Search

Adaptive tensor product wavelet methods are applied for solving Poisson’s equation, as well as anisotropic generalizations, in high space dimensions. It will be demonstrated that the resulting approximations converge in energy norm with the same rate as the best approximations from the span of the best N tensor product wavelets, where moreover the constant factor that we may lose is independent of the space dimension n. The cost of producing these approximations will be proportional to their length with a constant factor that may grow with n, but only linearly.

Physiology offers insights into fundamental aspects of the circulation, even when the circulation is nonphysiological, as during cardiac arrest and cardiopulmonary resuscitation (CPR). Human and animal experiments offer only limited insights as they are susceptible to uncontrollable variables. Mathematical models offer quantitative results for flows, pressures and volumes under clearly defined conditions, chosen by the experimenters. This report describes the left ventricle and its immediate environment as the core of a larger mathematical model. The model works normally under physiological conditions, and is specifically designed to allow understanding of flow, pressure, and volume phenomena under CPR as an extreme pathophysiological scenario. Using impedance-defined flow, the importance of valves for both the contracting as well as the asystolic ventricle is quantified. It demonstrates the role of venous pressure, sloshing of blood and flow around the cardiovascular circuit. The flow of 8 mL/s, in a pathological situation is demonstrated to be due not to cardiac compression, but to venous pressure. The principal conclusions are that the model functions in the physiological situation and that in an asystolic left ventricle with competent valves, intrathoracic pressure can modestly replace contractile properties. In an asystolic ventricle without competent valves, intrathoracic pressure variations accomplish little. The cardiac pump theory in CPR has limited applicability.

We construct a wavelet basis on the unit interval with respect to which both the (infinite) mass and stiffness matrix corresponding to the one-dimensional Laplacian are (truly) sparse and boundedly invertible. As a consequence, the (infinite) stiffness matrix corresponding to the Laplacian on the n-dimensional unit box with respect to the n-fold tensor product wavelet basis is also sparse and boundedly invertible. This greatly simplifies the implementation and improves the quantitative properties of an adaptive wavelet scheme to solve the multi-dimensional Poisson equation. The results extend to any second order partial differential operator with constant coefficients that defines a boundedly invertible operator.