expEikB

expEikB is a theoretical construct used in applied mathematics and computational physics to denote the exponential of a composite operator that blends an Eikonal-type operator with a bilinear form B. The term arises in discussions of high-frequency wave propagation and in numerical methods for hyperbolic and parabolic partial differential equations where phase and energy interactions must be captured together.

Definition and interpretation: Let Eik denote a discretized Eikonal operator that encodes phase information and B denote a symmetric bilinear form on a Hilbert space H. The expEikB operator is defined as the exponential of the combined operator A = EikB (or, in some conventions, of Eik and B taken additively), often written as exp(tA) for a time parameter t. In many contexts, expEikB refers to the map f mapsto exp(A) f, which advances a wavefield or state through the operator exponential.

Properties: If A is sectorial, the family {exp(tA)} forms a C0-semigroup on H, yielding a stable evolution under appropriate discretization. The action of expEikB typically couples oscillatory phase information from the Eikonal component with smoothing or energy-transfer effects from B, producing propagators that are nonlocal and context-dependent.

Applications: expEikB appears in seismic imaging, acoustics, and quantum semiclassical approximations, where it models combined phase evolution and energy distribution. It also features in exponential integrator schemes for PDEs, enabling larger time steps with controlled error.

Computational aspects: Practical implementations use Krylov subspace methods, Padé or rational approximants, or contour integral representations to approximate exp(tA) acting on a vector, often with specialized preconditioning for high-dimensional discretizations.

History and terminology: The exact definition and notation of expEikB vary across sources, reflecting an evolving concept rather than a universally standardized operator. It remains a niche topic within wave propagation theory and numerical analysis.