A discrepancy principle for Tikhonov regularization with approximately specified data

M. Thamban Nair; Eberhard Schock

Displaying similar documents to “A discrepancy principle for Tikhonov regularization with approximately specified data”

On some properties of three-dimensional minimal sets in $ℝ^{4}$

Tien Duc Luu (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

Similarity:

We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in $ℝ^{4}$ around a $𝕐$ -point and the existence of a point of particular type of a Mumford-Shah minimal set in $ℝ^{4}$ , which is very close to a $𝕋$ . This will give a local description of minimal sets of dimension 3 in $ℝ^{4}$ around a singular point and a property of Mumford-Shah minimal sets in $ℝ^{4}$ .

Zero-set property of o-minimal indefinitely Peano differentiable functions

Andreas Fischer (2008)

Annales Polonici Mathematici

Similarity:

Given an o-minimal expansion ℳ of a real closed field R which is not polynomially bounded. Let $^{\infty}$ denote the definable indefinitely Peano differentiable functions. If we further assume that ℳ admits $^{\infty}$ cell decomposition, each definable closed subset A of Rⁿ is the zero-set of a $^{\infty}$ function f:Rⁿ → R. This implies $^{\infty}$ approximation of definable continuous functions and gluing of $^{\infty}$ functions defined on closed definable sets.

Exceptional sets in Waring's problem: two squares and s biquadrates

Lilu Zhao (2014)

Acta Arithmetica

Similarity:

Let $R_{s} (n)$ denote the number of representations of the positive number n as the sum of two squares and s biquadrates. When $s = 3$ or 4, it is established that the anticipated asymptotic formula for $R_{s} (n)$ holds for all $n \leq X$ with at most $O (X^{(9 - 2 s) / 8 + ε})$ exceptions.

Renormings of $c_{0}$ and the minimal displacement problem

Łukasz Piasecki (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

The aim of this paper is to show that for every Banach space $(X, ∥ \cdot ∥)$ containing asymptotically isometric copy of the space $c_{0}$ there is a bounded, closed and convex set $C \subset X$ with the Chebyshev radius $r (C) = 1$ such that for every $k \geq 1$ there exists a $k$ -contractive mapping $T : C \to C$ with $∥ x - T x ∥ > 1 - 1 / k$ for any $x \in C$ .

Linear maps preserving $A$ -unitary operators

Abdellatif Chahbi, Samir Kabbaj, Ahmed Charifi (2016)

Mathematica Bohemica

Similarity:

Let $ℋ$ be a complex Hilbert space, $A$ a positive operator with closed range in $ℬ (ℋ)$ and $ℬ_{A} (ℋ)$ the sub-algebra of $ℬ (ℋ)$ of all $A$ -self-adjoint operators. Assume $φ : ℬ_{A} (ℋ)$ onto itself is a linear continuous map. This paper shows that if $φ$ preserves $A$ -unitary operators such that $φ (I) = P$ then $ψ$ defined by $ψ (T) = P φ (P T)$ is a homomorphism or an anti-homomorphism and $ψ (T^{♯}) = ψ {(T)}^{♯}$ for all $T \in ℬ_{A} (ℋ)$ , where $P = A^{+} A$ and $A^{+}$ is the Moore-Penrose inverse of $A$ . A similar result is also true if $φ$ preserves $A$ -quasi-unitary operators in both directions such that there...

Theoretical analysis for $ℓ_{1}$ - $ℓ_{2}$ minimization with partial support information

Haifeng Li, Leiyan Guo (2025)

Applications of Mathematics

Similarity:

We investigate the recovery of $k$ -sparse signals using the $ℓ_{1}$ - $ℓ_{2}$ minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume $k$ -sparse signals $𝐱$ with the prior support $T$ which is composed of $g$ true indices and $b$ wrong...

On the range-kernel orthogonality of elementary operators

Said Bouali, Youssef Bouhafsi (2015)

Mathematica Bohemica

Similarity:

Let $L (H)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$ . For $A, B \in L (H)$ , the generalized derivation $δ_{A, B}$ and the elementary operator $Δ_{A, B}$ are defined by $δ_{A, B} (X) = A X - X B$ and $Δ_{A, B} (X) = A X B - X$ for all $X \in L (H)$ . In this paper, we exhibit pairs $(A, B)$ of operators such that the range-kernel orthogonality of $δ_{A, B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of $Δ_{A, B}$ with respect to the wider class of unitarily invariant...

Best constants for the isoperimetric inequality in quantitative form

Marco Cicalese, Gian Paolo Leonardi (2013)

Journal of the European Mathematical Society

Similarity:

We prove some results in the context of isoperimetric inequalities with quantitative terms. In the $2$ -dimensional case, our main contribution is a method for determining the optimal coefficients $c_{1}, ..., c_{m}$ in the inequality $δ P (E) \geq \sum_{k = 1}^{m} c_{k} α {(E)}^{k} + o (α {(E)}^{m})$ , valid for each Borel set $E$ with positive and finite area, with $δ P (E)$ and $α (E)$ being, respectively, the $𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑑𝑒𝑓𝑖𝑐𝑖𝑡$ and the $𝐹𝑟𝑎𝑒𝑛𝑘𝑒𝑙𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦$ of $E$ . In $n$ dimensions, besides proving existence and regularity properties of minimizers for a wide class of $𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑎𝑡𝑖𝑣𝑒𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡𝑠$ including the lower semicontinuous extension of $\frac{δ P (E)}{α {(E)}^{2}}$ , we...

$H^{\infty}$ calculus and dilatations

Andreas M. Fröhlich, Lutz Weis (2006)

Bulletin de la Société Mathématique de France

Similarity:

We characterise the boundedness of the $H^{\infty}$ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $- A$ generates a bounded analytic $C_{0}$ semigroup $(T_{t})$ on a UMD space, then the $H^{\infty}$ calculus of $A$ is bounded if and only if $(T_{t})$ has a dilation to a bounded group on $L^{2} ([0, 1], X)$ . This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^{p}$ space we can choose another $L^{p}$ space in place of $L^{2} ([0, 1], X)$ .

More on exposed points and extremal points of convex sets in $ℝ^{n}$ and Hilbert space

Stoyu T. Barov (2023)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let $𝕍$ be a separable real Hilbert space, $k \in ℕ$ with $k < dim 𝕍$ , and let $B$ be convex and closed in $𝕍$ . Let $𝒫$ be a collection of linear $k$ -subspaces of $𝕍$ . A point $w \in B$ is called exposed by $𝒫$ if there is a $P \in 𝒫$ so that $(w + P) \cap B = {w}$ . We show that, under some natural conditions, $B$ can be reconstituted as the convex hull of the closure of all its exposed by $𝒫$ points whenever $𝒫$ is dense and $G_{δ}$ . In addition, we discuss the question when the set of exposed by some $𝒫$ points forms a $G_{δ}$ -set.

On upper bounds for total $k$ -domination number via the probabilistic method

Saylí Sigarreta, Saylé Sigarreta, Hugo Cruz-Suárez (2023)

Kybernetika

Similarity:

For a fixed positive integer $k$ and $G = (V, E)$ a connected graph of order $n$ , whose minimum vertex degree is at least $k$ , a set $S \subseteq V$ is a total $k$ -dominating set, also known as a $k$ -tuple total dominating set, if every vertex $v \in V$ has at least $k$ neighbors in $S$ . The minimum size of a total $k$ -dominating set for $G$ is called the total $k$ -domination number of $G$ , denoted by $γ_{k t} (G)$ . The total $k$ -domination problem is to determine a minimum total $k$ -dominating set of $G$ . Since the exact problem is in general quite difficult...

Product property for capacities in $ℂ^{N}$

Mirosław Baran, Leokadia Bialas-Ciez (2012)

Annales Polonici Mathematici

Similarity:

The paper deals with logarithmic capacities, an important tool in pluripotential theory. We show that a class of capacities, which contains the L-capacity, has the following product property: $C_{ν} (E ₁ \times E ₂) = m i n (C_{ν ₁} (E ₁), C_{ν ₂} (E ₂))$ , where $E_{j}$ and $ν_{j}$ are respectively a compact set and a norm in $ℂ^{N_{j}}$ (j = 1,2), and ν is a norm in $ℂ^{N ₁ + N ₂}$ , ν = ν₁⊕ₚ ν₂ with some 1 ≤ p ≤ ∞. For a convex subset E of $ℂ^{N}$ , denote by C(E) the standard L-capacity and by $ω_{E}$ the minimal width of E, that is, the minimal Euclidean distance between two supporting hyperplanes...

Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions

Catherine Bandle, Joachim von Below, Wolfgang Reichel (2008)

Journal of the European Mathematical Society

Similarity:

We consider linear elliptic equations $- Δ u + q (x) u = λ u + f$ in bounded Lipschitz domains $D \subset ℝ^{N}$ with mixed boundary conditions $\partial u / \partial n = σ (x) λ u + g$ on $\partial D$ . The main feature of this boundary value problem is the appearance of $λ$ both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient $σ (x)$ . We study positivity principles and anti-maximum principles. One of our main results states that if $σ$ is somewhere negative, $q \geq 0$ and $\int_{D} q (x) d x > 0$ then there exist two eigenvalues $λ_{- 1}$ , $λ_{1}$ such the positivity...

On a magnetic characterization of spectral minimal partitions

Bernard Helffer, Thomas Hoffmann-Ostenhof (2013)

Journal of the European Mathematical Society

Similarity:

Given a bounded open set $Ω$ in $ℝ^{n}$ (or in a Riemannian manifold) and a partition of $Ω$ by $k$ open sets $D_{j}$ , we consider the quantity ${𝚖𝚊𝚡}_{j} λ (D_{j})$ where $λ (D_{j})$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_{j}$ . If we denote by $ℒ_{k} (Ω)$ the infimum over all the $k$ -partitions of ${𝚖𝚊𝚡}_{j} λ (D_{j})$ , a minimal $k$ -partition is then a partition which realizes the infimum. When $k = 2$ , we find the two nodal domains of a second eigenfunction, but the analysis of higher $k$ ’s is non trivial and quite interesting. In this...

Moser's Inequality for a class of integral operators

Finbarr Holland, David Walsh (1995)

Studia Mathematica

Similarity:

Let 1 < p < ∞, q = p/(p-1) and for $f \in L^{p} (0, \infty)$ define $F (x) = (1 / x) ʃ_{0}^{x} f (t) d t$ , x > 0. Moser’s Inequality states that there is a constant $C_{p}$ such that $s u p_{a \leq 1} s u p_{f \in B_{p}} ʃ_{0}^{\infty} e x p [a x^{q} {| F (x) |}^{q} - x] d x = C_{p}$ where $B_{p}$ is the unit ball of $L^{p}$ . Moreover, the value a = 1 is sharp. We observe that $F = K_{1}$ f where the integral operator $K_{1}$ has a simple kernel K. We consider the question of for what kernels K(t,x), 0 ≤ t, x < ∞, this result can be extended, and proceed to discuss this when K is non-negative and homogeneous of degree -1. A sufficient condition on K is found for...