Computing Sensitivities in Evolutionary Systems: A Real-Time Reduced Order Modeling Strategy

Donello, Michael; Carpenter, Mark G.; Babaee, Hessam

doi:10.48550/arxiv.2012.14028

Cited by 3 publications

(6 citation statements)

References 34 publications

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“…heat and mass diffusivities. Most importantly, as it was shown recently [28], f-OTD can be used in solving PDEs for multi-dimensional combustion problems in a cost-effective manner -by exploiting the correlations between the spatiotemporal sensitivities of different species with respect to different parameters. This analysis can be especially insightful for problems containing rare events e.g.…”

Section: Discussionmentioning

confidence: 99%

“…As it was shown in Refs. [28,32], any skew-symmetric choice of matrix ϕ will lead to equivalent f-OTD subspaces. Here we choose ϕ = 0.…”

Section: Modeling the Sensitivity Matrixmentioning

confidence: 99%

“…Model reduction with local SA contains methods such as principal component analysis (PCA) [25,26] and construction of a species ranking [27]. In local SA, the sensitivities are commonly computed via finite difference (FD) discretization, or by directly solving a sensitivity equation (SE), or by an adjoint equation (AE) [28]. The computational cost of using FD or SE, which are forward in time methods, scales linearly with the number of parameters -making them impracticable when sensitivities with respect to a large number of parameters are needed.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, forced optimally time dependent (f-OTD) decomposition was introduced for computing sensitivities in evolutionary systems using a model driven low-rank approximation [28]. This methodology is the extension of OTD decomposition in which a mathematical framework is laid out for the extraction of the low-rank subspace associated with transient instability of the dynamical system [32].…”

Section: Introductionmentioning

confidence: 99%

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Skeletal Model Reduction with Forced Optimally Time Dependent Modes

Nouri¹,

Babaee²,

Givi³

et al. 2021

Preprint

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Sensitivity analysis with forced optimally time dependent (f-OTD) modes is introduced and its application for skeletal model reduction is demonstrated. f-OTD expands the sensitivity coefficient matrix into a lowdimensional, time dependent, orthonormal basis which captures directions of the phase space associated with most dominant sensitivities. These directions highlight the instantaneous active species and reaction paths.Evolution equations for the orthonormal basis and the projections of sensitivity matrix onto the basis are derived, and the application of f-OTD for skeletal reduction is described. In this framework, the sensitivity matrix is modeled, stored in a factorized manner, and never reconstructed at any time during the calculations.For demonstration purposes, sensitivity analysis is conducted of constant pressure ethylene-air burning in a zero-dimensional reactor and new skeletal models are generated. The flame speed, the ignition delay, and the extinction curve of resulted models are compared against some of the existing skeletal models. The results demonstrate the ability of f-OTD approach to eliminate unimportant reactions and species in a systematic, efficient and accurate manner.

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Section: Discussionmentioning

confidence: 99%

“…As it was shown in Refs. [28,32], any skew-symmetric choice of matrix ϕ will lead to equivalent f-OTD subspaces. Here we choose ϕ = 0.…”

Section: Modeling the Sensitivity Matrixmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Skeletal Model Reduction with Forced Optimally Time Dependent Modes

Nouri¹,

Babaee²,

Givi³

et al. 2021

Preprint

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“…The application of TDB is not limited to SPDEs. The TDB has been recently applied to deterministic problems for various applications, for example, reduced description of transient instabilities [36][37][38][39], flow control [40], prediction of extreme events [41], computing sensitivities [42], skeletal model reduction of detailed kinetics [43] as well as reduced-order modeling of passive and reactive species transport [44]. TDB has also been introduced independently in quantum mechanics, chemistry, dynamic low rank matrix and tensor approximations [45][46][47][48].…”

Section: Introductionmentioning

confidence: 99%

Reduced order modeling with time-dependent bases for PDEs with stochastic boundary conditions

Patil¹,

Babaee²

2021

Preprint

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Low-rank approximation using time-dependent bases (TDBs) has proven effective for reduced-order modeling of stochastic partial differential equations (SPDEs). In these techniques, the random field is decomposed to a set of deterministic TDBs and time-dependent stochastic coefficients. When applied to SPDEs with non-homogeneous stochastic boundary conditions (BCs), appropriate BC must be specified for each of the TDBs. However, determining BCs for TDB is not trivial because: (i) the dimension of the random BCs is different than the rank of the TDB subspace; (ii) TDB in most formulations must preserve orthonormality or orthogonality constraints and specifying BCs for TDB should not violate these constraints in the space-discretized form. In this work, we present a methodology for determining the boundary conditions for TDBs at no additional computational cost beyond that of solving the same SPDE with homogeneous BCs. Our methodology is informed by the fact the TDB evolution equations are the optimality conditions of a variational principle. We leverage the same variational principle to derive an evolution equation for the value of TDB at the boundaries. The presented methodology preserves the orthonormality or orthogonality constraints of TDBs. We present the formulation for both the dynamically bi-orthonormal (DBO) decomposition [1] as well as the dynamically orthogonal (DO) decomposition [2]. We show that the presented methodology can be applied to stochastic Dirichlet, Neumann, and Robin boundary conditions. We assess the performance of the presented method for linear advection-diffusion equation, Burgers' equation, and two-dimensional advection-diffusion equation with constant and temperature-dependent conduction coefficient.

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On-the-fly Reduced Order Modeling of Passive and Reactive Species via Time-Dependent Manifolds

Ramezanian,

Nouri,

Babaee

2021

Preprint

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One of the principal barriers in developing accurate and tractable predictive models in turbulent flows with a large number of species is to track every species by solving a separate transport equation, which can be computationally impracticable. In this paper, we present an on-the-fly reduced order modeling of reactive as well as passive transport equations to reduce the computational cost. The presented approach seeks a low-rank decomposition of the species to three time-dependent components: (i) a set of orthonormal spatial modes, (ii) a low-rank factorization of the instantaneous species correlation matrix, and (iii) a set of orthonormal species modes, which represent a low-dimensional time-dependent manifold. Our approach bypasses the need to solve the full-dimensional species to generate high-fidelity data -as it is commonly performed in data-driven dimension reduction techniques such as the principle component analysis. Instead, the low-rank components are directly extracted from the species transport equation. The evolution equations for the three components are obtained from optimality conditions of a variational principle. The time-dependence of the three components enables an on-the-fly adaptation of the low-rank decomposition to transient changes in the species. Several demonstration cases of reduced order modeling of passive and reactive transport equations are presented.

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Computing Sensitivities in Evolutionary Systems: A Real-Time Reduced Order Modeling Strategy

Cited by 3 publications

References 34 publications

Skeletal Model Reduction with Forced Optimally Time Dependent Modes

Skeletal Model Reduction with Forced Optimally Time Dependent Modes

Reduced order modeling with time-dependent bases for PDEs with stochastic boundary conditions

On-the-fly Reduced Order Modeling of Passive and Reactive Species via Time-Dependent Manifolds

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