## What is/are Active Subspace?

Active Subspace - As for the conventional GAS procedure, the preselection of the truncated wave function is based on the selection of multiple active subspaces while imposing restrictions on the interspace excitations.^{[1]}In this work, we present an extension of the genetic algorithm (GA) which exploits the active subspaces (AS) property to evolve the individuals on a lower dimensional space.

^{[2]}State-of-the-art parameter space dimension reduction methods, such as active subspace, aim to identify a subspace of the original input space that is sufficient to explain the output response.

^{[3]}Proper orthogonal decomposition (POD) and active subspace genetic algorithm (ASGA) are applied for a dimensional reduction of the original (high fidelity) model and for an efficient genetic optimization based on active subspace property.

^{[4]}For this purpose, we firstly appeal to Active Subspace Method (ASM) and develop an ASM-based regional sensitivity analysis, which investigates parametric sensitivity in local regions of the design space and aids conducing to parameters’ intra-sensitivity.

^{[5]}It implements several advanced numerical analysis techniques such as Active Subspaces (AS), Kernel-based Active Subspaces (KAS), and Nonlinear Level-set Learning (NLL) method.

^{[6]}We perform an exploratory analysis of coexistence behavior by approximating active subspaces to identify low-dimensional structure in the optimization criteria, i.

^{[7]}The active subspace method is applied to Burrows–Kurkov supersonic wall-jet flame to quantify modeling uncertainties associated with chemical kinetics, turbulence combustion models, and boundary co.

^{[8]}The problem is formulated as the identification of an unknown number of active subspaces in a large family of subspaces, accounting for the possible positions of potential targets in the delay-Doppler domain: the resulting testing problem is composite and multi-hypothesis.

^{[9]}For fast input dimension reduction, we utilize an approximate global sensitivity measure, for function-valued outputs, motivated by ideas from the active subspace methods.

^{[10]}The active subspace method together with sensitivity analysis is first employed to identify extreme low-dimensional active subspace of input parameter space and to facilitate the construction of response surfaces with small size of samples.

^{[11]}As for the conventional GAS procedure, the preselection of the truncated wave function is based on the selection of multiple active subspaces while imposing restrictions on the interspace excitations.

^{[12]}In this study, a predictive model that combined the active subspace method and the quantitative structure–property relationship (QSPR) method was developed to evaluate the ONs of hydrocarbons.

^{[13]}This is enhanced with a reactive subspace buffer that tracks concept drift occurrences in previously seen classes and adapts clusters accordingly.

^{[14]}Herein, we employ an adaptive active subspace method to efficiently handle the large dimensionality of the design space of a typical PSP-based material design problem within our multi-fidelity BO framework.

^{[15]}An efficient reliability method, termed as the AL-AS-GPR-PDEM, is proposed in this study, which effectively combines the probability density evolution method (PDEM) and the Gaussian process regression (GPR) surrogate model enhanced by both the active subspace (AS)-based dimension reduction and the active learning (AL)-based sampling strategy.

^{[16]}To overcome this difficulty, we propose herein the active subspace-based surrogate model (ASSM) method, which uses the active subspace to transform the high-dimensional input to low-dimensional features and then maps these features to the model target.

^{[17]}Active subspace (AS) method is adopted to identify the important direction of the system-parameter space in neural network, reduce the number of neurons in the input and middle layers, and simplify the network structure.

^{[18]}To overcome these challenges, in this paper, a cutting-edge parameter reduction (PR) approach for WECC CMLD based on active subspace method (ASM) is proposed.

^{[19]}Additionally, one can identify attractive subspaces from the causal-ANN by leveraging the structure of the causal-ANN and the theory of Bayesian Networks.

^{[20]}When the Lipschitz matrix is approximately low-rank, we can perform parameter reduction by constructing a ridge approximation whose active subspace is the span of the dominant eigenvectors of the Lipschitz matrix.

^{[21]}The problem translates itself into finding analytic conditions that characterize invariant and attractive subspaces.

^{[22]}Therefore, computational effort is minimized in this paper by reducing the dimensionality of the input space of the surrogate model by first computing the so-called active subspace.

^{[23]}Motivated by the idea of turbomachinery active subspace performance maps, this paper studies dimension reduction in turbomachinery 3D CFD simulations.

^{[24]}Uncertainty analysis within the CDF matching optimization loop is also investigated, including the identification of active subspaces and the propagation of constructed surrogate model in reduced dimensions.

^{[25]}Here, we show this problem can be defeated by generalizing DMET to use a multiconfigurational wave function as a bath without sacrificing practically attractive features of DMET, such as a second-quantization form of the embedded subsystem Hamiltonian, by dividing the active space into unentangled active subspaces each localized to one fragment.

^{[26]}The active subspace is useful in reducing the dimensionality of the input space of a function.

^{[27]}In this work we use the method of active subspaces to perform a global sensitivity analysis.

^{[28]}In this paper, a cross-pose face recognition algorithm which integrates the regression iterative method and the interactive subspace method was proposed, and through the regression iteration, the target function converges rapidly and important characteristics of the sample were extracted.

^{[29]}Transmitting information without agreeing the common active subspace may incur in a performance loss due to noise enhancement, energy loss and inter-system interference.

^{[30]}temperature, irradiance) with polynomial chaos and active subspace methods.

^{[31]}We also establish a lower bound for the Kolmorogov n-width for parametric LTI systems and illustrate that the method of active subspaces can be viewed as the dual concept to the minimizing subspace for the Kolmogorov n-width.

^{[32]}We employ the active subspaces approach to reduce the dimension of the parameter space, such that building a response surface on the resulting low-dimensional subspace requires many fewer runs of the expensive simulation, rendering the approach suitable for various turbulent combustion models.

^{[33]}Conditions on invariant and attractive subspaces are investigated, which ensure the stabilization of the target state/subspace.

^{[34]}We achieve this by combining Gaussian process machine learning with active subspaces; we then embed this into a parallelized discrete-time dynamic programming algorithm.

^{[35]}In this paper we begin by drawing parallels between ridge subspaces, sufficient dimension reduction and active subspaces; contrasting between techniques rooted in statistical regression to those rooted in approximation theory.

^{[36]}In order to overcome this issue, we adopt a snapshot active subspace method to reduce the input dimension.

^{[37]}In this work we use the method of active subspaces to perform a global sensitivity analysis.

^{[38]}For this, we focus on a convenient special case where we use Gaussian process regression coupled to active subspaces to model agents' forecasts.

^{[39]}The active subspace method and the quantitative structure-property relationship strategy were implemented to predict the cetane numbers of hydrocarbons.

^{[40]}For the first time in karst hydrology, we use the active subspace method to find directions in the parameter space that dominate the Bayesian update from the prior to the posterior distribution in order to effectively reduce the dimension of the problem and for computational efficiency.

^{[41]}In this work, we aim at increasing the precision of the model also for few input solutions by coupling the proper orthogonal decomposition with interpolation (PODI)—a data-driven reduced order method—with the active subspace (AS) property, an emerging tool for reduction in parameter space.

^{[42]}This study presents a matrix completion based approach for big data and large matrices facilitating local smoothness constraints combined with active subspace computing.

^{[43]}Parallels are drawn between our proposed approach, active subspaces and vector-valued dimension reduction.

^{[44]}A recent dimensionality reduction technique that has shown great promise is the method of `active subspaces'.

^{[45]}In this work, we aim at increasing the precision of the model also for few input solutions by coupling the proper orthogonal decomposition with interpolation (PODI) --- a data-driven reduced order method --- with the active subspace (AS) property, an emerging tool for reduction in parameter space.

^{[46]}Furthermore, we introduce a measure for the magnetic relevance of orbitals in the active subspace and a concept for the quantitative comparison of different chemical species.

^{[47]}In order to break the curse, active subspace methods (Constantine et al, J Sci Comput, 2013) and efficient surrogate techniques are used to assess the risk factor of the system, i.

^{[48]}Recent developments in the field of reduced-order modeling---and, in particular, active subspace construction---have made it possible to efficiently approximate complex models by constructing low-o.

^{[49]}In this article, we consider scenarios in which traditional estimates for the active subspace method based on probabilistic Poincar\'e inequalities are not valid due to unbounded Poincar\'e constants.

^{[50]}

## Multiple Active Subspace

As for the conventional GAS procedure, the preselection of the truncated wave function is based on the selection of multiple active subspaces while imposing restrictions on the interspace excitations.^{[1]}As for the conventional GAS procedure, the preselection of the truncated wave function is based on the selection of multiple active subspaces while imposing restrictions on the interspace excitations.

^{[2]}

## active subspace method

For this purpose, we firstly appeal to Active Subspace Method (ASM) and develop an ASM-based regional sensitivity analysis, which investigates parametric sensitivity in local regions of the design space and aids conducing to parameters’ intra-sensitivity.^{[1]}The active subspace method is applied to Burrows–Kurkov supersonic wall-jet flame to quantify modeling uncertainties associated with chemical kinetics, turbulence combustion models, and boundary co.

^{[2]}For fast input dimension reduction, we utilize an approximate global sensitivity measure, for function-valued outputs, motivated by ideas from the active subspace methods.

^{[3]}The active subspace method together with sensitivity analysis is first employed to identify extreme low-dimensional active subspace of input parameter space and to facilitate the construction of response surfaces with small size of samples.

^{[4]}In this study, a predictive model that combined the active subspace method and the quantitative structure–property relationship (QSPR) method was developed to evaluate the ONs of hydrocarbons.

^{[5]}Herein, we employ an adaptive active subspace method to efficiently handle the large dimensionality of the design space of a typical PSP-based material design problem within our multi-fidelity BO framework.

^{[6]}To overcome these challenges, in this paper, a cutting-edge parameter reduction (PR) approach for WECC CMLD based on active subspace method (ASM) is proposed.

^{[7]}In this paper, a cross-pose face recognition algorithm which integrates the regression iterative method and the interactive subspace method was proposed, and through the regression iteration, the target function converges rapidly and important characteristics of the sample were extracted.

^{[8]}temperature, irradiance) with polynomial chaos and active subspace methods.

^{[9]}In order to overcome this issue, we adopt a snapshot active subspace method to reduce the input dimension.

^{[10]}The active subspace method and the quantitative structure-property relationship strategy were implemented to predict the cetane numbers of hydrocarbons.

^{[11]}For the first time in karst hydrology, we use the active subspace method to find directions in the parameter space that dominate the Bayesian update from the prior to the posterior distribution in order to effectively reduce the dimension of the problem and for computational efficiency.

^{[12]}In order to break the curse, active subspace methods (Constantine et al, J Sci Comput, 2013) and efficient surrogate techniques are used to assess the risk factor of the system, i.

^{[13]}In this article, we consider scenarios in which traditional estimates for the active subspace method based on probabilistic Poincar\'e inequalities are not valid due to unbounded Poincar\'e constants.

^{[14]}