Pub Date : 2022-07-05DOI: 10.48550/arXiv.2207.02116
C. Cotter, R. Kirby, Hunter Morris
We derive a linearized rotating shallow water system modeling tides, which can be discretized by mixed finite elements. Unlike previous models, this model allows for multiple layers stratified by density. Like the single-layer case~cite{kirby2021preconditioning} a weighted-norm preconditioner gives a (nearly) parameter-robust method for solving the resulting linear system at each time step, but the all-to-all coupling between the layers in the model poses a significant challenge to efficiency. Neglecting the inter-layer coupling gives a preconditioner that degrades rapidly as the number of layers increases. By a careful analysis of the matrix that couples the layers, we derive a robust method that requires solving a reformulated system that only involves coupling between adjacent layers. Numerical results obtained using Firedrake confirm the theory.
{"title":"Weighted-norm preconditioners for a multi-layer tide model","authors":"C. Cotter, R. Kirby, Hunter Morris","doi":"10.48550/arXiv.2207.02116","DOIUrl":"https://doi.org/10.48550/arXiv.2207.02116","url":null,"abstract":"We derive a linearized rotating shallow water system modeling tides, which can be discretized by mixed finite elements. Unlike previous models, this model allows for multiple layers stratified by density. Like the single-layer case~cite{kirby2021preconditioning} a weighted-norm preconditioner gives a (nearly) parameter-robust method for solving the resulting linear system at each time step, but the all-to-all coupling between the layers in the model poses a significant challenge to efficiency. Neglecting the inter-layer coupling gives a preconditioner that degrades rapidly as the number of layers increases. By a careful analysis of the matrix that couples the layers, we derive a robust method that requires solving a reformulated system that only involves coupling between adjacent layers. Numerical results obtained using Firedrake confirm the theory.","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"73 1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83584028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide a unifying framework for $mathcal{L}_2$-optimal reduced-order modeling for linear time-invariant dynamical systems and stationary parametric problems. Using parameter-separable forms of the reduced-model quantities, we derive the gradients of the $mathcal{L}_2$ cost function with respect to the reduced matrices, which then allows a non-intrusive, data-driven, gradient-based descent algorithm to construct the optimal approximant using only output samples. By choosing an appropriate measure, the framework covers both continuous (Lebesgue) and discrete cost functions. We show the efficacy of the proposed algorithm via various numerical examples. Furthermore, we analyze under what conditions the data-driven approximant can be obtained via projection.
{"title":"(boldsymbol{mathcal{L}_2})-Optimal Reduced-Order Modeling Using Parameter-Separable Forms","authors":"Petar Mlinaric, S. Gugercin","doi":"10.1137/22m1500678","DOIUrl":"https://doi.org/10.1137/22m1500678","url":null,"abstract":"We provide a unifying framework for $mathcal{L}_2$-optimal reduced-order modeling for linear time-invariant dynamical systems and stationary parametric problems. Using parameter-separable forms of the reduced-model quantities, we derive the gradients of the $mathcal{L}_2$ cost function with respect to the reduced matrices, which then allows a non-intrusive, data-driven, gradient-based descent algorithm to construct the optimal approximant using only output samples. By choosing an appropriate measure, the framework covers both continuous (Lebesgue) and discrete cost functions. We show the efficacy of the proposed algorithm via various numerical examples. Furthermore, we analyze under what conditions the data-driven approximant can be obtained via projection.","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"134 1","pages":"554-"},"PeriodicalIF":0.0,"publicationDate":"2022-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86822029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-06DOI: 10.48550/arXiv.2206.02882
Q. Cheng, Jie Shen
We develop in this paper two classes of length preserving schemes for the Landau-Lifshitz equation based on two different Lagrange multiplier approaches. In the first approach, the Lagrange multiplier $lambda(bx,t)$ equals to $|nabla m(bx,t)|^2$ at the continuous level, while in the second approach, the Lagrange multiplier $lambda(bx,t)$ is introduced to enforce the length constraint at the discrete level and is identically zero at the continuous level. By using a predictor-corrector approach, we construct efficient and robust length preserving higher-order schemes for the Landau-Lifshitz equation, with the computational cost dominated by the predictor step which is simply a semi-implicit scheme. Furthermore, by introducing another space-independent Lagrange multiplier, we construct energy dissipative, in addition to length preserving, schemes for the Landau-Lifshitz equation, at the expense of solving one nonlinear algebraic equation. We present ample numerical experiments to validate the stability and accuracy for the proposed schemes, and also provide a performance comparison with some existing schemes.
{"title":"Length preserving numerical schemes for Landau-Lifshitz equation based on Lagrange multiplier approaches","authors":"Q. Cheng, Jie Shen","doi":"10.48550/arXiv.2206.02882","DOIUrl":"https://doi.org/10.48550/arXiv.2206.02882","url":null,"abstract":"We develop in this paper two classes of length preserving schemes for the Landau-Lifshitz equation based on two different Lagrange multiplier approaches. In the first approach, the Lagrange multiplier $lambda(bx,t)$ equals to $|nabla m(bx,t)|^2$ at the continuous level, while in the second approach, the Lagrange multiplier $lambda(bx,t)$ is introduced to enforce the length constraint at the discrete level and is identically zero at the continuous level. By using a predictor-corrector approach, we construct efficient and robust length preserving higher-order schemes for the Landau-Lifshitz equation, with the computational cost dominated by the predictor step which is simply a semi-implicit scheme. Furthermore, by introducing another space-independent Lagrange multiplier, we construct energy dissipative, in addition to length preserving, schemes for the Landau-Lifshitz equation, at the expense of solving one nonlinear algebraic equation. We present ample numerical experiments to validate the stability and accuracy for the proposed schemes, and also provide a performance comparison with some existing schemes.","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"62 1","pages":"530-"},"PeriodicalIF":0.0,"publicationDate":"2022-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76296725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work proposes a hyper-reduction method for nonlinear parametric dynamical systems characterized by gradient fields such as Hamiltonian systems and gradient flows. The gradient structure is associated with conservation of invariants or with dissipation and hence plays a crucial role in the description of the physical properties of the system. Traditional hyper-reduction of nonlinear gradient fields yields efficient approximations that, however, lack the gradient structure. We focus on Hamiltonian gradients and we propose to first decompose the nonlinear part of the Hamiltonian, mapped into a suitable reduced space, into the sum of d terms, each characterized by a sparse dependence on the system state. Then, the hyper-reduced approximation is obtained via discrete empirical interpolation (DEIM) of the Jacobian of the derived d-valued nonlinear function. The resulting hyper-reduced model retains the gradient structure and its computationally complexity is independent of the size of the full model. Moreover, a priori error estimates show that the hyper-reduced model converges to the reduced model and the Hamiltonian is asymptotically preserved. Whenever the nonlinear Hamiltonian gradient is not globally reducible, i.e. its evolution requires high-dimensional DEIM approximation spaces, an adaptive strategy is performed. This consists in updating the hyper-reduced Hamiltonian via a low-rank correction of the DEIM basis. Numerical tests demonstrate the applicability of the proposed approach to general nonlinear operators and runtime speedups compared to the full and the reduced models.
{"title":"Gradient-preserving hyper-reduction of nonlinear dynamical systems via discrete empirical interpolation","authors":"C. Pagliantini, Federico Vismara","doi":"10.1137/22M1503890","DOIUrl":"https://doi.org/10.1137/22M1503890","url":null,"abstract":"This work proposes a hyper-reduction method for nonlinear parametric dynamical systems characterized by gradient fields such as Hamiltonian systems and gradient flows. The gradient structure is associated with conservation of invariants or with dissipation and hence plays a crucial role in the description of the physical properties of the system. Traditional hyper-reduction of nonlinear gradient fields yields efficient approximations that, however, lack the gradient structure. We focus on Hamiltonian gradients and we propose to first decompose the nonlinear part of the Hamiltonian, mapped into a suitable reduced space, into the sum of d terms, each characterized by a sparse dependence on the system state. Then, the hyper-reduced approximation is obtained via discrete empirical interpolation (DEIM) of the Jacobian of the derived d-valued nonlinear function. The resulting hyper-reduced model retains the gradient structure and its computationally complexity is independent of the size of the full model. Moreover, a priori error estimates show that the hyper-reduced model converges to the reduced model and the Hamiltonian is asymptotically preserved. Whenever the nonlinear Hamiltonian gradient is not globally reducible, i.e. its evolution requires high-dimensional DEIM approximation spaces, an adaptive strategy is performed. This consists in updating the hyper-reduced Hamiltonian via a low-rank correction of the DEIM basis. Numerical tests demonstrate the applicability of the proposed approach to general nonlinear operators and runtime speedups compared to the full and the reduced models.","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"69 1","pages":"2725-"},"PeriodicalIF":0.0,"publicationDate":"2022-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90891939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Well-Balanced Asymptotic Preserving Scheme for the Two-Dimensional Rotating Shallow Water Equations with Nonflat Bottom Topography","authors":"A. Kurganov, Yongle Liu, M. Lukácová-Medvidová","doi":"10.1137/21m141573x","DOIUrl":"https://doi.org/10.1137/21m141573x","url":null,"abstract":"","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"3 1","pages":"1655-"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78910637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Gaussian Process Subspace Prediction for Model Reduction","authors":"Ruda Zhang, Simon Mak, D. Dunson","doi":"10.1137/21m1432739","DOIUrl":"https://doi.org/10.1137/21m1432739","url":null,"abstract":"","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"35 1","pages":"1428-"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85992984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we present a new family of semi-discrete and fully-discrete finite volume schemes for overdetermined, hyperbolic and thermodynamically compatible PDE systems. In the following we will denote these methods as HTC schemes. In particular, we consider the Euler equations of compressible gasdynamics, as well as the more complex Godunov-Peshkov-Romenski (GPR) model of continuum mechanics, which, at the aid of suitable relaxation source terms, is able to describe nonlinear elasto-plastic solids at large deformations as well as viscous fluids as two special cases of a more general first order hyperbolic model of continuum mechanics. The main novelty of the schemes presented in this paper lies in the fact that we solve the textit{entropy inequality} as a primary evolution equation rather than the usual total energy conservation law. Instead, total energy conservation is achieved as a mere consequence of a thermodynamically compatible discretization of all the other equations. For this, we first construct a discrete framework for the compressible Euler equations that mimics the continuous framework of Godunov's seminal paper textit{An interesting class of quasilinear systems} of 1961 textit{exactly} at the discrete level. All other terms in the governing equations of the more general GPR model, including non-conservative products, are judiciously discretized in order to achieve discrete thermodynamic compatibility, with the exact conservation of total energy density as a direct consequence of all the other equations. As a result, the HTC schemes proposed in this paper are provably marginally stable in the energy norm and satisfy a discrete entropy inequality by construction. We show some computational results obtained with HTC schemes in one and two space dimensions, considering both the fluid limit as well as the solid limit of the governing partial differential equations.
{"title":"On thermodynamically compatible finite volume schemes for continuum mechanics","authors":"S. Busto, M. Dumbser, I. Peshkov, E. Romenski","doi":"10.1137/21M1417508","DOIUrl":"https://doi.org/10.1137/21M1417508","url":null,"abstract":"In this paper we present a new family of semi-discrete and fully-discrete finite volume schemes for overdetermined, hyperbolic and thermodynamically compatible PDE systems. In the following we will denote these methods as HTC schemes. In particular, we consider the Euler equations of compressible gasdynamics, as well as the more complex Godunov-Peshkov-Romenski (GPR) model of continuum mechanics, which, at the aid of suitable relaxation source terms, is able to describe nonlinear elasto-plastic solids at large deformations as well as viscous fluids as two special cases of a more general first order hyperbolic model of continuum mechanics. The main novelty of the schemes presented in this paper lies in the fact that we solve the textit{entropy inequality} as a primary evolution equation rather than the usual total energy conservation law. Instead, total energy conservation is achieved as a mere consequence of a thermodynamically compatible discretization of all the other equations. For this, we first construct a discrete framework for the compressible Euler equations that mimics the continuous framework of Godunov's seminal paper textit{An interesting class of quasilinear systems} of 1961 textit{exactly} at the discrete level. All other terms in the governing equations of the more general GPR model, including non-conservative products, are judiciously discretized in order to achieve discrete thermodynamic compatibility, with the exact conservation of total energy density as a direct consequence of all the other equations. As a result, the HTC schemes proposed in this paper are provably marginally stable in the energy norm and satisfy a discrete entropy inequality by construction. We show some computational results obtained with HTC schemes in one and two space dimensions, considering both the fluid limit as well as the solid limit of the governing partial differential equations.","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"47 6 1","pages":"1723-"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73005009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Chattopadhyay, Marek Cygan, Noga Ron-Zewi, Christian Wulff-Nilsen
{"title":"Special Section on the Fifty-Second Annual ACM Symposium on the Theory of Computing (STOC 2020)","authors":"A. Chattopadhyay, Marek Cygan, Noga Ron-Zewi, Christian Wulff-Nilsen","doi":"10.1137/22n975494","DOIUrl":"https://doi.org/10.1137/22n975494","url":null,"abstract":"","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"60 25 1","pages":"20-"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78728043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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