769 research outputs found
Adaptive Regularization for Nonconvex Optimization Using Inexact Function Values and Randomly Perturbed Derivatives
A regularization algorithm allowing random noise in derivatives and inexact
function values is proposed for computing approximate local critical points of
any order for smooth unconstrained optimization problems. For an objective
function with Lipschitz continuous -th derivative and given an arbitrary
optimality order , it is shown that this algorithm will, in
expectation, compute such a point in at most
inexact evaluations of and its derivatives whenever , where
is the tolerance for th order accuracy. This bound becomes at
most
inexact evaluations if and all derivatives are Lipschitz continuous.
Moreover these bounds are sharp in the order of the accuracy tolerances. An
extension to convexly constrained problems is also outlined.Comment: 22 page
Adaptive Regularization Algorithms with Inexact Evaluations for Nonconvex Optimization
A regularization algorithm using inexact function values and inexact
derivatives is proposed and its evaluation complexity analyzed. This algorithm
is applicable to unconstrained problems and to problems with inexpensive
constraints (that is constraints whose evaluation and enforcement has
negligible cost) under the assumption that the derivative of highest degree is
-H\"{o}lder continuous. It features a very flexible adaptive mechanism
for determining the inexactness which is allowed, at each iteration, when
computing objective function values and derivatives. The complexity analysis
covers arbitrary optimality order and arbitrary degree of available approximate
derivatives. It extends results of Cartis, Gould and Toint (2018) on the
evaluation complexity to the inexact case: if a th order minimizer is sought
using approximations to the first derivatives, it is proved that a suitable
approximate minimizer within is computed by the proposed algorithm
in at most iterations and at most
approximate
evaluations. An algorithmic variant, although more rigid in practice, can be
proved to find such an approximate minimizer in
evaluations.While
the proposed framework remains so far conceptual for high degrees and orders,
it is shown to yield simple and computationally realistic inexact methods when
specialized to the unconstrained and bound-constrained first- and second-order
cases. The deterministic complexity results are finally extended to the
stochastic context, yielding adaptive sample-size rules for subsampling methods
typical of machine learning.Comment: 32 page
Updating constraint preconditioners for KKT systems in quadratic programming via low-rank corrections
This work focuses on the iterative solution of sequences of KKT linear
systems arising in interior point methods applied to large convex quadratic
programming problems. This task is the computational core of the interior point
procedure and an efficient preconditioning strategy is crucial for the
efficiency of the overall method. Constraint preconditioners are very effective
in this context; nevertheless, their computation may be very expensive for
large-scale problems, and resorting to approximations of them may be
convenient. Here we propose a procedure for building inexact constraint
preconditioners by updating a "seed" constraint preconditioner computed for a
KKT matrix at a previous interior point iteration. These updates are obtained
through low-rank corrections of the Schur complement of the (1,1) block of the
seed preconditioner. The updated preconditioners are analyzed both
theoretically and computationally. The results obtained show that our updating
procedure, coupled with an adaptive strategy for determining whether to
reinitialize or update the preconditioner, can enhance the performance of
interior point methods on large problems.Comment: 22 page
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Experimental and Numerical Analysis of Single Phase Flow in a micro T-junction
This paper was presented at the 4th Micro and Nano Flows Conference (MNF2014), which was held at University College, London, UK. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute, ASME Press, LCN London Centre for Nanotechnology, UCL University College London, UCL Engineering, the International NanoScience Community, www.nanopaprika.eu.In this work the fluid-dynamic behaviour of a micro T-junction has been investigated both
numerically and experimentally for low Reynolds numbers (Re<14) with water as working fluid. The
velocity profiles within the T-junction has been experimentally determined by using the micro Particle Image
Velocimetry (μPIV). The experimental data have been compared with the numerical results obtained by
means of a 3D model implemented in Comsol Multiphysics® environment for incompressible, isothermal,
laminar flows with constant properties. The comparison between the experimental and the numerical data
puts in evidence a perfect agreement among the results. In the central region of the T-junction where the
velocity profiles of the inlet branches interact, the maximum difference is less than 5.8% for different flow
rates imposed at the inlet (with the ratio 1:2) and less than 4.4% in the case of the same flow rate at the inlets
(1:1). Since the estimated uncertainty of the experimental velocity is about 3%, the obtained result can be
considered very good and it demonstrates that no significant scaling effects influences the liquid mixing for
low Reynolds numbers (Re<14) and the behaviour of the micro T-junction can be considered as
conventional. The detailed analysis of the velocity profile evolution within the central region of the mixer
has allowed to determine where the fully developed laminar profile is reached (for instance 260 mm far from
the centre of the T-junction when a maximum water flow rate of 8 ml/h is considered)
A matrix-free preconditioner for sparse symmetric positive definite systems and least-squares problems
The strange case of negative reflection
In this paper, we show the phenomenon of negative reflection occurring in a mechanical phononic structure, consisting of a grating of fixed inclusions embedded in a linear elastic matrix. The negative reflection is not due to the introduction of a subwavelength metastructure or materials with negative mechanical properties. Numerical analyses for out-of-plane shear waves demonstrate that there exist frequencies at which most of the incident energy is reflected at negative angles. The effect is symmetric with respect to a line that is not parallel to the normal direction to the grating structure. Simulations at different angles of incidence and computations of the energy fluxes show that negative reflection is achievable in a wide range of loading conditions
Solving Nonlinear Systems of Equations Via Spectral Residual Methods: Stepsize Selection and Applications
Spectral residual methods are derivative-free and low-cost per iteration procedures for solving nonlinear systems of equations. They are generally coupled with a nonmonotone linesearch strategy and compare well with Newton-based methods for large nonlinear systems and sequences of nonlinear systems. The residual vector is used as the search direction and choosing the steplength has a crucial impact on the performance. In this work we address both theoretically and experimentally the steplength selection and provide results on a real application such as a rolling contact problem
An optimally fast objective-function-free minimization algorithm using random subspaces
An algorithm for unconstrained non-convex optimization is described, which
does not evaluate the objective function and in which minimization is carried
out, at each iteration, within a randomly selected subspace. It is shown that
this random approximation technique does not affect the method's convergence
nor its evaluation complexity for the search of an -approximate
first-order critical point, which is , where
is the order of derivatives used. A variant of the algorithm using
approximate Hessian matrices is also analyzed and shown to require at most
evaluations. Preliminary numerical tests show that
the random-subspace technique can significantly improve performance on some
problems, albeit, unsurprisingly, not for all.Comment: 23 page
Interfacial cracks in bi-material solids: Stroh formalism and skew-symmetric weight functions
A new general approach for deriving the weight functions for 2D interfacial cracks in anisotropic bimaterials has been developed.For perfect interface conditions, the new method avoid the use of Wiener-Hopf technique and the challenging factorization problem connected. Both symmetric and skew-symmetric weight functions can be derived by means of the new approach. Weight functions can be used for deriving singular integral formulation of interfacial cracks in anisotropic media. The proposed method can be applied for studying interfacial cracks problems in many materials:monoclinic, orthotropic, cubic, piezoelectrics, poroelastics, quasicrystals
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