Augmented Lagrangian Scheme Research Articles

Efficient method for symmetric nonnegative matrix factorization with an approximate augmented Lagrangian scheme

In this paper, we propose an efficient method for solving symmetric nonnegative matrix factorization following an approximate augmented Lagrangian scheme. The augmented Lagrangian subproblem was solved column by column under the block coordinate descent (BCD) framework. In particular, we extend the recursive formula for rank-k nonnegative least squares problems by Chu et al. (2021) to subproblems generated by BCD framework in our method. Thereafter we derive the closed-form solution for k=4. Experiments for clustering show that the proposed method is noticeably efficient and achieves competitive performance compared with state-of-the-art methods.

An augmented Lagrangian approach for cardinality constrained minimization applied to variable selection problems

To solve convex constrained minimization problems, that also include a cardinality constraint, we propose an augmented Lagrangian scheme combined with alternating projection ideas. Optimization problems that involve a cardinality constraint are NP-hard mathematical programs and typically very hard to solve approximately. Our approach takes advantage of a recently developed and analyzed continuous formulation that relaxes the cardinality constraint. Based on that formulation, we solve a sequence of smooth convex constrained minimization problems, for which we use projected gradient-type methods. In our setting, the convex constraint region can be written as the intersection of a finite collection of convex sets that are easy and inexpensive to project. We apply our approach to a variety of over and under determined constrained linear least-squares problems, with both synthetic and real data that arise in variable selection, and demonstrate its effectiveness.

Atypical plug formation in internal elastoviscoplastic fluid flows over non-smooth topologies

An experimental and computational investigation of the internal flow of elastoviscoplastic fluids over non-smooth topologies is presented in two complimentary studies. In the first study, we visualize the creeping flow of a Carbopol gel over a cavity embedded in a thin slot using Optical Coherence Tomography (OCT) and confocal microscopy. We measure the size and shape of the plug as a function of Bingham and Weissenberg numbers. An asymmetry in the plug shape is observed which is also evident in our second study—numerical simulations using adaptive finite element method based upon an augmented Lagrangian scheme. We quantify the asymmetry and present the results as a function of the product of the Weissenberg and Bingham numbers which collapse onto a single curve for each of these geometries. These findings underscore the theoretical underpinnings of the synergy between elasticity and plasticity of these complex fluids.

Structural Topology Optimization with Local Finite-Life Fatigue Constraints

To improve the fatigue resistance of engineering structures, topology optimization has always been an effective design strategy. The direct calculation of large-scale local fatigue constraints remains a challenge due to high computational cost. In the past, the constraint aggregation techniques, such as the P-norm method, were often applied to aggregate local fatigue constraints into a global constraint, whereas the resultant optimal solution was not consistent with the original problem. In order to meet the local fatigue constraints accurately and reduce the number of constraints, the augmented Lagrangian scheme is employed to transform the original problem into the unconstrained problem. To evaluate the fatigue strength at every material point of structures under the proportional load with variable amplitude, we adopt the Sines fatigue criterion based on the Palmgren–Miner linear damage assumption. In addition, we solve the fatigue-constrained topology optimization problem on the unstructured polygonal meshes, which are not sensitive to numerical instabilities, such as checkerboard patterns, compared with lower-order triangular and bilateral meshes. We provide some numerical examples to validate the potential of the presented method to solve the fatigue-constrained topology optimization problem. Numerical results demonstrate that the optimized designs considering local fatigue constraints have a higher ratio of fatigue resistance to material consumption than those obtained through the traditional P-norm method. Therefore, the proposed approach retaining the local nature of fatigue constraints is more beneficial for realizing the efficient material utilization in structural topology.

Constrained composite optimization and augmented Lagrangian methods

AbstractWe investigate finite-dimensional constrained structured optimization problems, featuring composite objective functions and set-membership constraints. Offering an expressive yet simple language, this problem class provides a modeling framework for a variety of applications. We study stationarity and regularity concepts, and propose a flexible augmented Lagrangian scheme. We provide a theoretical characterization of the algorithm and its asymptotic properties, deriving convergence results for fully nonconvex problems. It is demonstrated how the inner subproblems can be solved by off-the-shelf proximal methods, notwithstanding the possibility to adopt any solvers, insofar as they return approximate stationary points. Finally, we describe our matrix-free implementation of the proposed algorithm and test it numerically. Illustrative examples show the versatility of constrained composite programs as a modeling tool and expose difficulties arising in this vast problem class.

Information-based seismic amplitude inversion

Amplitude variation with offset (AVO) inversion, particularly for more than two model parameters, is a highly ill-posed problem and, hence, regularization is indispensable. For common regularization such as norm-regularization, learning can be forged ahead via gradient descent. Alternatively, learning can be achieved by performing gradient descent on the objective and projects the result back into the space of the regularizer; this allows the use of atypical regularization functions. Here, we propose a regularized inverse problem to mitigate the Ill-posedness of the amplitude inversion. The regularization is added to measure the difference in information between the a priori probability density function and the predicted probability density of the inverted parameters, and therefore enhances the consistency between the inversion results and the a priori geological information. Information theory provides a collection of contrast functions which quantify the divergence from one probability distribution to another, such as the relative entropy. The a priori density is approximated by a Gaussian mixture model, obtained from well logs and rock physics data. The mixture model is a density estimator, providing the statistical properties of the model parameters of interest. The likelihood of the data and the divergence are combined in an augmented Lagrangian scheme, to obtain a stable solution that best generate the recorded seismic data and satisfy the geological constraints conveyed by the a priori probability density function. The proposed inversion scheme is applied to the anisotropy AVO inversion, for estimating the elastic and seismic anisotropy parameters of shale formations. Compared to the unconstrained minimization, the P- and S-wave velocity, and ε are better recovered, moreover, density and Thomsen's δ are well-constrained.

An augmented Lagrangian method for multiple nodal displacement-constrained topology optimization

This article proposes an augmented Lagrangian-based topology optimization approach to minimize the volume fraction subject to multiple nodal displacement constraints. The proposed method puts the multiple constraint equations in the objective function and transforms it into a sequence of unconstrained optimization problems. This study explains the theoretical aspects of the employed augmented Lagrangian approach in depth. Sensitivity expression in terms of design variables is identified by exercising the differentiate-then-discretize mechanism, and utilizes a moving asymptote algorithm to sort out a sequence of optimization subproblems. The optimized results are compared and analysed with those from conventional aggregation-based topology optimization. However, the predefined aggregation approach indicates a dependency on the supplied parameters. The numerical two- and three-dimensional examples reveal the viability and reliability of the suggested augmented Lagrangian scheme, which shows advantages in terms of robustness and efficiency.

Computational rheometry of yielding and viscoplastic flow in vane-and-cup rheometer fixtures

A planar two-dimensional computational analysis is presented to qualify traditional and fractal vane-in-cup geometries for accurate rheometry of simple viscoplastic fluids with and without slip. Numerical simulations based on an adaptive augmented Lagrangian scheme are used to study the two-dimensional flow field of yield-stress fluids within and around vane tools with N=3 to 24 arms for a wide range of Bingham numbers, B (i.e. the ratio of the yield stress over the characteristic viscous stress). This allows for accurate calculations of the velocity and stress fields around vanes with various geometries, as well as direct comparison to experimental observations of the output torque measured by a rheometer, revealing sources of variation and error. We describe the impact of the vane structure on the fluid velocity field, from few-arm cruciform vanes (N≤6) that significantly perturb the flow away from ideal azimuthal kinematics, to many-arm fractal vanes (N≥12) in which the internal structural features are successfully “cloaked” by a yield surface. This results in the shearing of an almost-circular ring of viscoplastic fluid that is indistinguishable from the annular ring of fluid deformed around a slip-free rotating cylindrical bob and leads to more accurate rheometric measurements of the material flow curve. Moreover, in direct comparison with data from previous literature, we show that slip conditions on the vane surface do not impact the velocity field or measured overall torque T, whereas slip conditions on the smooth outer wall have significant impact on data, even when using a vane geometry. Finally, we describe the impact of vane topography and Bingham number, B, on the measured torque and rheometric accuracy of vane-in-cup geometries for “simple” (inelastic) yield-stress fluids described by either the Bingham plastic or Herschel–Bulkley constitutive model.

A level-set-based topology optimization strategy using radial basis functions and a Hilbertian velocity extension

This work addresses the structural compliance minimization problem through a level-set-based strategy that rests upon radial basis functions with compact support combined with Hilbertian velocity extensions. A consistent augmented Lagrangian scheme is adopted to handle the volume constraint. The linear elasticity model and the variational problem associated with the computation of the velocity field are tackled by the finite element method using resources from the FEniCS project. The parameterization mesh constituted by the centers of the radial basis functions may be decoupled from the finite element mesh. A numerical investigation is conducted employing a Python implementation and five benchmark structures. Aimed at isolating the distinct aspects that compose the proposed strategy, the experiments provide grounds to analyze and put into perspective the inherent decisions and the related elements.

Correction to: On the best achievable quality of limit points of augmented Lagrangian schemes

A Correction to this paper has been published: https://doi.org/10.1007/s11075-021-01241-3

On the best achievable quality of limit points of augmented Lagrangian schemes

On the best achievable quality of limit points of augmented Lagrangian schemes

A Finite Element Algorithm for Large Deformation Biphasic Frictional Contact Between Porous-Permeable Hydrated Soft Tissues.

The frictional response of porous and permeable hydrated biological tissues such as articular cartilage is significantly dependent on interstitial fluid pressurization. To model this response, it is common to represent such tissues as biphasic materials, consisting of a binary mixture of a porous solid matrix and an interstitial fluid. However, no computational algorithms currently exist in either commercial or open-source software that can model frictional contact between such materials. Therefore, this study formulates and implements a finite element algorithm for large deformation biphasic frictional contact in the open-source finite element software FEBio. This algorithm relies on a local form of a biphasic friction model that has been previously validated against experiments, and implements the model into our recently-developed surface-to-surface (STS) contact algorithm. Contact constraints, including those specific to pressurized porous media, are enforced with the penalty method regularized with an active-passive augmented Lagrangian scheme. Numerical difficulties specific to challenging finite deformation biphasic contact problems are overcome with novel smoothing schemes for fluid pressures and Lagrange multipliers. Implementation accuracy is verified against semi-analytical solutions for biphasic frictional contact, with extensive validation performed using canonical cartilage friction experiments from prior literature. Essential details of the formulation are provided in this paper, and the source code of this biphasic frictional contact algorithm is made available to the general public.

Joint ptycho-tomography with deep generative priors

Joint ptycho-tomography is a powerful computational imaging framework to recover the refractive properties of a 3D object while relaxing the requirements for probe overlap that is common in conventional phase retrieval. We use an augmented Lagrangian scheme for formulating the constrained optimization problem and employ an alternating direction method of multipliers (ADMM) for the joint solution. ADMM allows the problem to be split into smaller and computationally more efficient subproblems: ptychographic phase retrieval, tomographic reconstruction, and regularization of the solution. We extend our ADMM framework with plug-and-play (PnP) denoisers by replacing the regularization subproblem with a general denoising operator based on machine learning. While the PnP framework enables integrating such learned priors as denoising operators, tuning of the denoiser prior remains challenging. To overcome this challenge, we propose a denoiser parameter to control the effect of the denoiser and to accelerate the solution. In our simulations, we demonstrate that our proposed framework with parameter tuning and learned priors generates high-quality reconstructions under limited and noisy measurement data.

The Effect of Iterative Procedures on the Robustness and Fidelity of Augmented Lagrangian SPH

The Augmented Lagrangian Smoothed Particle Hydrodynamics (ALSPH) method is a novel incompressible Smoothed Particle Hydrodynamics (SPH) approach that solves Navier–Stokes equations by an iterative augmented Lagrangian scheme through enforcing the divergence-free coupling of velocity and pressure fields. This study aims to systematically investigate the time step size and the number of inner iteration parameters to boost the performance of the ALSPH method. Additionally, the effects of computing spatial derivatives with two alternative schemes on the accuracy of numerical results are also scrutinized. Namely, the first scheme computes spatial derivatives on the updated particle positions at each iteration, whereas the second one employs the updated pressure and velocity fields on the initial particle positions to compute the gradients and divergences throughout the iterations. These two schemes are implemented to the solution of a flow over a circular cylinder at Reynolds numbers of 200 in two dimensions. Initially, simulations are performed in order to determine the optimum time step sizes by utilizing a maximum number of five iterations per time step. Subsequently, the optimum number of inner iterations is investigated by employing the predetermined optimum time step size under the same flow conditions. Finally, the schemes are tested on the same flow problem with different Reynolds numbers using the best performing combination of the aforementioned parameters. It is observed that the ALSPH method can enable one to increase the time step size without deteriorating the numerical accuracy as a consequence of imposing larger ALSPH penalty terms in larger time step sizes, which, overall, leads to improved computational efficiency. When considering the hydrodynamic flow characteristics, it can be stated that two spatial derivative schemes perform very similarly. However, the results indicate that the derivative operation with the updated particle positions produces slightly lower velocity divergence magnitudes at larger time step sizes.

An energy-relaxation-based framework for the modeling of magnetic shape memory alloys—Simulation of three-dimensional effects under homogeneous loading conditions

It is known from experimental findings that three-dimensional effects can have a strong influence on magnetic shape memory behavior. Such phenomena are, however, often neglected in MSMA constitutive models, as they only become meaningful under complex loading conditions. The extensions of our original modeling framework, cf. Bartel et al. (2020), to include 3D-effects is threefold: (i) vector-valued microstructural variables are now elements in R3, i.e. no longer parameterizable in polar coordinates, (ii) a third tetragonal martensite variant may form/vanish by switching from/back into both other variants, and (iii) a more general and robust algorithmic treatment is necessary. The latter includes the implementation of a staggered Augmented Lagrangian scheme to handle the now much larger and numerically more advanced sets of equality and inequality constraints. In this context, two extended model formulations are presented. The first considers a first-order, two-variant laminate approach (rank-one convexification), in which domain magnetizations, interface orientations etc. are now three-dimensional vectors. The second model is based on a convexification approach, for which the incorporation of the third martensitic variant is quite natural. Numerical examples are investigated to test the generalized modeling framework. Firstly, it is confirmed that both extended models recover the solution of the previously established two-dimensional model for a simple loading case. Secondly, response predictions for more complex loading scenarios (non-proportional bi-axial stresses, orthogonal magnetic field), motivated by experiments, are investigated. It is found that capturing the formation, elimination and mutual interaction of all martensitic variants as well as general three-dimensional magnetization vector orientations is of key importance under these conditions. The extended convexification model and modified algorithmic formulation are shown to reliably handle even such general cases.

Assessment of simultaneous and nested conservative augmented Lagrangian schemes for constrained multibody dynamics

Two alternative schemes used for the solution of multibody systems are reported and compared between them to evaluate their performance. Within the context of kinematic constraints imposed via augmented Lagrangian technique, one scheme proposes a simultaneous iterative solution for the computation of the Lagrangian multipliers and the nonlinear Newton-Raphson iterations. The second scheme uses a more classical Uzawa approach to compute the Lagrangian multipliers, and the nonlinear iterations are nested into the multiplier update loop. The performance of the methodology is tested by solving numerical experiments: simple, double, triple, and 10 bars pendulums with revolute joints. Moreover, 2D and 3D analyses are performed. The computed results using both the simultaneous iterative, and the nested iterative techniques, are reported to evaluate: kinematic responses, energy conservation, constraints verification, and iterations reduction.

Discrete Phase Coded Sequence Set Design for Waveform-Agile Radar Based on Alternating Direction Method of Multipliers

With increased degrees of freedom of the transmitter, a coherent waveform-agile radar system can change its interpulse waveform to enhance the capability against increasingly sophisticated jamming and spoofing threats. Furthermore, radar agile waveform can provide better detection performance than a single waveform. In this article, we consider discrete phase coded sequence set design (DPCSSD) problem based on range-Doppler discrete ambiguity function for coherent waveform-agile radar system. DPCSSD problem for optimizing discrete phase coded sequence set under constant module constraint with desired minimized sidelobes on range-Doppler plane, is a quartic function and the constraint is nonconvex. To solve this quartic optimization problem, we propose an effective algorithm based on alternating direction method of multipliers (ADMM), which is a powerful mathematical tool of the augmented Lagrangian scheme for dealing with separable objective functions. As the quartic optimization problem is hard to solve, majorization-minimization (MM) is also considered to introduce a quadratic auxiliary function which is the upperbound of the original function. By minimizing the auxiliary function, the original DPCSSD optimization problem can be solved through MM in an iterative way. Meanwhile, the problem of minimizing the quadratic auxiliary function with constraints can be solved through ADMM algorithm. In particular, fast implementation for the most computationally demanding step of matrix inversion is investigated. The effectiveness of the proposed approach is demonstrated via computer simulations.

Plug-and-Play Based Optimization Algorithm for New Crime Density Estimation

Different from a general density estimation, the crime density estimation usually has one important factor: the geographical constraint. In this paper, a new crime density estimation model is formulated, in which the regions where crime is impossible to happen, such as mountains and lakes, are excluded. To further optimize the estimation method, a learning-based algorithm, named Plug-and-Play, is implanted into the augmented Lagrangian scheme, which involves an off-the-shelf filtering operator. Different selections of the filtering operator make the algorithm correspond to several classical estimation models. Therefore, the proposed Plug-and-Play optimization based estimation algorithm can be regarded as the extended version and general form of several classical methods. In the experiment part, synthetic examples with different invalid regions and samples of various distributions are first tested. Then under complex geographic constraints, we apply the proposed method with a real crime dataset to recover the density estimation. The state-of-the-art results show the feasibility of the proposed model.

Iterative scheme-inspired network for impulse noise removal

This paper presents a supervised data-driven algorithm for impulse noise removal via iterative scheme-inspired network (IIN). IIN is defined over a data flow graph, which is derived from the iterative procedures in Alternating Direction Method of Multipliers (ADMM) algorithm for optimizing the L1-guided variational model. In the training phase, the L1-minimization is reformulated into an augmented Lagrangian scheme through adding a new auxiliary variable. In the testing phase, it has computational overhead similar to ADMM but uses optimized parameters learned from the training data for restoration task. Experimental results demonstrate that the newly proposed method can obtain very significantly superior performance than current state-of-the-art variational and dictionary learning-based approaches for salt-and-pepper noise removal.

Stress-constrained level set topology optimization for design-dependent pressure load problems

This work presents a level set framework to solve topology optimization problems subject to local stress constraints considering design-dependent pressure loads. Two technical difficulties are related to this problem. The first one is the local nature of stresses. To deal with this issue, stress constraints are included to the problem by means of an Augmented Lagrangian scheme. The second is the adequate association between the moving boundary and the pressure acting on it. This difficulty is easily overcome by the level set method that allows for a clear tracking of the boundary along the optimization process. In the present approach, a reaction–diffusion equation substitutes the classical Hamilton–Jacobi equation to control the level set evolution. This choice has the advantage of allowing the nucleation of holes inside the domain and the elimination of the undesirable level set reinitializations. In addition, the optimization algorithm allows the rupture of loading boundaries, that is, the crossing of the pressured (wet) boundary with the traction free boundary is not avoided. This gives more freedom to the algorithm for topological changes. In order to validate the proposed scheme against stress concentrations, all numerical examples are performed on constrained domains containing singularities. Moreover, optimized designs obtained from stress-constrained and compliance problems are compared.