An analysis of stochastic variance reduced gradient for linear inverse problems

Stochastic variance reduced gradient (SVRG) is a popular variance reduction technique for stochastic gradient descent (SGD). We provide a first analysis of the method for solving a class of linear inverse problems in the lens of the classical regularization theory. We prove that for a suitable constant step size schedule, the method can achieve an optimal convergence rate in terms of the noise level (under suitable regularity condition) and the variance of the SVRG iterate error is smaller than that by SGD. These theoretical findings are corroborated by a set of numerical experiments.

Relative Entropy of Coherent States on General CCR Algebras

For a subalgebra of a generic CCR algebra, we consider the relative entropy between a general (not necessarily pure) quasifree state and a coherent excitationthereof. We give a unified formula for this entropy in terms of single-particle modular data. Further, we investigate changes of the relative entropy along subalgebras arising from an increasing family of symplectic subspaces; here convexity of the entropy (as usually considered for the Quantum Null Energy Condition) is replaced with lower estimates for the second derivative, composed of “bulk terms” and “boundary terms”. Our main assumption is that the subspaces are in differential modular position, a regularity condition that generalizes the usual notion of half-sided modular inclusions. We illustrate our results in relevant examples, including thermal states for the conformal U(1)-current.

KMS states on crossed products by abelian groups

We provide a general description of the KMS states for flows whose fixed point algebra satisfies a certain regularity condition. This is then applied to crossed products by discrete groups, and in particular to certain flows on crossed products by discrete abelian groups where the methods can be combined with spectral analysis for abelian automorphism groups.

Staticity and regularity for zero rest-mass fields near spatial infinity on flat spacetime

Linear zero-rest-mass fields generically develop logarithmic singularities at the critical sets where spatial infinity meets null infinity. Friedrich's representation of spatial infinity is ideally suited to study this phenomenon. These logarithmic singularities are an obstruction to the smoothness of the zero-rest-mass field at null infinity and, in particular, to peeling. In the case of the spin-2 field it has been shown that these logarithmic singularities can be precluded if the initial data for the field satisfies a certain regularity condition involving the vanishing, at spatial infinity, of a certain spinor (the linearised Cotton spinor) and its totally symmetrised derivatives. In this article we investigate the relation between this regularity condition and the staticity of the spin-2 field. It is shown that while any static spin-2 field satisfies the regularity condition, not every solution satisfying the regularity condition is static. This result is in contrast with what happens in the case of General Relativity where staticity in a neighbourhood of spatial infinity and the smoothness of the field at future and past null infinities are much more closely related.

Ground states and associated path measures in the renormalized Nelson model

We prove the existence, uniqueness, and strict positivity of ground states of the possibly massless renormalized Nelson operator under an infrared regularity condition and for Kato decomposable electrostatic potentials fulfilling a binding condition. If the infrared regularity condition is violated, then we show non-existence of ground states of the massless renormalized Nelson operator with an arbitrary Kato decomposable potential. Furthermore, we prove the existence, uniqueness, and strict positivity of ground states of the massless renormalized Nelson operator in a non-Fock representation where the infrared condition is unnecessary. Exponential and superexponential estimates on the pointwise spatial decay and the decay with respect to the boson number for elements of spectral subspaces below localization thresholds are provided. Moreover, some continuity properties of ground state eigenvectors are discussed. Byproducts of our analysis are a hypercontractivity bound for the semigroup and a new remark on Nelson’s operator theoretic renormalization procedure. Finally, we construct path measures associated with ground states of the renormalized Nelson operator. Their analysis entails improved boson number decay estimates for ground state eigenvectors, as well as upper and lower bounds on the Gaussian localization with respect to the field variables in the ground state. As our results on uniqueness, positivity, and path measures exploit the ergodicity of the semigroup, we restrict our attention to one matter particle. All results are non-perturbative.

Trajectory approximately controllability and optimal control for noninstantaneous impulsive inclusions without compactness

In this paper, a noninstantaneous impulsive differential inclusion model is established based on the heating phenomenon of the rod. The controllability problem for this system governed by a semilinear differential inclusion with noninstantaneous impulses is studied in a Banach space and in this differential inclusion system we assume that the semigroup generated by the linear part of the inclusion is not compact. We suppose that the set-valued nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness and some sufficient conditions for approximately controllability for both upper and almost lower semicontinuous types of nonlinearity are presented. Also we discuss existence and the stability of optimal control. As an application, the controllability for a differential inclusion system governed by a heat equation is considered.

Higher Order Coercive Inequalities

We study the higher order q- Poincaré and other coercive inequalities for a class probability measures satisfying Adam’s regularity condition.

A robust discontinuous Galerkin scheme on anisotropic meshes

Discontinuous Galerkin (DG) methods are extensions of the usual Galerkin finite element methods. Although there are vast amount of studies on DG methods, most of them have assumed shape-regularity conditions on meshes for both theoretical error analysis and practical computations. In this paper, we present a new symmetric interior penalty DG scheme with a modified penalty term. We show that, without imposing the shape-regularity condition on the meshes, the new DG scheme inherits all of the good properties of standard DG methods, and is thus robust on anisotropic meshes. Numerical experiments confirm the theoretical error estimates obtained.

Semigroups of an inductive composition of terms

The set of all [Formula: see text]-ary terms of type [Formula: see text] together with a binary operation derived from a superposition [Formula: see text] forms various forms of semigroups. One may generalize such binary operation by deriving it from an inductive composition of terms and call it an inductive product. However, this operation is not associative on the same base set but it becomes associative when all elements of subterms of a fixed term used in an inductive product except itself are excluded from the base set. Hence, a semigroup is formed. In this paper, we mainly focus on the algebraic structures of this semigroup such as idempotent elements, elements associating with each type of regularity condition, and Green’s relations. The formulae of complexity of inducted terms are also under investigation.