A Rectangular Trust-Region Approach for Unconstrained and Box-Constrained Optimization Problems

Voglis, C.; Lagaris, Isaac E.

doi:10.1201/9780429081385-138

Cited by 20 publications

(20 citation statements)

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“…This functional form was chosen because it follows the shape of an asymmetric sigmoid, thereby fitting both the region of low W / h * where χ is flat at 0 and the gradual increase in χ for large W / h * . Equation was fit to the model results with the Python package scipy.optimize using nonlinear least squares and a trust region minimization algorithm (Voglis & Lagaris, ). The individual model runs were binned before fitting because the full data set is heavily weighted toward runs with W / h * ~2.…”

Section: Discussionmentioning

confidence: 99%

Controls on Mid‐ocean Ridge Normal Fault Seismicity Across Spreading Rates From Rate‐and‐State Friction Models

et al. 2018

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Recent seismic and geodetic observations have led to a growing realization that a significant amount of fault slip at plate boundaries occurs aseismically and that the amount of aseismic slip varies across tectonic settings. Seismic moment release rates measured along the fast‐spreading East Pacific Rise suggest that the majority of fault slip occurs aseismically. By contrast, at the slow‐spreading Mid‐Atlantic Ridge seismic slip may represent up to 60% of total fault displacement. In this study, we use rate‐and‐state friction models to quantify the seismic coupling coefficient, defined as the fraction of total fault slip that occurs seismically, on mid‐ocean ridge normal faults and investigate controls on fault behavior that might produce variations in coupling observed at oceanic spreading centers. We find that the seismic coupling coefficient scales with the ratio of the downdip width of the seismogenic area (W) to the critical earthquake nucleation size (h*). At mid‐ocean ridges, W is expected to increase with decreasing spreading rate. Thus, the relationship between seismic coupling and W/h* predicted from our models explains the first‐order variations in seismic coupling coefficient as a function of spreading rate.

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Section: Discussionmentioning

confidence: 99%

Controls on Mid‐ocean Ridge Normal Fault Seismicity Across Spreading Rates From Rate‐and‐State Friction Models

et al. 2018

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“…The convergence tolerance for the max-norm of the residual is set at the default ftol=6e-6. The resulting non-linear least-squares problem is solved using SciPy's least-squares solver with the 'dogbox' method [71]. The tolerance on the change to the cost function is set at ftol=1e-8.…”

Section: Solution Of the Reduced-order Modelsmentioning

confidence: 99%

The Adjoint Petrov–Galerkin method for non-linear model reduction

Parish

Wentland

Duraisamy

2020

Computer Methods in Applied Mechanics and Engineering

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We formulate a new projection-based reduced-ordered modeling technique for non-linear dynamical systems. The proposed technique, which we refer to as the Adjoint Petrov-Galerkin (APG) method, is derived by decomposing the generalized coordinates of a dynamical system into a resolved coarse-scale set and an unresolved fine-scale set. A Markovian finite memory assumption within the Mori-Zwanzig formalism is then used to develop a reduced-order representation of the coarse-scales. This procedure leads to a closed reduced-order model that displays commonalities with the adjoint stabilization method used in finite elements. The formulation is shown to be equivalent to a Petrov-Galerkin method with a non-linear, time-varying test basis, thus sharing some similarities with the Least-Squares Petrov-Galerkin method. Theoretical analysis examining a priori error bounds and computational cost is presented. Numerical experiments on the compressible Navier-Stokes equations demonstrate that the proposed method can lead to improvements in numerical accuracy, robustness, and computational efficiency over the Galerkin method on problems of practical interest. Improvements in numerical accuracy and computational efficiency over the Least-Squares Petrov-Galerkin method are observed in most cases.(G ROM) has been used successfully in a variety of problems. When applied to general non-self-adjoint and non-linear problems, however, theoretical analysis and numerical experiments have shown that Galerkin ROM lacks a priori guarantees of stability, accuracy, and convergence [9]. This last issue is particularly challenging as it demonstrates that enriching a ROM basis does not necessarily improve the solution [10]. The development of stable and accurate reduced-order modeling techniques for complex non-linear systems is the motivation for the current work.A significant body of research aimed at producing accurate and stable ROMs for complex non-linear problems exists in the literature. These efforts include, but are not limited to, "energy-based" inner products [9,11], symmetry transformations [12], basis adaptation [13,14], L 1 -norm minimization [15], projection subspace rotations [16], and least-squares residual minimization approaches [17,18,19,20,21,22,23,24]. The Least-Squares Petrov-Galerkin (LSPG) [22] method comprises a particularly popular leastsquares residual minimization approach and has been proven to be an effective tool for non-linear model reduction. Defined at the fully-discrete level (i.e., after spatial and temporal discretization), LSPG relies on least-squares minimization of the FOM residual at each time-step. While the method lacks a priori stability guarantees for general non-linear systems, it has been shown to be effective for complex problems of interest [24,23,25]. Additionally, as it is formulated as a minimization problem, physical constraints such as conservation can be naturally incorporated into the ROM formulation [26]. At the fully-discrete level, LSPG is sensitive to both the time integration scheme as ...

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“…A discretization transforms (Q) into a finite-dimensional linear-least squares problem which can be solved with standard algorithms for convex optimization problems with box constraints. We name the references [3,34] which are implemented in the Open-Source library SciPy, see [33], which we use for the computational results in this section.…”

Section: Computational Experimentsmentioning

confidence: 99%

Compactness and convergence rates in the combinatorial integral approximation decomposition

2020

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The combinatorial integral approximation decomposition splits the optimization of a discrete-valued control into two steps: solving a continuous relaxation of the discrete control problem, and computing a discrete-valued approximation of the relaxed control. Different algorithms exist for the second step to construct piecewise constant discrete-valued approximants that are defined on given decompositions of the domain. It is known that the resulting discrete controls can be constructed such that they converge to a relaxed control in the $$\hbox {weak}^*$$ weak ∗ topology of $$L^\infty $$ L ∞ if the grid constant of this decomposition is driven to zero. We exploit this insight to formulate a general approximation result for optimization problems, which feature discrete and distributed optimization variables, and which are governed by a compact control-to-state operator. We analyze the topology induced by the grid refinements and prove convergence rates of the control vectors for two problem classes. We use a reconstruction problem from signal processing to demonstrate both the applicability of the method outside the scope of differential equations, the predominant case in the literature, and the effectiveness of the approach.

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A Rectangular Trust-Region Approach for Unconstrained and Box-Constrained Optimization Problems

Cited by 20 publications

References 1 publication

Controls on Mid‐ocean Ridge Normal Fault Seismicity Across Spreading Rates From Rate‐and‐State Friction Models

Controls on Mid‐ocean Ridge Normal Fault Seismicity Across Spreading Rates From Rate‐and‐State Friction Models

The Adjoint Petrov–Galerkin method for non-linear model reduction

Compactness and convergence rates in the combinatorial integral approximation decomposition

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