Nonconvex perturbations of maximal monotone differential inclusions

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Multiplicative Noise Removal: Nonlocal Low-Rank Model and It\u27s Proximal Alternating Reweighted Minimization Algorithm

The goal of this paper is to develop a novel numerical method for efficient multiplicative noise removal. The nonlocal self-similarity of natural images implies that the matrices formed by their nonlocal similar patches are low-rank. By exploiting this low-rank prior with application to multiplicative noise removal, we propose a nonlocal low-rank model for this task and develop a proximal alternating reweighted minimization (PARM) algorithm to solve the optimization problem resulting from the model. Specifically, we utilize a generalized nonconvex surrogate of the rank function to regularize the patch matrices and develop a new nonlocal low-rank model, which is a nonconvex non-smooth optimization problem having a patchwise data fidelity and a generalized nonlocal low-rank regularization term. To solve this optimization problem, we propose the PARM algorithm, which has a proximal alternating scheme with a reweighted approximation of its subproblem. A theoretical analysis of the proposed PARM algorithm is conducted to guarantee its global convergence to a critical point. Numerical experiments demonstrate that the proposed method for multiplicative noise removal significantly outperforms existing methods, such as the benchmark SAR-BM3D method, in terms of the visual quality of the denoised images, and of the peak-signal-to-noise ratio (PSNR) and the structural similarity index measure (SSIM) values

Convexity and Liberation at Large Spin

We consider several aspects of unitary higher-dimensional conformal field theories (CFTs). We first study massive deformations that trigger a flow to a gapped phase. Deep inelastic scattering in the gapped phase leads to a convexity property of dimensions of spinning operators of the original CFT. We further investigate the dimensions of spinning operators via the crossing equations in the light-cone limit. We find that, in a sense, CFTs become free at large spin and 1/s is a weak coupling parameter. The spectrum of CFTs enjoys additivity: if two twists tau_1, tau_2 appear in the spectrum, there are operators whose twists are arbitrarily close to tau_1+tau_2. We characterize how tau_1+tau_2 is approached at large spin by solving the crossing equations analytically. We find the precise form of the leading correction, including the prefactor. We compare with examples where these observables were computed in perturbation theory, or via gauge-gravity duality, and find complete agreement. The crossing equations show that certain operators have a convex spectrum in twist space. We also observe a connection between convexity and the ratio of dimension to charge. Applications include the 3d Ising model, theories with a gravity dual, SCFTs, and patterns of higher spin symmetry breaking.Comment: 61 pages, 13 figures. v2: added reference and minor correctio

Measure control of a semilinear parabolic equation with a nonlocal time delay

We study a control problem governed by a semilinear parabolic equation. The control is a measure that acts as the kernel of a possibly nonlocal time delay term and the functional includes a nondifferentiable term with the measure norm of the control. Existence, uniqueness, and regularity of the solution of the state equation, as well as differentiability properties of the control-to-state operator are obtained. Next, we provide first order optimality conditions for local solutions. Finally, the control space is suitably discretized and we prove convergence of the solutions of the discrete problems to the solutions of the original problem. Several numerical examples are included to illustrate the theoretical results.The first two authors were partially supported by the Spanish Ministerio de Economía y Competitividad under projects MTM2014-57531-P and MTM2017-83185-P. The third author was supported by the collaborative research center SFB 910, TU Berlin, project B6

Numerical Reconstruction of Radiative Sources from Partial Boundary Measurements

We consider an inverse source problem in the stationary radiative transport through an absorbing and scattering medium in two dimensions. Using the angularly resolved radiation measured on an arc of the boundary, we propose a numerical algorithm to recover the source in the convex hull of this arc. The method involves an unstable step of inverting a bounded operator whose range is not closed. We show that the continuity constant of the discretized inverse grows at most linearly with the discretization step, thus stabilizing the problem. Numerical examples presented show the effectiveness of the proposed method

Hypocoercivity properties of adaptive Langevin dynamics

International audienceAdaptive Langevin dynamics is a method for sampling the Boltzmann-Gibbs distribution at prescribed temperature in cases where the potential gradient is subject to stochastic perturbation of unknown magnitude. The method replaces the friction in underdamped Langevin dynamics with a dynamical variable, updated according to a negative feedback loop control law as in the Nose-Hoover thermostat. Using a hypocoercivity analysis we show that the law of Adaptive Langevin dynamics converges exponentially rapidly to the stationary distribution, with a rate that can be quantified in terms of the key parameters of the dynamics. This allows us in particular to obtain a central limit theorem with respect to the time averages computed along a stochastic path. Our theoretical findings are illustrated by numerical simulations involving classification of the MNIST data set of handwritten digits using Bayesian logistic regression

Theoretical and numerical studies of some ill-posed problems in partial differential equations

Three nonlinear initial-boundary value problems are considered. A potential energy well theory applies for solutions of the hyperbolic problem u(,tt) = (DELTA)(,n) u in D x (0,T), u = 0 on (sigma) x (0,T), (PAR-DIFF)u/(PAR-DIFF)n = f(u) on (SIGMA) x (0,T), u(x,0) = U(x) and u(,t)(x,0) = V(x) in D, when the nonlinearity f is suitably restricted. Here D is a bounded, open, connected subset of R(\u27n); the boundary of D, (PAR-DIFF)D, consists of the disjoint (n-1)-dimen- sional submanifolds (sigma), (SIGMA), and their confluence; (DELTA)(,n) denotes the n-dimensional Laplacian; and (PAR-DIFF)/(PAR-DIFF)n denotes the outward normal derivative. The problem has a global weak solution in each dimen- sion n (GREATERTHEQ) 1 provided U lies in the potential well and the total initial energy is small. The global solution is obtained by expanding in normal modes in terms of the Helmholtz eigenfunctions and the eigenfunctions for a modified Steklov problem. Solutions of the hyperbolic problem which start in a region exterior to the potential well with sufficiently small total initial energy can only exist for a finite time;An analogous existence-nonexistence criterion obtains for glo- bal solutions of the parabolic problem u(,t) = (DELTA)(,n) u in D x (0,T), u = 0 on (sigma) x (0,T), (PAR-DIFF)u/(PAR-DIFF)n = f(u) on (SIGMA) x (0,T), and u(x,0) = U(x) in D;Let (phi) (ELEM) C(\u271)(- (INFIN),M) be nonnegative, increasing and satisfy;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);The problem u(,tt) = (DELTA)(,n) u + (epsilon)(phi)(u) in D x (0,T), u = 0 on (PAR-DIFF)D x (0,T), u(x,0) = u(,0)(x) and u(,t)(x,0) = v(,0)(x) in D, has a unique local continuous solution for (epsilon) \u3e 0 sufficiently small in dimensions n = 1,2,3 under appropriate assumptions on (phi), u(,0), v(,0), and (PAR-DIFF)D. The solution u can be continued as long as u \u3c M. A potential well theory is unobtainable for this problem in the Sobolev space H(,0)(\u271)(D) for n (GREATERTHEQ) 2; however, an a priori inequality for solutions guarantees global existence via energy considerations. Numerical evidence indicates that such an a priori inequality is sometimes satisfied by solutions when n (GREATERTHEQ) 2