This paper presents an abstract analysis of bounded variation (BV) methods for ill-posed operator equations Au = z. Let T(u) def = kAu ? zk 2 + J(u); where the penalty, or \regularization", parameter > 0 and the functional J(u) is the BV norm or seminorm of u, also known as the total variation of u. Under mild restrictions on the operator A and the functional J(u), it is shown that the function...

When discrete ill-posed problems are analyzed and solved by various numerical regularization techniques, a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector. In particular, the graph associated with Tikhonov regularization plays a central role. The main purpose of this paper is to advo...

We consider the nonlinear delay differential evolution equation { u′(t) ∈ Au(t)+ f (t, ut ), t ∈ R+, u(t) = g(u)(t), t ∈ [−τ, 0], where τ ≥ 0, X is a real Banach space, A is the infinitesimal generator of a nonlinear semigroup of contractions whose Lipschitz seminorm decays exponentially as t → e−ωt when t → +∞ and f : R+×C([−τ, 0]; D(A)) → X is jointly continuous. We prove that if f Lipschitz ...

It is shown that a) it is possible to define the topology of any topological algebra by a collection of F -seminorms, b) every complete locally uniformly absorbent (complete locally A-pseudoconvex) Hausdorff algebra is topologically isomorphic to a projective limit of metrizable locally uniformly absorbent algebras (respectively, A-(k-normed) algebras, where k ∈ (0, 1] varies, c) every complete...

We consider the series expansion of [Formula: see text]-Hardy inequality [G. Barbatis, S. Filippas and A. Tertikas, Series for text] Hardy inequalities, Indiana Univ. Math. J. 52 (2003) 171–190], in particular case where distance is taken from an interior point a bounded domain text]. For we improve it by adding as remainder term optimally weighted critical Sobolev norm, generalizing result [S....

Let $$\mathcal {H}$$ be a complex Hilbert space with inner product $$\langle \cdot , \rangle $$ and let A non-zero bounded positive linear operator on {H}.$$ $$\mathbb {B}_A(\mathcal {H})$$ denote the algebra of all operators which admit A-adjoint, $$N_A(\cdot )$$ seminorm . The generalized A-numerical radius $$T\in \mathbb is defined as $$\begin{aligned} \omega _{N_A}(T)=\displaystyle {\sup _{...

The classical shadows protocol, recently introduced by Huang, Kueng, and Preskill [Nat. Phys. 16, 1050 (2020)], is a quantum-classical protocol to estimate properties of an unknown quantum state. Unlike full state tomography, the can be implemented on near-term hardware requires few measurements make many predictions with high success probability. In this paper, we study effects noise protocol....

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We study Dirichlet boundary control of Stokes flows in 2D polygonal domains. consider cost functionals with two different regularization terms: the $L^2$ norm and an energy space seminorm. prove well-posedness regularity results for both problems, develop finite element discretizations error estimates latter problem. The motivation to problem follows from our analysis: we that choice ${\bm L}^2...

We consider infinite-horizon γ-discounted Markov Decision Processes, for which it is known that there exists a stationary optimal policy. We consider the algorithm Value Iteration and the sequence of policies π1, . . . , πk it implicitely generates until some iteration k. We provide performance bounds for non-stationary policies involving the last m generated policies that reduce the state-of-t...