The pyramid wavefront sensor(PWFS)can provide the sensitivity needed for demanding adaptive optics applications,such as imaging exoplanets using the future extremely large telescopes of over 30 m of diameter(D).However,its exquisite sensitivity has a limited linear range of operation,or dynamic range,although it can be extended through the use of beam modulation—despite sacrificing sensitivity and requiring additional optical hardware.Inspired by artificial intelligence techniques,this work proposes to train an optical layer—comprising a passive diffractive element placed at a conjugated Fourier plane of the pyramid prism—to boost the linear response of the pyramid sensor without the need for cumbersome modulation.We develop an end-2-end simulation to train the diffractive element,which acts as an optical preconditioner to the traditional least-square modal phase estimation process.Simulation results with a large range of turbulence conditions show a noticeable improvement in the aberration estimation performance equivalent to over 3λ∕D of modulation when using the optically preconditioned deep PWFS(DPWFS).Experimental results validate the advantages of using the designed optical layer,where the DPWFS can pair the performance of a traditional PWFS with 2λ∕D of modulation.Designing and adding an optical preconditioner to the PWFS is just the tip of the iceberg,since the proposed deep optics methodology can be used for the design of a completely new generation of wavefront sensors that can better fit the demands of sophisticated adaptive optics applications such as ground-to-space and underwater optical communications and imaging through scattering media.
FELIPE GUZMÁNJORGE TAPIACAMILO WEINBERGERNICOLÁS HERNÁNDEZJORGE BACCABENOIT NEICHELESTEBAN VERA
Based on the Crank-Nicolson and the weighted and shifted Grunwald operators,we present an implicit difference scheme for the Riesz space fractional reaction-dispersion equations and also analyze the stability and the convergence of this implicit difference scheme.However,after estimating the condition number of the coefficient matrix of the discretized scheme,we find that this coefficient matrix is ill-conditioned when the spatial mesh-size is sufficiently small.To overcome this deficiency,we further develop an effective banded M-matrix splitting preconditioner for the coefficient matrix.Some properties of this preconditioner together with its preconditioning effect are discussed.Finally,Numerical examples are employed to test the robustness and the effectiveness of the proposed preconditioner.
Krylov subspace methods are widely used for solving sparse linear algebraic equations,but they rely heavily on preconditioners,and it is difficult to find an effective preconditioner that is efficient and stable for all problems.In this paper,a novel projection strategy including the orthogonal and the oblique projection is proposed to improve the preconditioner,which can enhance the efficiency and stability of iteration.The proposed strategy can be considered as a minimization process,where the orthogonal projection minimizes the energy norm of error and the oblique projection minimizes the 2-norm of the residual,meanwhile they can be regarded as approaches to correct the approximation by solving low-rank inverse of the matrices.The strategy is a wide-ranging approach and provides a way to transform the constant preconditioner into a variable one.The paper discusses in detail the projection strategy for sparse approximate inverse(SPAI)preconditioner applied to flexible GMRES and conducts the numerical test for problems from different applications.The results show that the proposed projection strategy can significantly improve the solving process,especially for some non-converging and slowly convergence systems.
This paper proposes a two-parameter block triangular splitting(TPTS)preconditioner for the general block two-by-two linear systems.The eigenvalues of the corresponding preconditioned matrix are proved to cluster around 0 or 1 under mild conditions.The limited numerical results show that the TPTS preconditioner is more efficient than the classic block-diagonal and block-triangular preconditioners when applied to the flexible generalized minimal residual(FGMRES)method.
A new cyclic pseudo-elimination(CPE,in brief)preconditioner,which combines the pseudoelimination(PE)technique and LU factorization together,is proposed for a kind of cyclic structured matrices.For the case of M-matrices,some theoretical results of convergence and estimation of the condition number are presented.Numerical experiments show that the CPE preconditioner performs the best with respect to the reduction of number of iterations.Moreover,it costs much less time than the ILUT and block Jacobi(BJ)preconditioners in a whole in all tested cases.
