Wavelet boundary element methods – Adaptivity and goal-oriented error estimation

This article is dedicated to the adaptive wavelet boundary element method. It computes an approximation to the unknown solution of the boundary integral equation under consideration with a rate $N^{−s}_{dof}$, whenever the solution can be approximated with this rate in the setting determined by the underlying wavelet basis. The computational cost scale linearly in the number $N_{dof}$ of degrees of freedom. Goal-oriented error estimation for evaluating linear output functionals of the solution is also considered. An algorithm is proposed that approximately evaluates a linear output functional with a rate $N^{−(s+t)}_{dof}$, whenever the primal solution can be approximated with a rate $N^{-s}_{dof}$ and the dual solution can be approximated with a rate $N^{−t}_{dof}$, while the cost still scale linearly in $N_{dof}$. Numerical results for an acoustic scattering problem and for the point evaluation of the potential in case of the Laplace equation are reported to validate and quantify the approach

Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast

We consider divergence-form scalar elliptic equations and vectorial equations for elasticity with rough ($L^\infty(\Omega)$, $\Omega \subset \R^d$) coefficients $a(x)$ that, in particular, model media with non-separated scales and high contrast in material properties. We define the flux norm as the $L^2$ norm of the potential part of the fluxes of solutions, which is equivalent to the usual $H^1$-norm. We show that in the flux norm, the error associated with approximating, in a properly defined finite-dimensional space, the set of solutions of the aforementioned PDEs with rough coefficients is equal to the error associated with approximating the set of solutions of the same type of PDEs with smooth coefficients in a standard space (e.g., piecewise polynomial). We refer to this property as the {\it transfer property}. A simple application of this property is the construction of finite dimensional approximation spaces with errors independent of the regularity and contrast of the coefficients and with optimal and explicit convergence rates. This transfer property also provides an alternative to the global harmonic change of coordinates for the homogenization of elliptic operators that can be extended to elasticity equations. The proofs of these homogenization results are based on a new class of elliptic inequalities which play the same role in our approach as the div-curl lemma in classical homogenization.Comment: Accepted for publication in Archives for Rational Mechanics and Analysi

A genomic storm in critically injured humans

Critical injury in humans induces a genomic storm with simultaneous changes in expression of innate and adaptive immunity genes

Multi-omic analysis in injured humans: Patterns align with outcomes and treatment responses

Trauma is a leading cause of death and morbidity worldwide. Here, we present the analysis of a longitudinal multi-omic dataset comprising clinical, cytokine, endotheliopathy biomarker, lipidome, metabolome, and proteome data from severely injured humans. A "systemic storm" pattern with release of 1,061 markers, together with a pattern suggestive of the "massive consumption" of 892 constitutive circulating markers, is identified in the acute phase post-trauma. Data integration reveals two human injury response endotypes, which align with clinical trajectory. Prehospital thawed plasma rescues only endotype 2 patients with traumatic brain injury (30-day mortality: 30.3 versus 75.0%; p = 0.0015). Ubiquitin carboxy-terminal hydrolase L1 (UCHL1) was identified as the most predictive circulating biomarker to identify endotype 2-traumatic brain injury (TBI) patients. These response patterns refine the paradigm for human injury, while the datasets provide a resource for the study of critical illness, trauma, and human stress responses

Injury to the spleen

Curr Probl Surg 2001;38:921-1008