Reduced Solution Space Research Articles

A novel LSTM-integrated non-intrusive ROM for reliability analysis of hysteretic systems with large stochastic dimension

Failure probability estimation of hysteretic systems subjected to random process excitation appears in many practical engineering applications, such as structures subjected to earthquake and vibration control. This task is computationally challenging due to the need for repeated solutions of a high-resolution model. This cost grows significantly with the stochastic dimension — characterized by the number of random variables, in this case, arising from the discretization of the random excitation. To address this issue, a novel non-intrusive reduced order model (ROM) is proposed in this paper. Accordingly, the complexity due to spatial discretization is first reduced using proper orthogonal decomposition. However, interpolation among response time histories in the resulting reduced solution space is computationally infeasible due to large stochastic dimensionality. An existing ROM developed by the authors is now improved by adopting a long short-term memory (LSTM) network for this interpolation. Furthermore, this adoption required two additional modifications in the mathematical structure of the ROM related to data compression and nonlinear coupling. The proposed methodology can be seamlessly integrated with black-box nonlinear solvers. Through detailed numerical studies on a beam on Winkler foundation and a multi-storied building frame subjected to stationary and non-stationary excitations, the proposed ROM is found to be accurate, efficient, and robust with respect to the frequency band of interest. The ROM has shown reduction of two orders of magnitude in the computational time. Furthermore, the proposed ROM is inherently parallelizable and holds promise to be scaled to other types of nonlinear systems with large stochastic dimensionality.

Semi-active damping optimization of vibrational systems using the reduced basis method

In this article, we consider vibrational systems with semi-active damping that are described by a second-order model. In order to minimize the influence of external inputs to the system response, we are optimizing some damping values. As minimization criterion, we evaluate the energy response, that is the H2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}_2$$\\end{document}-norm of the corresponding transfer function of the system. Computing the energy response includes solving Lyapunov equations for different damping parameters. Hence, the minimization process leads to high computational costs if the system is of large dimension. We present two techniques that reduce the optimization problem by applying the reduced basis method to the corresponding parametric Lyapunov equations. In the first method, we determine a reduced solution space on which the Lyapunov equations and hence the resulting energy response values are computed approximately in a reasonable time. The second method includes the reduced basis method in the minimization process. To evaluate the quality of the approximations, we introduce error estimators that evaluate the error in the controllability Gramians and the energy response. Finally, we illustrate the advantages of our methods by applying them to two different examples.

Compact Circuits for Efficient Möbius Transform

The Möbius transform is a linear circuit used to compute the evaluations of a Boolean function over all points on its input domain. The operation is very useful in finding the solution of a system of polynomial equations over GF(2) for obvious reasons. However the operation, although linear, needs exponential number of logic operations (around n · 2n−1 bit xors) for an n-variable Boolean function. As such, the only known hardware circuit to efficiently compute the Möbius Transform requires silicon area that is exponential in n. For Boolean functions whose algebraic degree is bound by some parameter d, recursive definitions of the Möbius Transform exist that requires only O(nd+1) space in software. However converting the mathematical definition of this space-efficient algorithm into a hardware architecture is a non-trivial task, primarily because the recursion calls notionally lead to a depth-first search in a transition graph that requires context switches at each recursion call for which straightforward mapping to hardware is difficult. In this paper we look to overcome these very challenges in an engineering sense. We propose a space efficient sequential hardware circuit for the Möbius Transform that requires only polynomial circuit area (i.e. O(nd+1)) provided the algebraic degree of the Boolean function is limited to d. We show how this circuit can be used as a component to efficiently solve polynomial equations of degree at most d by using fast exhaustive search. We propose three different circuit architectures for this, each of which uses the Möbius Transform circuit as a core component. We show that asymptotically, all the solutions of a system of m polynomials in n unknowns and algebraic degree d over GF(2) can be found using a circuit of silicon area proportional to m · nd+1 and circuit depth proportional to 2 · log2(n − d).In the second part of the paper we introduce a fourth hardware solver that additionally aims to achieve energy efficiency. The main idea is to reduce the solution space to a small enough value by parallel application of Möbius Transform circuits over the first few equations of the system. This is done so that one can check individually whether the vectors of this reduced solution space satisfy each of the remaining equations of the system using lower power consumption. The new circuit has area also bound by m · nd+1 and has circuit depth proportional to d · log2 n. We also show that further optimizations with respect to energy consumption may be obtained by using depth-bound Möbius circuits that exponentially decrease run time at the cost of additional logic area and depth.

Quantum-Inspired Digital Annealing for Join Ordering

Finding the optimal join order (JO) is one of the most important problems in query optimisation, and has been extensively considered in research and practise. As it involves huge search spaces, approximation approaches and heuristics are commonly used, which explore a reduced solution space at the cost of solution quality. To explore even large JO search spaces, we may consider special-purpose software, such as mixed-integer linear programming (MILP) solvers, which have successfully solved JO problems. However, even mature solvers cannot overcome the limitations of conventional hardware prompted by the end of Moore's law. We consider quantum-inspired digital annealing hardware, which takes inspiration from quantum processing units (QPUs). Unlike QPUs, which likely remain limited in size and reliability in the near and mid-term future, the digital annealer (DA) can solve large instances of mathematically encoded optimisation problems today. We derive a novel, native encoding for the JO problem tailored to this class of machines that substantially improves over known MILP and quantum-based encodings, and reduces encoding size over the state-of-the-art. By augmenting the computation with a novel readout method, we derive valid join orders for each solution obtained by the (probabilistically operating) DA. Most importantly and despite an extremely large solution space, our approach scales to practically relevant dimensions of around 50 relations and improves result quality over conventionally employed approaches, adding a novel alternative to solving the long-standing JO problem.

Solving the Single-Row Facility Layout Problem by K-Medoids Memetic Permutation Group

The single-row facility layout problem (SRFLP) is concerned with arranging facilities along a straight line so as to minimize the sum of the products of the flow costs and distances among all facility pairs. SRFLP has rich practical applications and is however NP-hard. In this article, we first investigate a dedicated symmetry-breaking approach based on the permutation group theory for reducing the solution space of SRFLP. Relevant symmetry properties are identified through the alternating group of the original solution space or the corresponding coordinate rotation space. Then, a memetic algorithm is proposed to explore promising search regions regarding the reduced solution space. The memetic algorithm employs a problem-specific crossover operator guided by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${k}$ </tex-math></inline-formula> -medoids clustering technique to produce meaningful offspring solutions. The algorithm additionally uses a simulated annealing procedure to intensively exploit a given search region and a distance-and-quality-based population management strategy to ensure a reasonable diversity of the population. Experimental results on commonly used benchmark instances and newly introduced large-scale instances with sizes up to 2000 facilities show that the proposed algorithm competes favorably with state-of-the-art SRFLP algorithms. It attains all but one previous best known upper bounds (BKS) and discovers new upper bounds for 33 instances out of the 93 popular benchmark instances.

Generalized Low Switching Frequency Modulation for Neutral-Point-Clamped and Flying-Capacitor Four-Level Converters

Four-level neutral-point-clamped (NPC) and flying capacitor (FC) converters are suitable candidates for high-power medium-voltage applications due to their desired output performance and high power density. The major challenge on their successful applications lies in the dc-link capacitor and FC imbalance issue, especially for converters without redundant states that result in a significantly reduced control flexibility. In this article, a generalized low switching frequency modulation scheme based on selective harmonic elimination pulsewidth modulation (SHE-PWM) is proposed that can be applied to all types of 4L-NPC and 4L-FC converters. To address the dc-link capacitor and FC imbalance problems, the capacitor charge objective is formulated and incorporated into the nonlinear equations of SHE-PWM in solving the switching angles, which would then possess natural capacitor balancing ability. Four effective solutions are provided to address the reduced solution space issue with this formulation in high modulation index range for natural capacitor balancing, thus the full range operation can be guaranteed. Simulation and experimental results are presented to verify the effectiveness of this generalized modulation for 4L-NPC and 4L-FC converters.

Data-driven uncertainty quantification in computational human head models

Computational models of the human head are promising tools for estimating the impact-induced response of the brain, and thus play an important role in the prediction of traumatic brain injury. The basic constituents of these models (i.e., model geometry, material properties, and boundary conditions) are often associated with significant uncertainty and variability. As a result, uncertainty quantification (UQ), which involves quantification of the effect of this uncertainty and variability on the simulated response, becomes critical to ensure reliability of model predictions. Modern biofidelic head model simulations are associated with very high computational cost and high-dimensional inputs and outputs, which limits the applicability of traditional UQ methods on these systems. In this study, a two-stage, data-driven manifold learning-based framework is proposed for UQ of computational head models. This framework is demonstrated on a 2D subject-specific head model, where the goal is to quantify uncertainty in the simulated strain fields (i.e., output), given variability in the material properties of different brain substructures (i.e., input). In the first stage, a data-driven method based on multi-dimensional Gaussian kernel-density estimation and diffusion maps is used to generate realizations of the input random vector directly from the available data. Computational simulations of a small number of realizations provide input–output pairs for training data-driven surrogate models in the second stage. The surrogate models employ nonlinear dimensionality reduction using Grassmannian diffusion maps, Gaussian process regression to create a low-cost mapping between the input random vector and the reduced solution space, and geometric harmonics models for mapping between the reduced space and the Grassmann manifold. It is demonstrated that the surrogate models provide highly accurate approximations of the computational model while significantly reducing the computational cost. Monte Carlo simulations of the surrogate models are used for uncertainty propagation. UQ of the strain fields highlights significant spatial variation in model uncertainty, and reveals key differences in uncertainty among commonly used strain-based brain injury predictor variables.

A reduced order model for nonlinear time history seismic analyzes of elasto‐plastic 3D frame structures

AbstractA reduced order model for the nonlinear dynamic seismic analysis of elasto‐plastic 3D frame structures is presented. It is based on an approximated solution space for the displacement field that is assumed as the sum of two subspaces. The first subspace is generated by the relevant linear elastic modes of the generalized stiffness/mass linear eigenvalue problem in terms of participation factor. The second one is defined by collapse mechanisms associated to appropriate static load distributions. The time history analyzes are carried out within the reduced solution space, while the internal forces are evaluated on the full model to guarantee an accurate description of the constitutive laws. The proposed reduction method was tested for different structural geometries varying frequency content, direction and amplitude of the ground motion. Numerical results show accuracy and robustness also employing a few tens of modes, including elastic modes and plastic collapse mechanisms. In particular, the need for plastic modes in capturing the structural dynamics in case of seismic actions leading to significant plastic excursions is emphasized. This highlights the limits of simplified approaches based on the hypothesis of shape conservation in nonlinear problems, regardless of the building regularity.

Unsteady physics-based reduced order modeling for large-scale compressible aerodynamic applications

A physics-based reduced order model is presented as a viable approach to accurately predict unsteady flowfield solutions at a fraction of the computational time that is required by high fidelity computational fluid dynamics solvers. Such a model could be useful during the preliminary phases of the aircraft design process when an accurate prediction of unsteady loads is crucial to reduce the uncertainty of design choices. This paper extends the formulation of a previously defined technique to unsteady aerodynamic problems involving motion excitation in both subsonic and transonic conditions. In particular, the proposed unsteady-residual-based model defines a reduced solution space by applying proper orthogonal decomposition to a set of snapshots collected from the unsteady high fidelity simulations. Approximate flow solutions for unseen motion signals is then obtained in a time-marching fashion where, at each time step, the prediction coefficients are computed via L2-norm minimization of the unsteady residual vector as provided from the underlying flow solver. As suggested by previous studies, a further reduction of computational complexity is attempted by including a hyperreduction node selection, specifically the missing point estimation routine, in the prediction framework. Different model reduction and hyperreduction levels for the flowfield prediction of harmonic pitching motion of a 2D airfoil and a 3D aircraft in both subsonic and transonic conditions are investigated to determine their influence on the solution accuracy as well as on the required computational cost. Results show that the proposed technique accurately predicts unsteady solutions (maximum prediction error of about 3%) with a computational cost that is around 10% of the time needed to perform the equivalent high fidelity simulations. This performance achievement substantiates the idea that a reduced order model based on unsteady residual minimization is viable replacement for expensive flow solver evaluations in various applications where the computational budget is constrained.

Hybrid MPC System for Platoon based Cooperative Lane change Control Using Machine Learning Aided Distributed Optimization

This study is devoted to developing a platoon-based cooperative lane-change control (PB-CLC). It coordinates the trajectories of a CAV platoon under a platoon-centered platooning control to accommodate the CAV lane-change requests from its adjacent lane, aiming to reduce the negative traffic impacts on the platoon resulting from lane-change maneuvers, on the premise of ensuring CAVs’ safety and mobility. Mathematically, the PB-CLC control is established using a hybrid model predictive control (MPC) system. The hybrid MPC system involves an MPC-based mixed integer nonlinear programming optimizer (MINLP-MPC) for optimal lane-change decisions, which considers multiple objectives such as traffic smoothness, driving comfort and lane-change response promptness subject to vehicle dynamics and safety constraints. To ensure the feasible lane-change, this study investigates and provides a lower bound of the lane-change time window by analyzing the MINLP-MPC model feasibility. Apart from the optimal lane-change decision consideration, the hybrid MPC system is well designed to ensure the control continuity and smoothness. In particular, the hybrid MPC system control feasibility and stability are proved to enable the platoon's back-and-forth state switchings between car-following and lane-change accommodation states. Next, we developed a machine learning aided distributed branch and bound algorithm (ML-DBB) to solve the MINLP-MPC model within a control sampling time interval (< 1 second). Specifically, built upon computer simulation and the c-LHS sampling technique, supervised machine learning models are developed offline to predict a reduced solution space of the integer variables, which is further integrated into the distributed branch and bound method to solve the MINLP-MPC model efficiently online. Extensive numerical experiments validate the effectiveness and applicability of the ML-DBB algorithm and the PB-CLC control.

Autonomous assembly planning of demonstrated skills with reinforcement learning in simulation

Industrial robots used to assemble customized products in small batches require a lot of reprogramming. With this work we aim to reduce the programming complexity by autonomously finding the fastest assembly plans without any collisions with the environment. First, a digital twin of the robot uses a gym in simulation to learn which assembly skills (programmed by demonstration) are physically possible (i.e. no collisions with the environment). Only from this reduced solution space will the physical twin look for the fastest assembly plans. Experiments show that the system indeed converges to the fastest assembly plans. Moreover, pre-training in simulation drastically reduces the number of interactions before convergence compared to directly learning on the physical robot. This two-step procedure allows for the robot to autonomously find correct and fast assembly sequences, without any additional human input or mismanufactured products.

An EIM-degradation free reduced basis method via over collocation and residual hyper reduction-based error estimation

The need for multiple interactive, real-time simulations using different parameter values has driven the design of fast numerical algorithms with certifiable accuracies. The reduced basis method (RBM) presents itself as such an option. RBM features a mathematically rigorous error estimator which drives the construction of a low-dimensional subspace. A surrogate solution is then sought in this low-dimensional space approximating the parameter-induced high fidelity solution manifold. However when the system is nonlinear or its parameter dependence nonaffine, this efficiency gain degrades tremendously, an inherent drawback of the application of the empirical interpolation method (EIM).In this paper, we augment and extend the EIM approach as a direct solver, as opposed to an assistant, for solving nonlinear partial differential equations on the reduced level. The resulting method, called Reduced Over-Collocation method (ROC), is stable and capable of avoiding the efficiency degradation. Two critical ingredients of the scheme are collocation at about twice as many locations as the number of basis elements for the reduced approximation space, and an efficient error indicator for the strategic building of the reduced solution space. The latter, the main contribution of this paper, results from an adaptive hyper reduction of the residuals for the reduced solution. Together, these two ingredients render the proposed R2-ROC scheme both offline- and online-efficient. A distinctive feature is that the efficiency degradation appearing in traditional RBM approaches that utilize EIM for nonlinear and nonaffine problems is circumvented, both in the offline and online stages. Numerical tests on different families of time-dependent and steady-state nonlinear problems demonstrate the high efficiency and accuracy of our R2-ROC and its superior stability performance.

A Blind Spectral Unmixing in Wavelet Domain

In this article, a wavelet-based energy minimization framework is developed for joint estimation of endmembers and abundances without assuming pure pixels while considering noisy scenario. Spectrally dense and overlapped hyperspectral data are represented using biorthogonal wavelet bases that yield a compact linear mixing model in the wavelet domain. It acts as the data term and helps to reduce solution space of the unmixed components. Three prior terms are incorporated to better handle the ill-posedness, i.e., the logarithm of determinant volume regularizer enforces minimum endmember simplex, smoothness (spatial) prior to individual abundance maps, and spectral constraint through learning dictionary of abundances. Alternating nonnegative least-squares is employed to optimize the regularized and constrained nonnegative matrix factorization functional in the wavelet domain. We conduct theoretical analysis, discuss convergence and algorithmic details. Experiments are conducted on synthetic and three real benchmark hyperspectral data AVIRIS Cuprite, HYDICE Urban, and AVIRIS Jasper Ridge. The efficacy of the proposed algorithm is evaluated by comparing results with state of the art.

Solving the Stochastic Generation and Transmission Capacity Planning Problem Applied to Large-Scale Power Systems Using Generalized Shift-Factors

In this study, we successfully develop the transmission planning problem of large-scale power systems based on generalized shift-factors. These distribution factors produce a reduced solution space which does not need the voltage bus angles to model new transmission investments. The introduced formulation copes with the stochastic generation and transmission capacity expansion planning problem modeling the operational problem using a 24-hourly load behaviour. Results show that this formulation achieves an important reduction of decision variables and constraints in comparison with the classical disjunctive transmission planning methodology known as the Big M formulation without sacrificing optimality. We test both the introduced and the Big M formulations to find out convergence and time performance using a commercial solver. Finally, several test power systems and extensive computational experiments are conducted to assess the capacity planning methodology. Solving deterministic and stochastic problems, we demonstrate a prominent reduction in the solver simulation time especially with large-scale power systems.

Lensless compressive sensing with annulus-sector-shaped pixel geometry in the photon-starved environment

To achieve fast compressive object classification and imaging in photon-starved environment, we put forward a novel lensless sensing configuration based on annulus-sector-shaped pixels, where pixels located at different annulus correspond to different temporal profile of photon-counting histogram, thus mapping the spatial distribution features of target to the signatures at the temporal domain. A model is established based on the superposition of different temporal contributions, and solved by exploiting the rotational symmetry of each sector and dramatically reduced solution space. Object classification among 62 classes is conducted with a single-pixel single-photon detector, achieving the classification accuracy of 94.56% at the total operating time of 1 s, corresponding to ~0.82 photons per pixel and a compressive ratio as small as 0.98%. Our method is verified to have great potential in the high-speed photon-limited sensing.

Comparing Corrective and Preventive Security-Constrained DCOPF Problems Using Linear Shift-Factors

This study compares two efficient formulations to solve corrective as well as preventive security-constrained (SC) DC-based optimal power flow (OPF) problems using linear sensitivity factors without sacrificing optimality. Both SCOPF problems are modelled using two frameworks based on these distribution factors. The main advantage of the accomplished formulation is the significant reduction of decision variables and—equality and inequality—constraints in comparison with the traditional DC-based SCOPF formulation. Several test power systems and extensive computational experiments are conducted using a commercial solver to clearly demonstrate the feasibility to carry out the corrective and the preventive SCOPF problems with a reduced solution space. Another point worth noting is the lower simulation time achieved by the introduced methodology. Additionally, this study presents advantages and disadvantages for the proposed shift-factor formulation solving both corrective and preventive formulations.

An adaptive domain-based POD/ECM hyper-reduced modeling framework without offline training

This work presents a reduced-order modeling framework that precludes the need for offline training and adaptively adjusts its lower-order solution space as the analysis progresses. The analysis starts with a fully-solved step and elements are clustered based on their strain response. Elements with the highest strains are solved with a local/global approach in which degrees of freedom from elements undergoing the highest amount of nonlinearity are fully-solved and the rest is approximated by a Proper Orthogonal Decomposition (POD) reduced model with full integration. Elements belonging to the remaining clusters are subjected to a hyper-reduction step using the Empirical Cubature Method (ECM). Online error estimators are used to trigger a retraining process once the reduced solution space becomes inadequate. The performance of the framework is assessed through a series of numerical examples featuring a material model with pressure-dependent plasticity.

Efficiency of the solution representations for the hybrid flow shop scheduling problem with makespan objective

In this paper we address the classical hybrid flow shop scheduling problem with makespan objective. As this problem is known to be NP-hard and a very common layout in real-life manufacturing scenarios, many studies have been proposed in the literature to solve it. These contributions use different solution representations of the feasible schedules, each one with its own advantages and disadvantages. Some of them do not guarantee that all feasible semiactive schedules are represented in the space of solutions –thus limiting in principle their effectiveness– but, on the other hand, these simpler solution representations possess clear advantages in terms of having consistent neighbourhoods with well-defined neighbourhood moves. Therefore, there is a trade-off between the solution space reduction and the ability to conduct an efficient search in this reduced solution space. This trade-off is determined by two aspects, i.e. the extent of the solution space reduction, and the quality of the schedules left aside by this solution space reduction. In this paper, we analyse the efficiency of the different solution representations employed in the literature for the problem. More specifically, we first establish the size of the space of semiactive schedules achieved by the different solution representations and, secondly, we address the issue of the quality of the schedules that can be achieved by these representations using the optimal solutions given by several MILP models and complete enumeration. The results obtained may contribute to design more efficient algorithms for the hybrid flow shop scheduling problem.

A Computation-Efficient Approach for Segment Routing Traffic Engineering

The unprecedented growth of network traffic has brought excessive challenges to network operators. To prevent network congestion, network operators conduct traffic engineering (TE) for their routing optimization. In recent years, segment routing traffic engineering (SRTE) has emerged as one of the promising approaches for its high scalability and low control overheads. However, conventional SRTE approaches in large-scale networks are computationally prohibitive, which may lead to delayed system operations and unsatisfactory service qualities. In this paper, we formulate a bi-objective mixed-integer nonlinear program (BOMINLP) to investigate the trade-off between link utilization and computation time in SRTE. Due to the difficulty in solving the original problem directly, we decompose it into two sequential sub-problems. The first sub-problem is to minimize computation time through node selection, and the second one is to minimize maximum link utilization via flow assignment. To this end, we first employ randomized sampling based on stretch bounding to obtain a reduced solution space and then solve a linear program (LP) using existing software tools for the sub-problems. To evaluate our proposed solution, we employ network topologies and traffic matrices from publicly available datasets. Our simulation results show that our proposed solution can effectively reduce computation time while retaining comparable maximum link utilization as compared with several comparison approaches.

Self-Adaptive Solution-Space Reduction Algorithm for Multi-Objective Evolutionary Design Optimization of Water Distribution Networks

An effective way to improve the computational efficiency of evolutionary algorithms is to make the solution space of the optimization problem under consideration smaller. A new reliability-based algorithm that does this was developed for water distribution networks. The objectives considered in the formulation of the optimization problem were minimization of the initial construction cost and maximization of the flow entropy as a resilience surrogate. After achieving feasible solutions, the active solution space of the optimization problem was re-set for each pipe in each generation until the end of the optimization. The algorithm re-sets the active solution space by reducing the number of pipe diameter options for each pipe, based on the most likely flow distribution. The main components of the methodology include an optimizer, a hydraulic simulator and an algorithm that calculates the flow entropy for any given network configuration. The methodology developed is generic and self-adaptive, and prior setting of the reduced solution space is not required. A benchmark network in the literature was investigated, and the results showed that the algorithm improved the computational efficiency and quality of the solutions achieved by a considerable margin.