Convergence analysis of an Inexact Infeasible Interior Point method for Semidefinite Programming

In this paper we present an extension to SDP of the well known infeasible Interior Point method for linear programming of Kojima,Megiddo and Mizuno (A primal-dual infeasible-interior-point algorithm for Linear Programming, Math. Progr., 1993). The extension developed here allows the use of inexact search directions; i.e., the linear systems defining the search directions can be solved with an accuracy that increases as the solution is approached. A convergence analysis is carried out and the global convergence of the method is prove

On affine scaling inexact dogleg methods for bound-constrained nonlinear systems

Within the framework of affine scaling trust-region methods for bound constrained problems, we discuss the use of a inexact dogleg method as a tool for simultaneously handling the trust-region and the bound constraints while seeking for an approximate minimizer of the model. Focusing on bound-constrained systems of nonlinear equations, an inexact affine scaling method for large scale problems, employing the inexact dogleg procedure, is described. Global convergence results are established without any Lipschitz assumption on the Jacobian matrix, and locally fast convergence is shown under standard assumptions. Convergence analysis is performed without specifying the scaling matrix used to handle the bounds, and a rather general class of scaling matrices is allowed in actual algorithms. Numerical results showing the performance of the method are also given

On affine scaling inexact dogleg methods for bound-constrained nonlinear systems

Within the framework of affine scaling trust-region methods for bound constrained problems, we discuss the use of a inexact dogleg method as a tool for simultaneously handling the trust-region and the bound constraints while seeking for an approximate minimizer of the model. Focusing on bound-constrained systems of nonlinear equations, an inexact affine scaling method for large scale problems, employing the inexact dogleg procedure, is described. Global convergence results are established without any Lipschitz assumption on the Jacobian matrix, and locally fast convergence is shown under standard assumptions. Convergence analysis is performed without specifying the scaling matrix used to handle the bounds, and a rather general class of scaling matrices is allowed in actual algorithms. Numerical results showing the performance of the method are also given

A PDE-constrained optimization formulation for discrete fracture network flows

We investigate a new numerical approach for the computation of the 3D flow in a discrete fracture network that does not require a conforming discretization of partial differential equations on complex 3D systems of planar fractures. The discretization within each fracture is performed independently of the discretization of the other fractures and of their intersections. Independent meshing process within each fracture is a very important issue for practical large scale simulations making easier mesh generation. Some numerical simulations are given to show the viability of the method. The resulting approach can be naturally parallelized for dealing with systems with a huge number of fractures

Flow simulations in porous media with immersed intersecting fractures

A novel approach for fully 3D flow simulations in porous media with immersed networks of fractures is presented. The method is based on the discrete fracture and matrix model, in which fractures are represented as two-dimensional objects in a three-dimensional porous matrix. The problem, written in primal formulation on both the fractures and the porous matrix, is solved resorting to the constrained minimization of a properly designed cost functional that expresses the matching conditions at fracture-fracture and fracture-matrix interfaces. The method, originally conceived for intricate fracture networks in impervious rock matrices, is here extended to fractures in a porous permeable rock matrix. The purpose of the optimization approach is to allow for an easy meshing process, independent of the geometrical complexity of the domain, and for a robust and efficient resolution tool, relying on a strong parallelism. The present work is devoted to the presentation of the new method and of its applicability to flow simulations in poro-fractured domains

Multi-Objective Optimisation of an Aerostatic Pad: Design of Position, Number and Diameter of the Supply Holes

ABSTRACTIn this paper, a rectangular aerostatic bearing with multiple supply holes is optimised with a multiobjective optimisation approach. The design variables taken into account are the supply holes position, their number and diameter, the supply pressure, while the objective functions are the load capacity, the air consumption and the stiffness and damping coefficients. A genetic algorithm is applied in order to find the Pareto set of solutions. The novelty with respect to other optimisations which can be found in literature is that number and location of the supply holes is completely free and not associated to a pre-defined scheme. A vector x associated with the supply holes location is introduced in the design parameters and given in input to the optimizer

Uncertainty quantification in Discrete Fracture Network models: stochastic fracture transmissivity

We consider flows in fractured media, described by Discrete Fracture Network (DFN) models. We perform an Uncertainty Quantification analysis, assuming the fractures' transmissivity coefficients to be random variables. Two probability distributions (log-uniform and log-normal) are used within different laws that express the coefficients in terms of a family of independent stochastic variables; truncated Karhunen-Loève expansions provide instances of such laws. The approximate computation of quantities of interest, such as mean value and variance for outgoing fluxes, is based on a stochastic collocation approach that uses suitable sparse grids in the range of the stochastic variables (whose number defines the stochastic dimension of the problem). This produces a non-intrusive computational method, in which the DFN flow solver is applied as a black-box. A very fast error decay, related to the analytical dependence of the observed quantities upon the stochastic variables, is obtained in the low dimensional cases using isotropic sparse grids; comparisons with Monte Carlo results show a clear gain in efficiency for the proposed method. For increasing dimensions attained via successive truncations of Karhunen-Loève expansions, results are still good although the rates of convergence are progressively reduced. Resorting to suitably tuned anisotropic grids is an effective way to contrast such curse of dimensionality: in the explored range of dimensions, the resulting convergence histories are nearly independent of the dimension

Coupling traffic models on networks and urban dispersion models for simulating sustainable mobility strategies

AbstractThe aim of the present paper is to investigate the viability of macroscopic traffic models for modeling and testing different traffic scenarios, in order to define the impact on air quality of different strategies for the reduction of traffic emissions. To this aim, we complement a well assessed traffic model on networks (Garavello and Piccoli (2006) [1]) with a strategy for estimating data needed from the model and we couple it with the urban dispersion model Sirane (Soulhac (2000) [2])

Numerical investigation on a block preconditioning strategy to improve the computational efficiency of DFN models

[EN] The simulation of underground flow across intricate fracture networks can be addressed by means of discrete fracture network models. The combination of such models with an optimization formulation allows for the use of nonconforming and independent meshes for each fracture. The arising algebraic problem produces a symmetric saddle-point matrix with a rank-deficient leading block. In our work, we investigate the properties of the system to design a block preconditioning strategy to accelerate the iterative solution of the linearized algebraic problem. The matrix is first permuted and then projected in the symmetric positive-definite Schur-complement space. The proposed strategy is tested in applications of increasing size, in order to investigate its capabilities.Gazzola, L.; Ferronato, M.; Berrone, S.; Pieraccini, S.; Scialò, S. (2022). Numerical investigation on a block preconditioning strategy to improve the computational efficiency of DFN models. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 346-354. https://doi.org/10.4995/YIC2021.2021.12234OCS34635