Runge–Kutta methodsIn numerical analysis, the Runge–Kutta methods (ˈrʊŋəˈkʊtɑː ) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta. The most widely known member of the Runge–Kutta family is generally referred to as "RK4", the "classic Runge–Kutta method" or simply as "the Runge–Kutta method".
Euler methodIn mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis (published 1768–1870).
Ergodic hypothesisIn physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a long period of time. Liouville's theorem states that, for a Hamiltonian system, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble (i.
Computational scienceComputational science, also known as scientific computing, technical computing or scientific computation (SC), is a division of science that uses advanced computing capabilities to understand and solve complex physical problems. This includes Algorithms (numerical and non-numerical): mathematical models, computational models, and computer simulations developed to solve sciences (e.
Inverse problemAn inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the density of the Earth from measurements of its gravity field. It is called an inverse problem because it starts with the effects and then calculates the causes. It is the inverse of a forward problem, which starts with the causes and then calculates the effects.
MethodologyIn its most common sense, methodology is the study of research methods. However, the term can also refer to the methods themselves or to the philosophical discussion of associated background assumptions. A method is a structured procedure for bringing about a certain goal, like acquiring knowledge or verifying knowledge claims. This normally involves various steps, like choosing a sample, collecting data from this sample, and interpreting the data. The study of methods concerns a detailed description and analysis of these processes.
PhylogeneticsIn biology, phylogenetics (ˌfaɪloʊdʒəˈnɛtɪks,_-lə-) is the study of the evolutionary history and relationships among or within groups of organisms. These relationships are determined by phylogenetic inference methods that focus on observed heritable traits, such as DNA sequences, protein amino acid sequences, or morphology. The result of such an analysis is a phylogenetic tree—a diagram containing a hypothesis of relationships that reflects the evolutionary history of a group of organisms.
Iterative methodIn computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones. A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of the iterative method.
Pushforward measureIn measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function. Given measurable spaces and , a measurable mapping and a measure , the pushforward of is defined to be the measure given by for This definition applies mutatis mutandis for a signed or complex measure. The pushforward measure is also denoted as , , , or .
Probabilistic numericsProbabilistic numerics is an active field of study at the intersection of applied mathematics, statistics, and machine learning centering on the concept of uncertainty in computation. In probabilistic numerics, tasks in numerical analysis such as finding numerical solutions for integration, linear algebra, optimization and simulation and differential equations are seen as problems of statistical, probabilistic, or Bayesian inference.
