TY - JOUR AB - We derive a posteriori estimates for the modeling error caused by the assumption of perfect incompressibility in the incompressible Navier-Stokes equation: Real fluids are never perfectly incompressible but always feature at least some low amount of compressibility. Thus, their behavior is described by the compressible Navier-Stokes equation, the pressure being a steep function of the density. We rigorously estimate the difference between an approximate solution to the incompressible Navier-Stokes equation and any weak solution to the compressible Navier-Stokes equation in the sense of Lions (without assuming any additional regularity of solutions). Heuristics and numerical results suggest that our error estimates are of optimal order in the case of "well-behaved" flows and divergence-free approximations of the velocity field. Thus, we expect our estimates to justify the idealization of fluids as perfectly incompressible also in practical situations. AU - Fischer, Julian L ID - 1314 IS - 5 JF - SIAM Journal on Numerical Analysis TI - A posteriori modeling error estimates for the assumption of perfect incompressibility in the Navier-Stokes equation VL - 53 ER - TY - JOUR AB - We present an algorithm for the derivation of lower bounds on support propagation for a certain class of nonlinear parabolic equations. We proceed by combining the ideas in some recent papers by the author with the algorithmic construction of entropies due to Jüngel and Matthes, reducing the problem to a quantifier elimination problem. Due to its complexity, the quantifier elimination problem cannot be solved by present exact algorithms. However, by tackling the quantifier elimination problem numerically, in the case of the thin-film equation we are able to improve recent results by the author in the regime of strong slippage n ∈ (1, 2). For certain second-order doubly nonlinear parabolic equations, we are able to extend the known lower bounds on free boundary propagation to the case of irregular oscillatory initial data. Finally, we apply our method to a sixth-order quantum drift-diffusion equation, resulting in an upper bound on the time which it takes for the support to reach every point in the domain. AU - Julian Fischer ID - 1313 IS - 1 JF - Interfaces and Free Boundaries TI - Estimates on front propagation for nonlinear higher-order parabolic equations: An algorithmic approach VL - 17 ER - TY - JOUR AB - In the present work we introduce the notion of a renormalized solution for reaction–diffusion systems with entropy-dissipating reactions. We establish the global existence of renormalized solutions. In the case of integrable reaction terms our notion of a renormalized solution reduces to the usual notion of a weak solution. Our existence result in particular covers all reaction–diffusion systems involving a single reversible reaction with mass-action kinetics and (possibly species-dependent) Fick-law diffusion; more generally, it covers the case of systems of reversible reactions with mass-action kinetics which satisfy the detailed balance condition. For such equations the existence of any kind of solution in general was an open problem, thereby motivating the study of renormalized solutions. AU - Julian Fischer ID - 1316 IS - 1 JF - Archive for Rational Mechanics and Analysis TI - Global existence of renormalized solutions to entropy-dissipating reaction–diffusion systems VL - 218 ER - TY - JOUR AB - In plants, vacuolar H+-ATPase (V-ATPase) activity acidifies both the trans-Golgi network/early endosome (TGN/EE) and the vacuole. This dual V-ATPase function has impeded our understanding of how the pH homeostasis within the plant TGN/EE controls exo- and endocytosis. Here, we show that the weak V-ATPase mutant deetiolated3 (det3) displayed a pH increase in the TGN/EE, but not in the vacuole, strongly impairing secretion and recycling of the brassinosteroid receptor and the cellulose synthase complexes to the plasma membrane, in contrast to mutants lacking tonoplast-localized V-ATPase activity only. The brassinosteroid insensitivity and the cellulose deficiency defects in det3 were tightly correlated with reduced Golgi and TGN/EE motility. Thus, our results provide strong evidence that acidification of the TGN/EE, but not of the vacuole, is indispensable for functional secretion and recycling in plants. AU - Yu, Luo AU - Scholl, Stefan AU - Doering, Anett AU - Yi, Zhang AU - Irani, Niloufer AU - Di Rubbo, Simone AU - Neumetzler, Lutz AU - Krishnamoorthy, Praveen AU - Van Houtte, Isabelle AU - Mylle, Evelien AU - Bischoff, Volker AU - Vernhettes, Samantha AU - Winne, Johan AU - Friml, Jirí AU - Stierhof, York AU - Schumacher, Karin AU - Persson, Staffan AU - Russinova, Eugenia ID - 1383 IS - 7 JF - Nature Plants TI - V-ATPase activity in the TGN/EE is required for exocytosis and recycling in Arabidopsis VL - 1 ER - TY - CONF AB - In this work we aim at extending the theoretical foundations of lifelong learning. Previous work analyzing this scenario is based on the assumption that learning tasks are sampled i.i.d. from a task environment or limited to strongly constrained data distributions. Instead, we study two scenarios when lifelong learning is possible, even though the observed tasks do not form an i.i.d. sample: first, when they are sampled from the same environment, but possibly with dependencies, and second, when the task environment is allowed to change over time in a consistent way. In the first case we prove a PAC-Bayesian theorem that can be seen as a direct generalization of the analogous previous result for the i.i.d. case. For the second scenario we propose to learn an inductive bias in form of a transfer procedure. We present a generalization bound and show on a toy example how it can be used to identify a beneficial transfer algorithm. AU - Pentina, Anastasia AU - Lampert, Christoph ID - 1425 TI - Lifelong learning with non-i.i.d. tasks VL - 2015 ER -