54 research outputs found
Forward-backward truncated Newton methods for convex composite optimization
This paper proposes two proximal Newton-CG methods for convex nonsmooth
optimization problems in composite form. The algorithms are based on a a
reformulation of the original nonsmooth problem as the unconstrained
minimization of a continuously differentiable function, namely the
forward-backward envelope (FBE). The first algorithm is based on a standard
line search strategy, whereas the second one combines the global efficiency
estimates of the corresponding first-order methods, while achieving fast
asymptotic convergence rates. Furthermore, they are computationally attractive
since each Newton iteration requires the approximate solution of a linear
system of usually small dimension
Generalized Newton's Method based on Graphical Derivatives
This paper concerns developing a numerical method of the Newton type to solve
systems of nonlinear equations described by nonsmooth continuous functions. We
propose and justify a new generalized Newton algorithm based on graphical
derivatives, which have never been used to derive a Newton-type method for
solving nonsmooth equations. Based on advanced techniques of variational
analysis and generalized differentiation, we establish the well-posedness of
the algorithm, its local superlinear convergence, and its global convergence of
the Kantorovich type. Our convergence results hold with no semismoothness
assumption, which is illustrated by examples. The algorithm and main results
obtained in the paper are compared with well-recognized semismooth and
-differentiable versions of Newton's method for nonsmooth Lipschitzian
equations
Truncated Nonsmooth Newton Multigrid for phase-field brittle-fracture problems
We propose the Truncated Nonsmooth Newton Multigrid Method (TNNMG) as a
solver for the spatial problems of the small-strain brittle-fracture
phase-field equations. TNNMG is a nonsmooth multigrid method that can solve
biconvex, block-separably nonsmooth minimization problems in roughly the time
of solving one linear system of equations. It exploits the variational
structure inherent in the problem, and handles the pointwise irreversibility
constraint on the damage variable directly, without penalization or the
introduction of a local history field. Memory consumption is significantly
lower compared to approaches based on direct solvers. In the paper we introduce
the method and show how it can be applied to several established models of
phase-field brittle fracture. We then prove convergence of the solver to a
solution of the nonsmooth Euler-Lagrange equations of the spatial problem for
any load and initial iterate. Numerical comparisons to an operator-splitting
algorithm show a speed increase of more than one order of magnitude, without
loss of robustness
A TFETI Domain Decomposition Solver for Elastoplastic Problems
We propose an algorithm for the efficient parallel implementation of
elastoplastic problems with hardening based on the so-called TFETI (Total
Finite Element Tearing and Interconnecting) domain decomposition method. We
consider an associated elastoplastic model with the von Mises plastic criterion
and the linear isotropic hardening law. Such a model is discretized by the
implicit Euler method in time and the consequent one time step elastoplastic
problem by the finite element method in space. The latter results in a system
of nonlinear equations with a strongly semismooth and strongly monotone
operator. The semismooth Newton method is applied to solve this nonlinear
system. Corresponding linearized problems arising in the Newton iterations are
solved in parallel by the above mentioned TFETI domain decomposition method.
The proposed TFETI based algorithm was implemented in Matlab parallel
environment and its performance was illustrated on a 3D elastoplastic
benchmark. Numerical results for different time discretizations and mesh levels
are presented and discussed and a local quadratic convergence of the semismooth
Newton method is observed
Generalized Hadamard Product and the Derivatives of Spectral Functions
In this work we propose a generalization of the Hadamard product between two
matrices to a tensor-valued, multi-linear product between k matrices for any . A multi-linear dual operator to the generalized Hadamard product is
presented. It is a natural generalization of the Diag x operator, that maps a
vector into the diagonal matrix with x on its main diagonal.
Defining an action of the orthogonal matrices on the space of
k-dimensional tensors, we investigate its interactions with the generalized
Hadamard product and its dual. The research is motivated, as illustrated
throughout the paper, by the apparent suitability of this language to describe
the higher-order derivatives of spectral functions and the tools needed to
compute them. For more on the later we refer the reader to [14] and [15], where
we use the language and properties developed here to study the higher-order
derivatives of spectral functions.Comment: 24 page
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