With respect to space-time tensor-product wavelet bases, parabolic initial boundary value problems are equivalently formulated as bi-infinite matrix problems. Adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate. In case the spatial domain is of product type, the use of spatial tensor product wavelet bases is proved to overcome the so-called curse of dimensionality, i.e., the reduction of the convergence rate with increasing spatial dimension.

https://www.hcm.uni-bonn.de/fileadmin/user_upload/sparcity_and_computation/Stevenson.pdf

Space-time adaptive wavelet methods for parabolic evolution problems

We propose stable variational formulations for certain linear, unsymmetric operators with first order transport equations in bounded domains serving as the primary focus of this paper. The central objective is to develop for such classes adaptive solution concepts with provable error reduction. To adaptively resolve anisotropic solution features such as propagating singularities, the presently proposed variational formulations allow, in particular, the employment of trial spaces spanned by directional representation systems. Since such systems, typically given as frames, are known to be stable only in $L_2$, special emphasis is placed on $L_2$-stable formulations. The proposed stability concept is based on perturbations of certain “ideal” test spaces in Petrov--Galerkin formulations; see also [L. F. Demkowicz and J. Gopalakrishnan, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 1558--1572], [L. Demkowicz and J. Gopalakrishnan, Numer. Methods Partial Differential Equations, 27 (2011), pp. 70--105], [J...

/pdf/adaptive-petrov-galerkin-methods-for-first-order-transport-3a7dodmn03.pdf

Adaptive Petrov-Galerkin methods for first order transport equations !

We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we conduct a rigorous convergence analysis, where all parameters required for the execution of the methods depend only on the underlying infinite-dimensional problem, but not on a concrete discretization. Under certain assumptions on the rates for the involved low-rank approximations and basis expansions, we can also give bounds on the computational complexity of the iteration as a function of the prescribed target error. Our theoretical findings are illustrated and supported by computational experiments. These demonstrate that problems in very high dimensions can be treated with controlled solution accuracy.

/pdf/adaptive-near-optimal-rank-tensor-approximation-for-high-2im0owozlu.pdf

Adaptive Near-Optimal Rank Tensor Approximation for High-Dimensional Operator Equations

In [Math. Comp, 70 (2001), 27–75] and [Found. Comput. Math., 2(3) (2002), 203–245], Cohen, Dahmen and DeVore introduced adaptive wavelet methods for solving operator equations. These papers meant a break-through in the field, because their adaptive methods were not only proven to converge, but also with a rate better than that of their non-adaptive counterparts in cases where the latter methods converge with a reduced rate due a lacking regularity of the solution. Until then, adaptive methods were usually assumed to converge via a saturation assumption. An exception was given by the work of Dorfler in [SIAM J. Numer. Anal., 33 (1996), 1106–1124], where an adaptive finite element method was proven to converge, with no rate though.

https://www.springer.com/cda/content/document/cda_downloaddocument/9783642034121-c13.pdf?SGWID=0-0-45-803415-p173925505

Adaptive wavelet methods for solving operator equations: An overview

In this paper, we introduce an adaptive wavelet method for operator equations on unbounded domains. We use wavelet bases on ? n to equivalently express the operator equation in terms of a well-conditioned discrete problem on sequence spaces. By realizing an approximate adaptive operator application also for unbounded domains, we obtain a scheme that is convergent at an asymptotically optimal rate. We show the quantitative performance of the scheme by various numerical experiments.

https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi/forschung/PreprintServer/2011/KestlerUrban_PreprintVersion.pdf

Adaptive Wavelet Methods on Unbounded Domains

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.

/pdf/an-adaptive-wavelet-method-for-solving-high-dimensional-1v8rabhbe3.pdf

An Adaptive Wavelet Method for Solving High-Dimensional Elliptic PDEs

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.

/pdf/a-sparse-laplacian-in-tensor-product-wavelet-coordinates-4w91259h9h.pdf

A sparse Laplacian in tensor product wavelet coordinates

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.

/pdf/modeling-in-cardiopulmonary-resuscitation-pumping-the-heart-2cj8tey4wt.pdf

Tammo Jan Dijkema

Papers

An Adaptive Wavelet Method for Solving High-Dimensional Elliptic PDEs

A sparse Laplacian in tensor product wavelet coordinates

Modeling in cardiopulmonary resuscitation : pumping the heart