arXiv subjects

B. F. Svaiter

Publications and source records attributed to B. F. Svaiter.

At least 19 recordsLinked to original sources

arXiv2026

Weak- and strong-disorder effects in the continuous random-field Ising model

We discuss the weak- and strong-disorder effects in the continuous random-field Ising model using the distributional zeta-function method. In the weak-disorder regime, the vacuum is located at the origin, whereas in the strong-disorder regime, the vacuum is shifted from the origin. By performing the quenched-disorder average at the level of the effective action, we derive the quadratic and interaction terms. In the weak-disorder limit, we show that the infrared structure of the two-point correlation functions yields a decomposition of the physical field into correlated components with distinct scaling dimensions. This mechanism exhibits the characteristic $1/p^4$ behavior, which shifts the upper critical dimension to $d_c^{+}=6$. The universal critical behavior of the RFIM near this dimension is governed by a minimal infrared effective action. In the strong-disorder regime, we obtain a diagonal quadratic action with a discrete spectrum of massive modes. The absence of massless modes implies that conventional criticality may be replaced by a symmetry-restoration transition. The resulting spectral representation of correlation functions converges rapidly and remains well controlled in the infrared limit.

cond-mat.dis-nn↗

arXiv2024

Bounds in partition functions of the continuous random field Ising model

We investigate the critical properties of continuous random field Ising model (RFIM). Using the distributional zeta-function method, we obtain a series representation for the quenched free energy. It is possible to show that for each moment of the partition function, the multiplet of $k$-fields the Gaussian contribution has one field with the contribution of the disorder and $(k-1)$-fields with the usual propagator. Although the non-gaussian contribution is non-perturbative we are able to show that the model is confined between two $\mathbb{Z}_2\times\mathcal{O}(k-1)$-symmetric models. Using arguments of lower critical dimension alongside with monotone operators, we show that the phase of the continuous RFIM can be restricted by an $\mathbb{Z}_2 \times \mathcal{O}(k-1) \to \mathcal{O}(k-2)$ phase transition.

cond-mat.dis-nn↗

arXiv2024

Critical Casimir effect in a disordered $O(2)$-symmetric model

Critical Casimir effect appears when critical fluctuations of an order parameter interact with classical boundaries. We investigate this effect in the setting of a Landau-Ginzburg model with continuous symmetry in the presence of quenched disorder. The quenched free energy is written as an asymptotic series of moments of the models partition function. Our main result is that, in the presence of a strong disorder, Goldstone modes of the system contribute either with an attractive or with a repulsive force. This result was obtained using the distributional zeta-function method without relying on any particular ansatz in the functional space of the moments of the partition function.

cond-mat.soft↗

arXiv2020

On a family of gradient type projection methods for nonlinear ill-posed problems

We propose and analyze a family of successive projection methods whose step direction is the same as Landweber method for solving nonlinear ill-posed problems that satisfy the Tangential Cone Condition (TCC). This family enconpasses Landweber method, the minimal error method, and the steepest descent method; thush providing an unified framework for the analysis of these methods. Moreover, we define in this family new methods which are convergent for the constant of the TCC in a range twice as large as the one required for the Landweber and other gradient type methods. The TCC is widely used in the analysis of iterative methods for solving nonlinear ill-posed problems. The key idea in this work is to use the TCC in order to construct special convex sets possessing a separation property, and to succesively project onto these sets. Numerical experiments are presented for a nonlinear 2D elliptic parameter identification problem, validating the efficiency of our method.

math.NA↗

arXiv2020

On projective Landweber-Kaczmarz methods for solving systems of nonlinear ill-posed equations

In this article we combine the projective Landweber method, recently proposed by the authors, with Kaczmarz's method for solving systems of non-linear ill-posed equations. The underlying assumption used in this work is the tangential cone condition. We show that the proposed iteration is a convergent regularization method. Numerical tests are presented for a non-linear inverse problem related to the Dirichlet-to-Neumann map, indicating a superior performance of the proposed method when compared with other well established iterations. Our preliminary investigation indicates that the resulting iteration is a promising alternative for computing stable solutions of large scale systems of nonlinear ill-posed equations.

math.NA↗

arXiv2020

Range-relaxed criteria for choosing the Lagrange multipliers in the Levenberg-Marquardt method

In this article we propose a novel strategy for choosing the Lagrange multipliers in the Levenberg-Marquardt method for solving ill-posed problems modeled by nonlinear operators acting between Hilbert spaces. Convergence analysis results are established for the proposed method, including: monotonicity of iteration error, geometrical decay of the residual, convergence for exact data, stability and semi-convergence for noisy data. Numerical experiments are presented for an elliptic parameter identification two-dimensional EIT problem. The performance of our strategy is compared with standard implementations of the Levenberg-Marquardt method (using a priori choice of the multipliers).

math.NA↗

arXiv2020

Range-relaxed criteria for choosing the Lagrange multipliers in nonstationary iterated Tikhonov method

In this article we propose a novel nonstationary iterated Tikhonov (NIT) type method for obtaining stable approximate solutions to ill-posed operator equations modeled by linear operators acting between Hilbert spaces. Geometrical properties of the problem are used to derive a new strategy for choosing the sequence of regularization parameters (Lagrange multipliers) for the NIT iteration. Convergence analysis for this new method is provided. Numerical experiments are presented for two distinct applications: I) A 2D elliptic parameter identification problem (Inverse Potential Problem); II) An image deblurring problem. The results obtained validate the efficiency of our method compared with standard implementations of the NIT method (where a geometrical choice is typically used for the sequence of Lagrange multipliers).

math.NA↗

arXiv2016

Disordered Field Theory in $d=0$ and Distributional Zeta-Function

Recently we introduced a new technique for computing the average free energy of a system with quenched randomness. The basic tool of this technique is a distributional zeta-function. The distributional zeta-function is a complex function whose derivative at the origin yields the average free energy of the system as the sum of two contributions: the first one is a series in which all the integer moments of the partition function of the model contribute; the second one, which can not be written as a series of the integer moments, can be made as small as desired. In this paper we present a mathematical rigorous proof that the average free energy of one disordered $λφ^{4}$ model defined in a zero-dimensional space can be obtained using the distributional zeta-function technique. We obtain an analytic expression for the average free energy of the model.

math-ph↗

arXiv2016

The Distributional Zeta-Function in Disordered Field Theory

In this paper we present a new mathematical rigorous technique for computing the average free energy of a disordered system with quenched randomness, using the replicas. The basic tool of this technique is a distributional zeta-function, a complex function whose derivative at the origin yields the average free energy of the system as the sum of two contributions: the first one is a series in which all the integer moments of the partition function of the model contribute; the second one, which can not be written as a series of the integer moments, can be made as small as desired. This result supports the use of integer moments of the partition function, computed via replicas, for expressing the average free energy of the system. One advantage of the proposed formalism is that it does not require the understanding of the properties of the permutation group when the number of replicas goes to zero. Moreover, the symmetry is broken using the saddle-point equations of the model. As an application for the distributional zeta-function technique, we obtain the average free energy of the disordered $λφ^{4}$ model defined in a $d$-dimensional Euclidean space.

cond-mat.stat-mech↗

arXiv2013

Riemann zeta zeros and prime number spectra in quantum field theory

The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line $\Re(s)=1/2$. Hilbert and Pólya suggested that one possible way to prove the Riemann hypothesis is to interpret the nontrivial zeros in the light of spectral theory. Following this approach, we discuss a necessary condition that such a sequence of numbers should obey in order to be associated with the spectrum of a linear differential operator of a system with countably infinite number of degrees of freedom described by quantum field theory. The sequence of nontrivial zeros is zeta regularizable. Then, functional integrals associated with hypothetical systems described by self-adjoint operators whose spectra is given by this sequence can be constructed. However, if one considers the same situation with primes numbers, the associated functional integral cannot be constructed, due to the fact that the sequence of prime numbers is not zeta regularizable. Finally, we extend this result to sequences whose asymptotic distributions are not "far away" from the asymptotic distribution of prime numbers.

math-ph↗

arXiv2012

Kantorovich's Theorem on Newton's Method

In this work we present a simplifyed proof of Kantorovich's Theorem on Newton's Method. This analysis uses a technique which has already been used for obtaining new extensions of this theorem.

math.NA↗

arXiv2012

On Gossez type (D) maximal monotone operators

Gossez type (D) operators are defined in non-reflexive Banach spaces and share with the subdifferential a topological related property, characterized by bounded nets. In this work we present new properties and characterizations of these operators. The class (NI) was defined after Gossez defined the class (D) and seemed to generalize the class (D). One of our main results is the proof that these classes, type (D) and (NI), are identical.

math.FA↗

arXiv2012

A Markovian growth dynamics on rooted binary trees evolving according to the Gompertz curve

Inspired by biological dynamics, we consider a growth Markov process taking values on the space of rooted binary trees, similar to the Aldous-Shields model. Fix $n\ge 1$ and $β>0$. We start at time 0 with the tree composed of a root only. At any time, each node with no descendants, independently from the other nodes, produces two successors at rate $β(n-k)/n$, where $k$ is the distance from the node to the root. Denote by $Z_n(t)$ the number of nodes with no descendants at time $t$ and let $T_n = β^{-1} n \ln(n /\ln 4) + (\ln 2)/(2 β)$. We prove that $2^{-n} Z_n(T_n + n τ)$, $τ\in\bb R$, converges to the Gompertz curve $\exp (- (\ln 2) e^{-βτ})$. We also prove a central limit theorem for the martingale associated to $Z_n(t)$.

q-bio.CB↗

arXiv2012

Geometric properties of maximal monotone operators and convex functions which may represent them

We study the relations between some geometric properties of maximal monotone operators and generic geometric and analytical properties of the functions on the associate Fitzpatrick family of convex representations. We also investigate under which conditions a convex function represents a maximal monotone operator with bounded range and provide an example of a non type (D) operator on this class.

math.FA↗

arXiv2012

A class of Fejer convergent algorithms, approximate resolvents and the Hybrid Proximal-Extragradient method

A new framework for analyzing Fejer convergent algorithms is presented. Using this framework we define a very general class of Fejer convergent algorithms and establish its convergence properties. We also introduce a new definition of approximations of resolvents which preserve some useful features of the exact resolvent, and use this concept to present an unifying view of the Forward-Backward splitting method, Tseng's Modified Forward-Backward splitting method and Korpelevich's method. We show that methods based on families of approximate resolvents fall within the aforementioned class of Fejer convergent methods. We prove that such approximate resolvents are the iteration maps of the Hybrid Proximal-Extragradient method.

math.OC↗

arXiv2012

A non-type (D) linear isometry

Previous constructions of non-type (D) maximal monotone operators were based on the non-type (D) operators introduced by Gossez, and the construction of such operators or the proof that they were non-type (D) were not straightforward. The aim of this paper is to present a very simple non-type (D) linear isometry.

math.FA↗

arXiv2011

A robust Kantorovich's theorem on inexact Newton method with relative residual error tolerance

We prove that under semi-local assumptions, the inexact Newton method with a fixed relative residual error tolerance converges Q-linearly to a zero of the non-linear operator under consideration. Using this result we show that Newton method for minimizing a self-concordant function or to find a zero of an analytic function can be implemented with a fixed relative residual error tolerance. In the absence of errors, our analysis retrieve the classical Kantorovich Theorem on Newton method.

math.NA↗

arXiv2011

Gauge functions for convex cones

We analyze a class of sublinear functionals which characterize the interior and the exterior of a convex cone in a normed linear space.

math.FA↗