. Steele and Gerald Jay Sussman, via a series of memos now known as the Lambda Papers. It was the first dialect of Lisp to choose lexical scope and the first to require implementations to perform tail-call optimization, giving stronger support for functional programming and associated techniques such as recursive algorithms.
Explicit and implicit methods
Explicit and implicit methods are approaches used in numerical analysis for obtaining numerical approximations to the solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical processes. Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while implicit methods find a solution by solving an equation involving both the current state of the system and the later one.
Low-rank approximation
In mathematics, low-rank approximation is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank. The problem is used for mathematical modeling and data compression. The rank constraint is related to a constraint on the complexity of a model that fits the data. In applications, often there are other constraints on the approximating matrix apart from the rank constraint, e.
Strong and weak typing
In computer programming, one of the many ways that programming languages are colloquially classified is whether the language's type system makes it strongly typed or weakly typed (loosely typed). However, there is no precise technical definition of what the terms mean and different authors disagree about the implied meaning of the terms and the relative rankings of the "strength" of the type systems of mainstream programming languages.
Lyapunov stability
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point stay near forever, then is Lyapunov stable. More strongly, if is Lyapunov stable and all solutions that start out near converge to , then is said to be asymptotically stable (see asymptotic analysis).
Structural stability
In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact C1-small perturbations). Examples of such qualitative properties are numbers of fixed points and periodic orbits (but not their periods). Unlike Lyapunov stability, which considers perturbations of initial conditions for a fixed system, structural stability deals with perturbations of the system itself.
Diffusion equation
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's laws of diffusion). In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The diffusion equation is a special case of the convection–diffusion equation, when bulk velocity is zero.
Numerical stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.