2024
http://publications.mfo.de/handle/mfo/4095
Sat, 20 Apr 2024 13:48:48 GMT2024-04-20T13:48:48ZOn Dykstra’s Algorithm with Bregman Projections
http://publications.mfo.de/handle/mfo/4134
On Dykstra’s Algorithm with Bregman Projections
Pinto, Pedro; Pischke, Nicholas
We provide quantitative results on the asymptotic behavior of Dykstra’s algorithm with Bregman projections, a combination of the well-known Dykstra’s algorithm and the method of cyclic Bregman projections, designed to find best approximations and solve the convex feasibility problem in a non-Hilbertian setting. The result we provide arise through the lens of proof mining, a program in mathematical logic which extracts computational information from non-effective proofs. Concretely, we provide a highly uniform and computable rate of metastability of low complexity and, moreover, we also specify general circumstances in which one can obtain full and effective rates of convergence. As a byproduct of our quantitative analysis, we also for the first time establish the strong convergence of Dykstra’s method with Bregman projections in infinite dimensional (reflexive) Banach spaces.
Tue, 16 Apr 2024 00:00:00 GMThttp://publications.mfo.de/handle/mfo/41342024-04-16T00:00:00ZPinto, PedroPischke, NicholasWe provide quantitative results on the asymptotic behavior of Dykstra’s algorithm with Bregman projections, a combination of the well-known Dykstra’s algorithm and the method of cyclic Bregman projections, designed to find best approximations and solve the convex feasibility problem in a non-Hilbertian setting. The result we provide arise through the lens of proof mining, a program in mathematical logic which extracts computational information from non-effective proofs. Concretely, we provide a highly uniform and computable rate of metastability of low complexity and, moreover, we also specify general circumstances in which one can obtain full and effective rates of convergence. As a byproduct of our quantitative analysis, we also for the first time establish the strong convergence of Dykstra’s method with Bregman projections in infinite dimensional (reflexive) Banach spaces.Ky Fan Theorem for Sphere Bundles
http://publications.mfo.de/handle/mfo/4131
Ky Fan Theorem for Sphere Bundles
Panina, Gaiane; Živaljević, Rade
The classic Ky Fan theorem is a combinatorial equivalent of Borsuk-Ulam theorem. It is a generalization and extension of Tucker’s lemma and, just like its predecessor, it pinpoints important properties of antipodal colorings of vertices of a triangulated sphere Sn. Here we describe generalizations of Ky Fan theorem for the case when the sphere is replaced by the total space of a triangulated sphere bundle.
Fri, 05 Apr 2024 00:00:00 GMThttp://publications.mfo.de/handle/mfo/41312024-04-05T00:00:00ZPanina, GaianeŽivaljević, RadeThe classic Ky Fan theorem is a combinatorial equivalent of Borsuk-Ulam theorem. It is a generalization and extension of Tucker’s lemma and, just like its predecessor, it pinpoints important properties of antipodal colorings of vertices of a triangulated sphere Sn. Here we describe generalizations of Ky Fan theorem for the case when the sphere is replaced by the total space of a triangulated sphere bundle.A Gentle Introduction to Interpolation on the Grassmann Manifold
http://publications.mfo.de/handle/mfo/4097
A Gentle Introduction to Interpolation on the Grassmann Manifold
Ciaramella, Gabriele; Gander, Martin J.; Vanzan, Tommaso
Wed, 10 Jan 2024 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40972024-01-10T00:00:00ZCiaramella, GabrieleGander, Martin J.Vanzan, TommasoArm Exponent for the Gaussian Free Field on Metric Graphs in Intermediate Dimensions
http://publications.mfo.de/handle/mfo/4096
Arm Exponent for the Gaussian Free Field on Metric Graphs in Intermediate Dimensions
Drewitz, Alexander; Prévost, Alexis; Rodriguez, Pierre-François
We investigate the bond percolation model on transient weighted graphs ${G}$ induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in ${G}$ have polynomial volume growth with growth exponent $\alpha$ and that the Green's function for the random walk on ${G}$ exhibits a power law decay with exponent $\nu$, in the regime $1\leq \nu \leq \frac{\alpha}{2}$. In particular, this includes the cases of ${G}=\mathbb Z^3$, for which $\nu=1$, and ${G}= \mathbb Z^4$, for which $\nu=\frac{\alpha}{2}=2$. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance $R$ like $R^{-\frac{\nu}{2}+o(1)}$. Our results are in fact more precise and yield logarithmic corrections when $\nu > 1$ as well as corrections of order $\log \log R$ when $\nu=1$. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when $\nu > 1$ and essentially optimal when $\nu=1$. This extends previous results from [16].
Mon, 08 Jan 2024 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40962024-01-08T00:00:00ZDrewitz, AlexanderPrévost, AlexisRodriguez, Pierre-FrançoisWe investigate the bond percolation model on transient weighted graphs ${G}$ induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in ${G}$ have polynomial volume growth with growth exponent $\alpha$ and that the Green's function for the random walk on ${G}$ exhibits a power law decay with exponent $\nu$, in the regime $1\leq \nu \leq \frac{\alpha}{2}$. In particular, this includes the cases of ${G}=\mathbb Z^3$, for which $\nu=1$, and ${G}= \mathbb Z^4$, for which $\nu=\frac{\alpha}{2}=2$. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance $R$ like $R^{-\frac{\nu}{2}+o(1)}$. Our results are in fact more precise and yield logarithmic corrections when $\nu > 1$ as well as corrections of order $\log \log R$ when $\nu=1$. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when $\nu > 1$ and essentially optimal when $\nu=1$. This extends previous results from [16].