Newton's method for polynomial root finding is one of mathematics' most well-known algorithms. The method also has its shortcomings: it is undefined at critical points, it could exhibit chaotic behavior and is only guaranteed to converge locally. Based on the {\it Geometric Modulus Principle} for a complex polynomial $p(z)$, together with a {\it Modulus Reduction Theorem} proved here, we develop the {\it Robust Newton's method} (RNM), defined everywhere with a step-size that guarantees an {\it a priori} reduction in polynomial modulus in each iteration. Furthermore, we prove RNM iterates converge globally, either to a root or a critical point. Specifically, given $\varepsilon $ and any seed $z_0$, in $t=O(1/\varepsilon^{2})$ iterations of RNM, independent of degree of $p(z)$, either $|p(z_t)| \leq \varepsilon$ or $|p(z_t) p'(z_t)| \leq \varepsilon$. By adjusting the iterates at {\it near-critical points}, we describe a {\it modified} RNM that necessarily convergence to a root. In combination with Smale's point estimation, RNM results in a globally convergent Newton's method having a locally quadratic rate. We present sample polynomiographs that demonstrate how in contrast with Newton's method RNM smooths out the fractal boundaries of basins of attraction of roots. RNM also finds potentials in computing all roots of arbitrary degree polynomials. A particular consequence of RNM is a simple algorithm for solving cubic equations.
A version of the convexification globally convergent numerical method is constructed for a coefficient inverse problem for a wave-like partial differential equation. The presence of the Carleman Weight Function in the corresponding Tikhonov-like cost functional ensures the global strict convexity of this functional. Numerical results are presented to illustrate the effectiveness and efficiency of the proposed method.
The comparative study of two globally convergent numerical methods for acoustic tomography is carried out in two dimensions. These are the boundary control method and the quasi-reversibility method. The novelty is that in the latter a nonlinear inverse problem is reduced to a family of the linear integral equations of the first kind via the Lavrentiev approach and this reduction is used within the quasi-reversibility method. The analysis of its stability is carried out via Carleman estimates. The computational effectiveness of these methods is tested in the numerical experiments with the smoot…
We present a globally convergent method to accelerate density-based topology optimization using projection-based reduced-order models (ROMs) and trust-region methods. To accelerate topology optimization, we replace the large-scale finite element simulation, which dominates the computational cost, with ROMs that reduce the cost of objective function and gradient evaluations by orders of magnitude. To guarantee convergence, we first introduce a trust-region method that employs generalized trust-region constraints and prove it is globally convergent. We then devise a class of globally convergent …
In this paper we derive $7^{th}$ order convergent integration formula in time which is weakly $L$-stable. To derive the method we use, Newton Cotes formula, fifth-order Hermite interpolation polynomial approximation (osculatory interpolation) and sixth-order explicit backward Taylor's polynomial approximation. The vector form of this formula is used to solve Burger's equation which is one dimensional form of Navier-Stokes equation. We observe that the method gives high accuracy results in the case of inconsistencies as well as for small values of viscosity, e.g., $10^{-3}$. Computations are pe…
We propose a fast and robust scheme for the direct minimization of the Ohta-Kawasaki energy that characterizes the microphase separation of diblock copolymer melts. The scheme employs a globally convergent modified Newton method with line search which is shown to be mass-conservative, energy-descending, asymptotically quadratically convergent, and three orders of magnitude more efficient than the commonly-used gradient flow approach. The regularity and the first-order condition of minimizers are analyzed. A numerical study of the chemical substrate guided directed self-assembly of diblock copo…
Globally convergent and adaptive finite element methods in imaging of buried objects from experimental backscattering radar measurements
Chalmers University of Technology and Gothenburg University（查尔姆斯理工大学和哥德堡大学）；Iowa State University（爱荷华州立大学）；University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）
To compute the spatially distributed dielectric constant from the backscattering data, we study a coefficient inverse problem for a 1D hyperbolic equation. To solve the inverse problem, we establish a new version of Carleman estimate and then employ this estimate to construct a cost functional which is strictly convex on a convex bounded set with an arbitrary diameter in a Hilbert space. The strict convexity property is rigorously proved. This result is called the convexification theorem and is considered as the central analytical result of this paper. Minimizing this convex functional by the …
Università della Svizzera italiana（瑞士意大利语大学）；Euler Institute（欧拉研究所）
We propose a globally convergent multilevel training method for deep residual networks (ResNets). The devised method can be seen as a novel variant of the recursive multilevel trust-region (RMTR) method, which operates in hybrid (stochastic-deterministic) settings by adaptively adjusting mini-batch sizes during the training. The multilevel hierarchy and the transfer operators are constructed by exploiting a dynamical system's viewpoint, which interprets forward propagation through the ResNet as a forward Euler discretization of an initial value problem. In contrast to traditional training appr…
A globally convergent Carleman-Picard method for an inverse initial-value problem for a nonlinear diffusive coagulation-fragmentation equation coagulation-fragmentation equation
California State University, Long Beach（加州州立大学长滩分校）；Texas A&M University（德克萨斯农工大学）；University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）
We develop and analyze a broad family of stochastic/randomized algorithms for inverting a matrix. We also develop specialized variants maintaining symmetry or positive definiteness of the iterates. All methods in the family converge globally and linearly (i.e., the error decays exponentially), with explicit rates. In special cases, we obtain stochastic block variants of several quasi-Newton updates, including bad Broyden (BB), good Broyden (GB), Powell-symmetric-Broyden (PSB), Davidon-Fletcher-Powell (DFP) and Broyden-Fletcher-Goldfarb-Shanno (BFGS). Ours are the first stochastic versions of t…
In this work we present a fast, globally convergent, iterative algorithm for computing the asymptotically stable states of nonlinear large--scale systems of quadratic autonomous Ordinary Differential Equations (ODEs) modeling, e.g., the dynamic of complex chemical reaction networks. Towards this aim, we reformulate the problem as a box--constrained optimization problem where the roots of a set of nonlinear equations need to be determined. Then, we propose to use a projected Newton's approach combined with a gradient descent algorithm so that every limit point of the sequence generated by the o…
Globally Convergent Jacobi-Type Algorithms for Robust Approximate Diagonalization of Higher-Order Cumulant Tensors
School of Science, Sun Yat-sen University（中山大学理学院）；Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong（香港理工大学系统工程及工程管理学系）；Centre de Recherche en Automatique de Nancy (CRAN), Université de Lorraine, CNRS（洛林大学国家科学研究中心南西自动化研究中心）
We propose to combine the Carleman estimate and the Newton method to solve an inverse source problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse source problem is conditionally logarithmic. Hence, numerical results due to the conventional least squares optimization might not be reliable. In order to enhance the stability, we approximate this problem by truncating the high frequency terms of the Fourier series that represents the solution to the governing equation. By this, we derive a system of nonlinear elliptic PDEs whose solution consists of F…
An efficient, globally convergent method for optimization under uncertainty using adaptive model reduction and sparse grids
Department of Mathematics, Lawrence Berkeley National Laboratory, University of California, Berkeley（加州大学伯克利分校劳伦斯伯克利国家实验室数学系）；University of California, Berkeley（加州大学伯克利分校）；Department of Aerospace and Mechanical Engineering, University of Notre Dame（圣母院大学航空航天与机械工程系）；University of Notre Dame（圣母院大学）；Sandia National Laboratories（桑迪亚国家实验室）
Provably convergent Newton-Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics
Finding roots of univariate polynomials is one of the fundamental tasks of numerics, and there is still a wide gap between root finders that are well understood in theory and those that perform well in practice. We investigate the root finding method of Weierstrass, a root finder that tries to approximate all roots of a given polynomial in parallel (in the Jacobi version, i.e., with parallel updates). This method has a good reputation for finding all roots in practice except in obvious cases of symmetry, but very little is known about its global dynamics and convergence properties. We show tha…
