Active Subspace Research Articles

NMPC in active subspaces: Dimensionality reduction with recursive feasibility guarantees

Dimensionality reduction of decision variables is a practical and classic method to reduce the computational burden in linear and Nonlinear Model Predictive Control (NMPC). Available results range from early move-blocking ideas to singular-value decomposition. For schemes more complex than move-blocking it is seemingly not straightforward to guarantee recursive feasibility of the receding-horizon optimization. Decomposing the space of decision variables related to the inputs into active and inactive complements, this paper proposes a general framework for effective feasibility-preserving dimensionality reduction in NMPC. We show how – independently of the actual choice of the subspaces – recursive feasibility can be established. Moreover, we propose the use of global sensitivity analysis to construct the active subspace in data-driven fashion based on user-defined criteria. Numerical examples illustrate the efficacy of the proposed scheme. Specifically, for a chemical reactor we obtain a significant reduction by factor 20−40 at a closed-loop performance decay of less than 0.05%.

Localized Active Space-State Interaction: a Multireference Method for Chemical Insight.

Multireference electronic structure methods, like the complete active space (CAS) self-consistent field model, have long been used to characterize chemically interesting processes. Important work has been done in recent years to develop modifications having a lower computational cost than CAS, but typically these methods offer no more chemical insight than that from the CAS solution being approximated. In this paper, we present the localized active space-state interaction (LASSI) method that can be used not only to lower the intrinsic cost of the multireference calculation but also to improve interpretability. The localized active space (LAS) approach utilizes the local nature of the electron-electron correlation to express a composite wave function as an antisymmetrized product of unentangled wave functions in local active subspaces. LASSI then uses these LAS states as a basis from which to express complete molecular wave functions. This not only makes the molecular wave function more compact but also permits flexibility in choosing those states to be included in the basis. Such selective inclusion of states translates to the selective inclusion of specific types of interactions, thereby allowing a quantitative analysis of these interactions. We demonstrate the use of LASSI to study charge migration and spin-flip excitations in multireference organic molecules. We also compute the J coupling parameter for a bimetallic compound using various LAS bases to construct the Hamiltonian to provide insights into the coupling mechanism.

Aerodynamic optimisation of civil aero-engine nacelles by dimensionality reduction and multi-fidelity techniques

PurposeAerodynamic shape optimisation is complex because of the high dimensionality of the problem, the associated non-linearity and its large computational cost. These three aspects have an impact on the overall time of the design process. To overcome these challenges, this paper aims to develop a method for transonic aerodynamic design with dimensionality reduction and multifidelity techniques.Design/methodology/approachThe developed methodology is used for the optimisation of an installed civil ultra-high bypass ratio aero-engine nacelle. As such, the effects of airframe-engine integration are considered during the optimisation routine. The active subspace method is applied to reduce the dimensionality of the problem from 32 to 2 design variables with a database compiled with Euler computational fluid dynamics (CFD) calculations. In the reduced dimensional space, a co-Kriging model is built to combine Euler lower-fidelity and Reynolds-averaged Navier stokes higher-fidelity CFD evaluations.FindingsRelative to a baseline aero-engine nacelle derived from an isolated optimisation process, the proposed method yielded a non-axisymmetric nacelle configuration with an increment in net vehicle force of 0.65% of the nominal standard net thrust.Originality/valueThis work investigates the viability of CFD optimisation through a combination of dimensionality reduction and multifidelity method and demonstrates that the developed methodology enables the optimisation of complex aerodynamic problems.

Kinetic subspace investigation using neural network for uncertainty quantification in nonpremixed flamelets

Propagating uncertainties in kinetic models through turbulent combustion simulations to properly quantify the uncertainties in the simulation results remains a challenging and numerically expensive problem. Efficient approaches have been proposed for certain flames in the flamelet region by reducing their uncertainty input from a high-dimensional kinetic parameter space to a one-dimensional variable. However, this one-dimensional assumption does not apply to all flamelet regimes. In the current work, we developed a systematic approach to discover low-dimensional active subspace reductions that apply to the entire mixture fraction space of the flamelet, and that function even in cases where the uncertainty response is not uniform across the entire solution domain and the one-dimensional assumption does not apply. In doing so, we are able to achieve uncertainty quantification with a tunable tradeoff between high accuracy and low computational cost through careful selection of subspace dimensionality. We facilitated computation in this method using a specifically designed deep neural network based surrogate model to compute the temperature gradients of the flamelet profile to the kinetic parameters. We presented, as a proof-of-concept, a two-stage active subspace reduction on the kinetic parameter space of a nonpremixed methane flamelet. In doing so we demonstrated that its uncertainty response cannot be represented by a one-dimensional kinetic variable due to its uncorrelated behavior across the mixture fraction domain. We instead proposed a four-dimensional active subspace that captures 98% of the uncertainty response in the flame profile at largely reduced computational cost compared to the full kinetic parameter space. The tunability, generality, and reduced computational cost of this method demonstrate its potential to facilitate uncertainty quantification of complex and large-scale combustion problems.

Uncertainty quantification of kinetic models using adjoint-driven active subspace algorithms

Simulations of chemically reacting flows are particularly sensitive to kinetic models, which in turn depend on a large number of parameters. Therefore, quantifying parametric uncertainties is essential in order to assess the applicability and performance of such models. The size of the parameter space renders conventional methods in quantifying uncertainties impractical. The active subspace methodology has proven to be very effective in such applications and has been adopted here to analyze ignition delay time in an isochoric adiabatic reactor. This method has been augmented with an adjoint-based algorithm to provide low-cost access to gradient information, necessary for dimensionality reduction. In addition, the predictions of the active subspace method have been compared to a linear approximation using the adjoint model (LAAM), where depending on the case, a satisfactory estimation is achieved at a fraction of the cost. Moreover, a strategy is presented for constructing a multi-dimensional response surface for cases, where contrary to previous studies, a single dominant direction is insufficient in predicting the corresponding PDF. Finally, the uncertainties with respect to initial conditions are analyzed, highlighting the limitations of the linear approach (LAAM) in cases with strong nonlinearities.

Kernel-based active subspaces with application to computational fluid dynamics parametric problems usingthe discontinuous Galerkin method.

Nonlinear extensions to the active subspaces method have brought remarkable results for dimension reduction in the parameter space and response surface design. We further develop a kernel-based nonlinear method. In particular, we introduce it in a broader mathematical framework that contemplates also the reduction in parameter space of multivariate objective functions. The implementation is thoroughly discussed and tested on more challenging benchmarks than the ones already present in the literature, for which dimension reduction with active subspaces produces already good results. Finally, we show a whole pipeline for the design of response surfaces with the new methodology in the context of a parametric computational fluid dynamics application solved with the discontinuous Galerkin method.

Using active subspace-based similarity analysis for design of combustion experiments

Experimental data under a wide range of conditions are essential for the optimization of combustion kinetic models. However, some laboratory measurements under given conditions may not be conducted due to the constraint of existing techniques. It is thus needed to employ the experimental data obtained under alternate conditions to improve the model predictions for the desired conditions. In this work, an active subspace-based similarity analysis is proposed as an experimental design method to find substitutes for experiments (or measurements) that are difficult to conduct. The eigenvalues and eigenvectors of the matrix that contains the gradient information of a model output with respect to inputs (matrix C of the active subspace) are used to calculate the cosine-based similarity of key reactions of two model targets. The method is demonstrated in three combustion systems, i.e., ignition of hydrogen/oxygen mixture, premixed flame of the dimethyl ether (DME), and C2H6/O2 systems in different reactors. The results show that if the similarity coefficient is large, the key reactions for the two model targets are similar, and the measurement of one target can improve the model prediction of the other target. In addition to designing experimental targets or conditions with strong constraint effects beforehand, this method can also be used to classify potential experimental targets/conditions.

A feature mapping strategy of metamodelling for nonlinear stochastic dynamical systems with low to high-dimensional input uncertainties

This paper deals with the issue on metamodelling (a.k.a. surrogate modelling) of nonlinear stochastic dynamical systems, which are often with multiple input uncertainties Θ∈Rn, viz., the dimension n may range from low to high (e.g., n≥10). In this paper, to circumvent the problem of “curse of dimensionality” of high-dimensional input uncertainties, the feature spaces of outputs and inputs are firstly extracted from the original output and input spaces, and thus a feature mapping strategy is proposed. To form the feature output space, the nonlinear autoregressive with exogenous inputs (NARX) and the proper orthogonal decomposition (POD) are adopted, while the feature input space is detected by the active subspace method (ASM). It is found that the dimension of feature input (output) space may be much less than the one of original input (output) space, thus the applicability of many metamodelling methods can be naturally enhanced. On the constructed input–output feature space, the procedure of metamodelling is completed by the polynomial chaos expansion (PCE) combined with Kriging, which can capture global behaviours as well as local characteristics of the computational model. Two techniques are introduced to accelerate the proposed feature mapping strategy, consisting of the GF-discrepancy minimization algorithm for the design of experiments (DoEs), and the manifold optimization technique for the parameter identification of ASM. Four benchmarks, including a mathematical function (n=2), a dynamical quarter car model (n=10), a Bouc–Wen nonlinear oscillator subjected to earthquake ground motions (n=30), and the first sub-system (as a black box) of the NASA UQ Challenge 2019 (n=100), are studied to demonstrate the accuracy and efficiency of the proposed method. Some problems to be further studied are also outlined.

Learning Active Multimodal Subspaces in the Brain.

Here we introduce a multimodal framework to identify subspaces in the human brain that are defined by collective changes in structural and functional measures and are actively linked to demographic, biological and cognitive indicators in a population. We determine the multimodal subspaces using principles of active subspace learning (ASL) and demonstrate its application on a sample learning task (biological ageing) on a schizophrenia dataset. The proposed multimodal ASL method successfully identifies latent brain representations as subsets of brain regions and connections forming covarying subspaces in association with biological age. We show that schizophrenia is characterized by different subspace patterns compared to those in a cognitively normal brain. The multimodal features generated by projecting structural and functional MRI components onto these active subspaces perform better than several PCA-based transformations and equally well when compared to non-transformed features on the studied learning task. In essence, the proposed method successfully learns active brain subspaces associated with a specific brain condition but inferred from the brain imaging data along with the biological/cognitive traits of interest. Clinical relevance- The work introduces a novel way to create multimodal brain biomarkers based on subspaces computed in association with cognitive or biological traits of interest. These subspaces collectively covary maximally in association with a given trait and successfully retain predictive information.

Assessment of critical species for differential mixing in transported PDF simulations of a non-premixed ethylene DNS flame

The difference in the mixing timescale among species, i.e., differential mixing, is a combined result due to the difference in molecular diffusivity and the difference in reaction-induced scalar gradient. Being a key component for micro-mixing, incorporating differential mixing is a desirable feature for a transported probability density function (TPDF) simulation. Nevertheless, the lack of understanding of differential mixing hinders its model development. This paper intends to address that whether there exist critical species for differential mixing, such that considering only the differential mixing of these species can reproduce the prediction with the full differential mixing of all species. To achieve so, TPDF simulations have been performed for a temporally evolving, non-premixed ethylene DNS flame, with the DNS-extracted mean velocity, turbulent diffusivity and species mixing timescales to enable a clean assessment on the critical species for differential mixing. An algorithm based on active subspace (AS) method is proposed to identify the critical species. It is found that the identified 10 critical species are neither the ones having the highest mixing frequency nor the ones causing the largest temperature change. The TPDF simulation with the differential mixing of the 10 critical species can well reproduce the mean and conditional mean quantities of the reference simulation with the full differential mixing of all species, illustrating the effectiveness of the AS method. Compared to the crude single-species trial method, the AS method better identifies the critical species as it accounts for the coupling nature of the chemical reaction system. By analyzing the reaction pathway in 1D opposed jet flames, it is found that the critical species all involve in the controlling reactions for the overall combustion process, thus providing a physical interpretation for the identified critical species. The effects of quantity of interest and micro-mixing model on critical species are also analyzed.

Dimensionality Reduction, Modelling, and Optimization of Multivariate Problems Based on Machine Learning

Simulation-based optimization design is becoming increasingly important in engineering. However, carrying out multi-point, multi-variable, and multi-objective optimization work is faced with the “Curse of Dimensionality”, which is highly time-consuming and often limited by computational burdens as in aerodynamic optimization problems. In this paper, an active subspace dimensionality reduction method and the adaptive surrogate model were proposed to reduce such computational costs while keeping a high precision. In this method, the active subspace dimensionality reduction technique, three-layer radial basis neural network approach, and polynomial fitting process were presented. For the model evaluation, a NASA standard test function problem and RAE2822 airfoil drag reduction optimization were investigated in the experimental design problem. The efficacy of the method was proved by both the experimental examples in which the adaptive surrogate model in a dominant one-dimensional active subspace is given and the optimization efficiency was improved by two orders. Furthermore, the results show that the constructed surrogate model reduced dimensionality and alleviated the complexity of conventional multivariate surrogate modeling with high precision.

Dimensionality reduction of independent influence factors in the objective evaluation of quality of experience

Big Data analytics and Artificial Intelligence (AI) technologies have become the focus of recent research due to the large amount of data. Dimensionality reduction techniques are recognized as an important step in these analyses. The multidimensional nature of Quality of Experience (QoE) is based on a set of Influence Factors (IFs) whose dimensionality is preferable to be higher due to better QoE prediction. As a consequence, dimensionality issues occur in QoE prediction models. This paper gives an overview of the used dimensionality reduction technique in QoE modeling and proposes modification and use of Active Subspaces Method (ASM) for dimensionality reduction. Proposed modified ASM (mASM) uses variance/standard deviation as a measure of function variability. A straightforward benefit of proposed modification is the possibility of its application in cases when discrete or categorical IFs are included. Application of modified ASM is not restricted to QoE modeling only. Obtained results show that QoE function is mostly flat for small variations of input IFs which is an additional motive to propose a modification of the standard version of ASM. This study proposes several metrics that can be used to compare different dimensionality reduction approaches. We prove that the percentage of function variability described by an appropriate linear combination(s) of input IFs is always greater or equal to the percentage that corresponds to the selection of input IF(s) when the reduction degree is the same. Thus, the proposed method and metrics are useful when optimizing the number of IFs for QoE prediction and a better understanding of IFs space in terms of QoE.

A high-dimension structural reliability method based on active learning Kriging and dimension reduction technique

For efficiently and accurately estimating the failure probability and the further sensitivity index of the high-dimension structures, a novel AS-AK-MCS method is proposed in this work. This method fully employs the merits of the active subspace-based dimension reduction technique, the active learning (AL) Kriging surrogate model, and the Monte Carlo simulation. In the proposed method, the intractable gradient information needed by the active subspace method is obtained by a crude Kriging model with initial training sample points. In the construction of the crude Kriging model, the proposed trend model selection criterion reduces the man-made error. Then the active subspace method converts the reliability analysis from the original high-dimension space into the low-dimension subspace, which facilities constructing an AL-Kriging model and also avoids the tricky “curse of dimension” problem. The state-of-the-art U learning function is applied as the points adding criterion in the active subspace. In order to demonstrate the effectivity and versatility of the proposed method, three representative examples including the linear/nonlinear and the explicit/implicit performance functions are studied for estimating the failure probability and the failure probability-based sensitivity index. Finally, the proposed method is applied to the reliability and sensitivity analysis of a composite radome structure.

Sensitivity Coefficient Evaluation of an Accelerator-Driven System Using ROM-Lasso Method

We propose the use of reduced-order modeling to improve the sensitivity coefficient evaluation method based on Lasso-type penalized linear regression. In this method, cross sections of interest are uniformly randomly sampled, and corresponding perturbed core analyses are performed. The sensitivity coefficients of the higher-dimensional model are expanded by the active subspace (AS) attained by the lower-dimensional model, and the expansion coefficients are estimated by the Lasso regression. In addition, AS bases can be flexibly chosen according to neutronics parameters of interest. We conducted a verification calculation for an accelerator-driven system and clarified that the proposed method successfully reduces the calculation cost by a couple of orders of magnitude compared with the direct method. The proposed method can be used to practically evaluate the sensitivity coefficients of various parameters.

Bi-Criteria Radio Spectrum Sharing With Subspace-Based Pareto Tracing

Radio spectrum is a scarce resource. To meet demands, new wireless technologies must operate in shared spectrum over unlicensed bands (coexist). We consider coexistence of Long-Term Evolution (LTE) License-Assisted Access (LAA) with incumbent Wi-Fi systems. Our scenario consists of multiple LAA and Wi-Fi links sharing an unlicensed band; we aim to simultaneously optimize performance of both coexistence systems. To do this, we present a technique to continuously estimate the Pareto frontier of parameter sets (traces) which approximately maximize all convex combinations of network throughputs over network parameters. We use a dimensionality reduction approach known as active subspaces to determine that this near-optimal parameter set is primarily composed of two physically relevant parameters. A choice of two-dimensional subspace enables visualizations augmenting explainability and the reduced-dimension convex problem results in approximations which dominate random grid search.

Global sensitivity analysis of structural models by active subspace and neural network

PurposeGlobal sensitivity can measure the influence of input variables on model responses and is of positive significance for the improvement design of structural systems. This work aims to study the global sensitivity of structural models by combining the active subspace theory and neural network.Design/methodology/approachThis study aims to improve the efficiency of global sensitivity analysis for high-dimensional structural systems, a novel method based on active subspace and surrogate model is proposed. Active subspace can reduce the dimension of input variables, and an adaptive scaling strategy is proposed to improve the accuracy in finding the active subspace. The uncertainty propagation of active variables and model response is performed through the artificial neural network. Then the global sensitivity analysis is carried out.FindingsSeveral examples are studied by using the Monte Carlo simulation method and the proposed method. Comparison of the results shows that the proposed method has preferable accuracy and low computational cost.Originality/valueThe proposed method provides a practicable tool for the variance-based sensitivity analysis of structural systems. Apart from sensitivity analysis, the method can be also extended for use in other fields relating to uncertainty propagation.

Global sensitivity analysis and uncertainty quantification of soot formation in an n-dodecane spray flame

Quantifying the uncertainty of soot formation in a turbulent combustion simulation remains a challenging task. In this work, global sensitivity analysis and uncertainty quantification of soot formation in the widely known Spray-A flame have been carried out. The numerical simulation reasonably predicts the measured penetration length, ignition delay, flame lift-off length, and soot volume fraction at quasi-steady state, providing a good baseline for the subsequent analysis. The active subspace method has been applied to perform global sensitivity analysis, with the peak soot volume fraction on centerline (PSVF-c) and the axial location of the peak soot volume fraction on centerline (LPSVF-c) at 4 ms being chosen as the quantities of interest (QoIs), and the parameters in soot and turbulence model being considered as the uncertainty inputs. It is found that at the early stage of the spray flame, e.g., 1 ms after the start of injection, more than one dimensional active subspace is needed to characterize the input space, while the active subspace tends to one dimension as the spray flame approaches the quasi-steady state. At the quasi-steady state of the spray flame, surface growth rate parameter is the controlling parameter for PSVF-c. Meanwhile, LPSVF-c is controlled by turbulence model parameters. Uncertainty quantification for soot formation is performed by constructing the response surfaces in the active subspace to map the input parameters to the QoIs. The 95% confidence interval due to the uncertainty of turbulence model, soot model and the overall uncertainty of these two models are [6.64,6.91], [6.59,8.05] and [6.52,8.02] ppm for PSVF-c, and [78.17,81.02], [79.39,80.21] and [78.26,81.22] mm for LPSVF-c. This work demonstrates the potential of using the active subspace method for global sensitivity analysis and uncertainty quantification of the soot formation in turbulent spray flames.

Artificial neural network based response surface for data-driven dimensional analysis

The classical dimensional analysis method has limitations in determining the uniqueness and relative importance of the dimensionless quantities. A machine-learning based dimensional analysis method is proposed to address the limitations. The proposed method identifies unique and relevant dimensionless quantities by combining an artificial neural network with the data-driven dimensional analysis. We employ a fully connected neural network to construct the ridge function for the response surface in a physical system. The gradient of the response surface for active subspace analysis is computed based on a finite difference approximation. An effective approach is proposed to determine the independent variables of experimental measurements or numerical simulations for computing the gradient of the response surface. The proposed method is validated by analyzing benchmark pipe flows and a fluid-structure interaction system. The dominant dimensionless quantities obtained by the proposed method are consistent with those reported in the literature. The proposed method has the advantage of identifying the relatively important dimensionless quantities without referring to the complex theoretical equations.

DIAS: A Data-Informed Active Subspace Regularization Framework for Inverse Problems

This paper presents a regularization framework that aims to improve the fidelity of Tikhonov inverse solutions. At the heart of the framework is the data-informed regularization idea that only data-uninformed parameters need to be regularized, while the data-informed parameters, on which data and forward model are integrated, should remain untouched. We propose to employ the active subspace method to determine the data-informativeness of a parameter. The resulting framework is thus called a data-informed (DI) active subspace (DIAS) regularization. Four proposed DIAS variants are rigorously analyzed, shown to be robust with the regularization parameter and capable of avoiding polluting solution features informed by the data. They are thus well suited for problems with small or reasonably small noise corruptions in the data. Furthermore, the DIAS approaches can effectively reuse any Tikhonov regularization codes/libraries. Though they are readily applicable for nonlinear inverse problems, we focus on linear problems in this paper in order to gain insights into the framework. Various numerical results for linear inverse problems are presented to verify theoretical findings and to demonstrate advantages of the DIAS framework over the Tikhonov, truncated SVD, and the TSVD-based DI approaches.

High dimensional reliability analysis based on combinations of adaptive Kriging and dimension reduction technique

AbstractHigh dimensional reliability analysis is unavoidable if a structural system involving stochastic process because Karhunen–Loève (K‐L) expansion is commonly used. Reliability analysis for structural systems with computationally intensive numerical models and high dimensions is challenging. In this study, an effective high dimensional reliability analysis method is proposed based on principal components and active subspace (PCAS) and active Kriging, and is termed as PCAS‐AK. The proposed method can address two shortcomings existing in PCAS: the latter select training samples randomly and cannot deal with parallel processing. The proposed method, however, selects training samples adaptively, subsequently, the training samples that have high contribution to probability of failure are selected, which can improve computational efficiency. To reduce overall computational time and allow parallelization, K‐weighted‐means combined with learning function are used to select multiple samples at each iteration. After the dimension reduction of both input and output spaces, the samples selected by adaptive method are used to construct Kriging model in the latent low‐dimensional space. Three examples are used to demonstrate the applicability and accuracy of the proposed method. Results show that the proposed method can deal with parallel processing and is, generally, more effective than PCAS for high dimensional reliability problems.