Search Results
Canonical sets of best L1-approximation
2012, Dryanov, D., Petrov, P.
In mathematics, the term approximation usually means either interpolation on a point set or approximation with respect to a given distance. There is a concept, which joins the two approaches together, and this is the concept of characterization of the best approximants via interpolation. It turns out that for some large classes of functions the best approximants with respect to a certain distance can be constructed by interpolation on a point set that does not depend on the choice of the function to be approximated. Such point sets are called canonical sets of best approximation. The present paper summarizes results on canonical sets of best L1-approximation with emphasis on multivariate interpolation and best L1-approximation by blending functions. The best L1-approximants are characterized as transfinite interpolants on canonical sets. The notion of a Haar-Chebyshev system in the multivariate case is discussed also. In this context, it is shown that some multivariate interpolation spaces share properties of univariate Haar-Chebyshev systems. We study also the problem of best one-sided multivariate L 1-approximation by sums of univariate functions. Explicit constructions of best one-sided L1-approximants give rise to well-known and new inequalities.
All functions are locally s-harmonic up to a small error
2014, Dipierro, Serena, Savin, Ovidiu, Valdinoci, Enrico
We show that we can approximate every function f Ck (B1) with a s-harmonic function in B1 that vanishes outside a compact set. That is, s-harmonic functions are dense in Ck loc. This result is clearly in contrast with the rigidity of harmonic functions in the classical case and can be viewed as a purely nonlocal feature.
Results of the third Marine Ice Sheet Model Intercomparison Project (MISMIP+)
2020, Cornford, Stephen L., Seroussi, Helene, Asay-Davis, Xylar S., Gudmundsson, G. Hilmar, Arthern, Rob, Borstad, Chris, Christmann, Julia, dos Santos, Thiago Dias, Feldmann, Johannes, Goldberg, Daniel, Hoffman, Matthew J., Humbert, Angelika, Kleiner, Thomas, Leguy, Gunter, Lipscomb, William H., Merino, Nacho, Durand, Gaël, Morlighem, Mathieu, Pollard, David, Rückamp, Martin, Williams, C. Rosie, Yu, Hongju
We present the result of the third Marine Ice Sheet Model Intercomparison Project, MISMIP+. MISMIP+ is intended to be a benchmark for ice-flow models which include fast sliding marine ice streams and floating ice shelves and in particular a treatment of viscous stress that is sufficient to model buttressing, where upstream ice flow is restrained by a downstream ice shelf. A set of idealized experiments first tests that models are able to maintain a steady state with the grounding line located on a retrograde slope due to buttressing and then explore scenarios where a reduction in that buttressing causes ice stream acceleration, thinning, and grounding line retreat. The majority of participating models passed the first test and then produced similar responses to the loss of buttressing. We find that the most important distinction between models in this particular type of simulation is in the treatment of sliding at the bed, with other distinctions - notably the difference between the simpler and more complete treatments of englacial stress but also the differences between numerical methods - taking a secondary role. © 2020 Wolters Kluwer Medknow Publications. All rights reserved.
Spectral Theory of Infinite Quantum Graphs
2018, Exner, Pavel, Kostenko, Aleksey, Malamud, Mark, Neidhardt, Hagen
We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph and the corresponding properties of a certain weighted discrete Laplacian on the underlying discrete graph. Using this connection together with spectral theory of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new results on spectral properties of quantum graphs. Namely, we prove several self-adjointness results including a Gaffney-type theorem. We investigate the problem of lower semiboundedness, prove several spectral estimates (bounds for the bottom of spectra and essential spectra of quantum graphs, CLR-type estimates) and study spectral types.
Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems
2016, Neidhardt, Hagen, Stephan, Artur, Zagrebnov, Valentin A.
In the present paper we advocate the Howland-Evans approach to solution of the abstract non-autonomous Cauchy problem (non-ACP) in a separable Banach space X. The main idea is to reformulate this problem as an autonomous Cauchy problem (ACP) in a new Banach space Lp(I;X), p 2 [1;1), consisting of X-valued functions on the time-interval I. The fundamental observation is a one-to-one correspondence between solution operators (propagators) for a non-ACP and the corresponding evolution semigroups for ACP in Lp(I;X). We show that the latter also allows to apply a full power of the operatortheoretical methods to scrutinise the non-ACP including the proof of the Trotter product approximation formulae with operator-norm estimate of the rate of convergence. The paper extends and improves some recent results in this direction in particular for Hilbert spaces.
Numerical analysis for nematic electrolytes
2020, Baňas, L'ubomír, Lasarzik, Robert, Prohl, Andreas
We consider a system of nonlinear PDEs modeling nematic electrolytes, and construct a dissipative solution with the help of its implementable, structure-inheriting space-time discretization. Computational studies are performed to study the mutual effects of electric, elastic, and viscous effects onto the molecules in a nematic electrolyte.
Local approximation of arbitrary functions by solutions of nonlocal equations
2016, Dipierro, Serena, Savin, Ovidiu, Valdinoci, Enrico
We show that any function can be locally approximated by solutions of prescribed linear equations of nonlocal type. In particular, we show that every function is locally s-caloric, up to a small error. The case of non-elliptic and non-parabolic operators is taken into account as well.