Beyond the Spectral Theorem: Spectrally Decomposing Arbitrary Functions of Nondiagonalizable Operators

Nonlinearities in finite dimensions can be linearized by projecting them into infinite dimensions. Unfortunately, often the linear operator techniques that one would then use simply fail since the operators cannot be diagonalized. This curse is well known. It also occurs for finite-dimensional linear operators. We circumvent it by developing a meromorphic functional calculus that can decompose arbitrary functions of nondiagonalizable linear operators in terms of their eigenvalues and projection operators. It extends the spectral theorem of normal operators to a much wider class, including circumstances in which poles and zeros of the function coincide with the operator spectrum. By allowing the direct manipulation of individual eigenspaces of nonnormal and nondiagonalizable operators, the new theory avoids spurious divergences. As such, it yields novel insights and closed-form expressions across several areas of physics in which nondiagonalizable dynamics are relevant, including memoryful stochastic processes, open non unitary quantum systems, and far-from-equilibrium thermodynamics. The technical contributions include the first full treatment of arbitrary powers of an operator. In particular, we show that the Drazin inverse, previously only defined axiomatically, can be derived as the negative-one power of singular operators within the meromorphic functional calculus and we give a general method to construct it. We provide new formulae for constructing projection operators and delineate the relations between projection operators, eigenvectors, and generalized eigenvectors. By way of illustrating its application, we explore several, rather distinct examples.Comment: 29 pages, 4 figures, expanded historical citations; http://csc.ucdavis.edu/~cmg/compmech/pubs/bst.ht

Simulation of stellar instabilities with vastly different timescales using domain decomposition

Strange mode instabilities in the envelopes of massive stars lead to shock waves, which can oscillate on a much shorter timescale than that associated with the primary instability. The phenomenon is studied by direct numerical simulation using a, with respect to time, implicit Lagrangian scheme, which allows for the variation by several orders of magnitude of the dependent variables. The timestep for the simulation of the system is reduced appreciably by the shock oscillations and prevents its long term study. A procedure based on domain decomposition is proposed to surmount the difficulty of vastly different timescales in various regions of the stellar envelope and thus to enable the desired long term simulations. Criteria for domain decomposition are derived and the proper treatment of the resulting inner boundaries is discussed. Tests of the approach are presented and its viability is demonstrated by application to a model for the star P Cygni. In this investigation primarily the feasibility of domain decomposition for the problem considered is studied. We intend to use the results as the basis of an extension to two dimensional simulations.Comment: 15 pages, 10 figures, published in MNRA

MADNESS: A Multiresolution, Adaptive Numerical Environment for Scientific Simulation

MADNESS (multiresolution adaptive numerical environment for scientific simulation) is a high-level software environment for solving integral and differential equations in many dimensions that uses adaptive and fast harmonic analysis methods with guaranteed precision based on multiresolution analysis and separated representations. Underpinning the numerical capabilities is a powerful petascale parallel programming environment that aims to increase both programmer productivity and code scalability. This paper describes the features and capabilities of MADNESS and briefly discusses some current applications in chemistry and several areas of physics