Generalized Minimal Residual
Generalized Minimal Residual - Related Videos
01:22
Generalized Hooke's Law
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2025
The generalized Hooke's Law is a broadened version of Hooke's Law, which extends to all types of stress and in every direction. Consider an isotropic material shaped into a cube subjected to multiaxial loading. In this scenario, normal stresses are exerted along the three coordinate axes. As a result of these stresses, the cubic shape deforms into a rectangular parallelepiped. Despite this deformation, the new shape maintains equal sides, and there is a normal strain in the direction of the...
01:30
Generalized Anxiety Disorder
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2025
Generalized Anxiety Disorder (GAD) is a chronic condition characterized by excessive and uncontrollable worry that persists for at least six months, significantly interfering with daily functioning. Unlike situational anxiety, which arises in response to specific stressors, GAD often occurs without a clear cause. Individuals may experience disproportionate worry about work, health, or relationships. For instance, a person might continuously fear poor health despite normal medical evaluations or...
01:24
Generalization, Discrimination, and Extinction
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2024
Generalization, discrimination, and extinction are key concepts in operant conditioning that influence how behaviors are learned and maintained. Generalization occurs when a behavior reinforced in one context is performed in similar situations. For instance, a student who studies diligently for calculus and receives excellent grades might apply the same study habits to psychology and history, expecting similar results. Generalization shows how learning in one setting can influence behavior in...
01:20
Social Foundations of Self II: The Generalized Other
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2025
According to George Herbert Mead, as children progress beyond the game stage, they develop a more comprehensive understanding of societal rules and norms. This cognitive and social development enables them to internalize the expectations of the broader community, refining their ability to regulate behavior.Consistent participation in organized activities is crucial in helping children recognize that their actions are not isolated but contribute to a more significant, interconnected group effort.
01:26
Residual Stresses
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2024
Residual stresses reside in a structure even after removing the original stress inducer. This phenomenon often arises from varied plastic deformations across different parts of a structure. Consider a rod stretched beyond its yield point. It will not regain its original length due to permanent deformation. Even after load removal, the rod does not entirely lose stress because of uneven plastic deformations, resulting in residual stresses. The computation of these stresses in structures is...
FAQs
The Arnoldi process generates an orthonormal basis for the Krylov subspace explored by GMRES. Orthonormality gives the method a structured set of directions for constructing successive approximations, while the expanding subspace allows the solution estimate to incorporate increasingly relevant information from the linear system. This basis-building step is central to applying GMRES without forming a matrix inverse.
The residual measures the discrepancy remaining after an approximate solution is substituted into the linear system. GMRES selects the approximation with the smallest residual norm among the candidates in its current Krylov subspace. This criterion provides a direct measure for choosing each iterate and helps the method target an approximation that better satisfies the original equations.
Preconditioning transforms the linear system into a more favorable form before or during the iterative solution process. The goal is to make the transformed problem easier for GMRES to handle, which can improve convergence. In engineering computations, this can be important because better convergence may reduce the computation time and resources required for large sparse systems.
GMRES is particularly appropriate when the system matrix is sparse or nonsymmetric, conditions that commonly arise in large computational models. Its iterative strategy avoids explicitly calculating a matrix inverse, while the Krylov-subspace approach builds approximations from the system's matrix information. These characteristics make it useful when direct handling of a large matrix would be computationally demanding.
A typical workflow begins with the linear system and, when appropriate, a preconditioned form that has more favorable behavior. GMRES then applies the Arnoldi process to construct an orthonormal Krylov-subspace basis. At each iteration, it evaluates candidate approximations in that subspace and selects the one with the smallest residual norm, continuing this process as the approximation improves.
Engineering applications include computational problems in fluid flow, structural analysis, and electromagnetics. These areas can produce large linear systems whose sparse or nonsymmetric structure makes iterative treatment valuable. By avoiding explicit matrix inversion and potentially benefiting from preconditioning, GMRES can help reduce memory use and computation time when solving demanding models.