Although Kraken's k-mer-based approach provides a fast taxonomic classification of metagenomic sequence data, its large memory requirements can be limiting for some applications. Kraken 2 improves ...upon Kraken 1 by reducing memory usage by 85%, allowing greater amounts of reference genomic data to be used, while maintaining high accuracy and increasing speed fivefold. Kraken 2 also introduces a translated search mode, providing increased sensitivity in viral metagenomics analysis.
In this paper, we consider the minimization for the Dirichlet energy of Sobolev homeomorphisms between two-dimensional annuli in R2 and R3, respectively. It should be noticed that in this case a ...Nitsche phenomenon occurs. The main result of this paper partly extends the corresponding result in Astala et al. (2010).
The class of non-autonomous functionals under study is characterized by the fact that the energy density changes its ellipticity and growth properties according to the point; some regularity results ...are proved for related minimizers. These results are the borderline counterpart of analogous ones previously derived for non-autonomous functionals with (p,q)-growth. Also, similar functionals related to Musielak-Orlicz spaces are discussed, in which basic properties like the density of smooth functions, the boundedness of maximal and integral operators, and the validity of Sobolev type inequalities are naturally related to the assumptions needed to prove the regularity of minima.
We have an M x N real-valued arbitrary matrix A (e.g. a dictionary) with M<N and data d describing the sought-after object with the help of A. This work provides an in-depth analysis of the (local ...and global) minimizers of an objective function F combining a quadratic data-fidelity term and an L_0 penalty applied to each entry of the sought-after solution, weighted by a regularization parameter b>0. For several decades, this objective has attracted a ceaseless effort to conceive algorithms approaching a good minimizer. Our theoretical contributions, summarized below, shed new light on the existing algorithms and can help the conception of innovative numerical schemes. To solve the normal equation associated with any M-row submatrix of A is equivalent to compute a local minimizer u* of F. (Local) minimizers u* of F are strict if and only if the submatrix, composed of those columns of A whose indexes form the support of u*, has full column rank. An outcome is that strict local minimizers of F are easily computed without knowing the value of b. Each strict local minimizer is linear in data. It is proved that F has global minimizers and that they are always strict. They are studied in more details under the (standard) assumption that rank(A)=M<N. The global minimizers with M-length support are seen to be impractical. Given d, critical values b_k for any k<M are exhibited such that if b>b_k, all global minimizers of F are k-sparse. An assumption on A is adopted and proved to fail only on a closed negligible subset. Then for all data d beyond a closed negligible subset, the objective F for b>b_k, k<M, has a unique global minimizer and this minimizer is k-sparse. Instructive small-size (5 x 10) numerical illustrations confirm the main theoretical results.
We study a nonlocal version of the total variation-based model with L1 fidelity for image denoising, where the regularizing term is replaced with the fractional s-total variation. We discuss ...regularity of the level sets and uniqueness of solutions, both for high and low values of the fidelity parameter. We analyse in detail the case of binary data given by the characteristic functions of convex sets.
I am presenting a survey of regularity results for both minima of variational integrals, and solutions to non-linear elliptic, and sometimes parabolic, systems of partial differential equations. I ...will try to take the reader to the Dark Side...PUBLICATION ABSTRACT
Local minimizers of integral functionals of the calculus of variations are analyzed under growth conditions dictated by different lower and upper bounds for the integrand. Growths of non-necessarily ...power type are allowed. The local boundedness of the relevant minimizers is established under a suitable balance between the lower and the upper bounds. Classical minimizers, as well as quasi-minimizers are included in our discussion. Functionals subject to so-called p,q-growth conditions are embraced as special cases and the corresponding sharp results available in the literature are recovered.
We study ground states of two-component Bose–Einstein condensates (BEC) with trapping potentials in R2, where the intraspecies interaction (−a1,−a2) and the interspecies interaction −β are both ...attractive, i.e, a1, a2 and β are all positive. The existence and non-existence of ground states are classified completely by investigating equivalently the associated L2-critical constraint variational problem. The uniqueness and symmetry-breaking of ground states are also analyzed under different types of trapping potentials as β↗β⁎=a⁎+(a⁎−a1)(a⁎−a2), where 0<ai<a⁎:=‖w‖22 (i=1,2) is fixed and w is the unique positive solution of Δw−w+w3=0 in R2. The semi-trivial limit behavior of ground states is tackled in the companion paper 12.
PDF
