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## POD/DEIM Reduced-Order Strategies for Efficient Four Dimensional Variational Data Assimilation

Venue: | Virginia Polytechnic Institute and State University |

Citations: | 2 - 1 self |

### Citations

622 |
der Vorst. Templates for the Solution of Linear Systems
- Barret, Berry, et al.
- 1993
(Show Context)
Citation Context .... For operations with sparse matrices we employed SPARSEKIT library Saad [87] and the sparse linear systems obtained during the quasi-Newton iterations were solved using MGMRES library Barrett et al. =-=[14]-=-, Kelley [62], Saad [88]. Here we did not decouple the model equations as in Stefanescu and Navon [32] where the Jacobian is either block cyclic tridiagonal or block tridiagonal. By keeping all discre... |

274 |
New Approach to Variable Metric Algorithms,”
- Fletcher, “A
- 1970
(Show Context)
Citation Context ...d optimality conditions (18) is a surrogate model that accurately represents the corresponding full order condition (accuracy). Iterative methods such as Broyden–Fletcher-âĂŞGoldfarbâĂŞ-Shanno (BFGS) =-=[21, 44, 45, 91]-=- uses feasible triplets (x,λ,∇x0) to solve (1), in the following sense Definition 4.2 The triplet (x,λ,∇x0) ∈ RNstate×N × RNstate×N × RNstate is said to be KKT-feasible if x is the solution of the for... |

208 |
The convergence of a class of double-rank minimization algorithms.
- Broyden
- 1970
(Show Context)
Citation Context ...d optimality conditions (18) is a surrogate model that accurately represents the corresponding full order condition (accuracy). Iterative methods such as Broyden–Fletcher-âĂŞGoldfarbâĂŞ-Shanno (BFGS) =-=[21, 44, 45, 91]-=- uses feasible triplets (x,λ,∇x0) to solve (1), in the following sense Definition 4.2 The triplet (x,λ,∇x0) ∈ RNstate×N × RNstate×N × RNstate is said to be KKT-feasible if x is the solution of the for... |

195 |
An ‘empirical interpolation’ method: Application to efficient reduced-basis discretization of partial differential equations,
- Barrault, Maday, et al.
- 2004
(Show Context)
Citation Context ...llem et al. [8] used a two-step gappy POD procedure to decrease the computational complexity of the reduced nonlinear terms in the solution of shape optimization problems. Reduced basis approximation =-=[13, 47, 78, 86, 38]-=- is known to be very efficient for parameterized problems and has recently been applied in the context of reduced order optimization [48, 70, 74, 85]. This paper develops a systematic approach to POD ... |

173 | An introduction to estimation theory.
- Cohn
- 1997
(Show Context)
Citation Context ...o ti+1. The state variables are xi ∈ RNstate and the data (observation) values are yi ∈ RNobs at times ti, i = 0, .., N . These observations are corrupted by instruments and representativeness errors =-=[31]-=- which are assumed to have a normal distribution N (0,Ri), where Ri ∈ RNobs×Nobs describes the observation error covariance matrix at time ti. The nonlinear observation operator Hi : RNstate → RNobs m... |

95 |
A Posteriori Error Bounds for Reduced-Basis Approximations of Parametrized Parabolic Partial Differential Equations,
- Grepl, Patera
- 2005
(Show Context)
Citation Context ...llem et al. [8] used a two-step gappy POD procedure to decrease the computational complexity of the reduced nonlinear terms in the solution of shape optimization problems. Reduced basis approximation =-=[13, 47, 78, 86, 38]-=- is known to be very efficient for parameterized problems and has recently been applied in the context of reduced order optimization [48, 70, 74, 85]. This paper develops a systematic approach to POD ... |

83 |
Nonlinear model reduction via discrete empirical interpolation.
- Chaturantabut, Sorensen
- 2010
(Show Context)
Citation Context ...polation points, V ∈ RNstate×m gathers the first m POD basis modes of the nonlinear term while P ∈ RNstate×m is the DEIM interpolation selection matrix (Chaturantabut [26], Chaturantabut and Sorensen =-=[27, 28]-=-). The systematic application of these techniques in the Petrov-Galerkin projection framework to (1b) leads to the following reduced order forward model x̃i+1 = M̃i,i+1 (x̃i) , M̃i,i+1 (x̃i) = W TMi,i... |

64 |
Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: Error estimates and suboptimal control
- Hinze, Volkwei
(Show Context)
Citation Context ...], Leibfritz and Volkwein [71], Sachs and Volkwein [89]. Other studies proposed to include time derivatives, nonlinear terms, and adjoint information (see Diwoky and Volkwein [40], Hinze and Volkwein =-=[52]-=-, Hinze [51], Gubisch and Volkwein [48], Hay et al. [50], Carlberg and Farhat [24], Zahr and Farhat [100]) into POD basis for reduced order optimization purposes. A-posteriori analysis for POD applied... |

63 |
A family of variable-metric method derived by variational means",
- Goldfarb
- 1970
(Show Context)
Citation Context ...d optimality conditions (18) is a surrogate model that accurately represents the corresponding full order condition (accuracy). Iterative methods such as Broyden–Fletcher-âĂŞGoldfarbâĂŞ-Shanno (BFGS) =-=[21, 44, 45, 91]-=- uses feasible triplets (x,λ,∇x0) to solve (1), in the following sense Definition 4.2 The triplet (x,λ,∇x0) ∈ RNstate×N × RNstate×N × RNstate is said to be KKT-feasible if x is the solution of the for... |

60 | Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations,”
- Atwell, King
- 2001
(Show Context)
Citation Context ... formulation of the cost function is employed in the “adjoint of reduced” case [39]. In the control literature these approaches are referred to as “design-then-reduce” and “reduce-thendesign” methods =-=[11, 12]-=-. The “reduced of adjoint” approach avoids the implementation of the adjoint of the tangent linear approximation of the original nonlinear model by replacing it with a reduced adjoint. This is obtaine... |

50 |
A reduced order approach to fourdimensional variational data assimilation using proper orthogonal decomposition
- Cao, Zhu, et al.
(Show Context)
Citation Context ...stems is available in Lass and Volkwein [69]. POD was successfully applied to solve strong constraint four dimensional variational (4D-Var) data assimilation problems for oceanic problems (Cao et al. =-=[23]-=-, Fang et al. [43]) and atmospheric models (Chen et al. [30, 29], Daescu and Navon [36, 37], Du et al. [41]). A strategy that formulates first order optimality conditions starting from POD models has ... |

42 | Trust-region proper orthogonal decomposition for optimal flow control
- Arian, Fahl, et al.
- 2000
(Show Context)
Citation Context ...in [65], Bergmann et al. [19], Ravindran [83], Yue and Meerbergen [99], Zahr and Farhat [100], Zahr et al. [101] proposed to update the basis according to the current optimal control. In Arian et al. =-=[9]-=-, Bergmann and Cordier [18], this updating technique was combined with a trust region (TR) strategy to determine whether after an optimization step an update of the POD-basis should be performed. Addi... |

41 |
Efficient Nonlinear Model Reduction Via a Least-Squares Petrov–Galerkin Projection and Compressive Tensor Approximations,”
- Carlberg, Bou-Mosleh, et al.
- 2011
(Show Context)
Citation Context ...all-scale convex optimization problem [7]. In addition we can constrain the construction of the left basis UTa to minimize a norm of the residual arising at each Newton iteration to promote stability =-=[25]-=-. It will be worth checking how this updated basis will affect the accuracy of the reduced adjoint model and the reduced gradient. An elegant solution to avoid stability issues while maintaining the c... |

39 | Optimal control of the cylinder wake in the laminar regime by trust-region methods and pod reduced-order models
- Bergmann, Cordier
(Show Context)
Citation Context ...19], Ravindran [83], Yue and Meerbergen [99], Zahr and Farhat [100], Zahr et al. [101] proposed to update the basis according to the current optimal control. In Arian et al. [9], Bergmann and Cordier =-=[18]-=-, this updating technique was combined with a trust region (TR) strategy to determine whether after an optimization step an update of the POD-basis should be performed. Additional work on TR/POD prove... |

37 |
Adaptive control of a wake flow using proper orthogonal decomposition
- Afanasiev, Hinze
- 2001
(Show Context)
Citation Context ...l POD based reduced optimization should represent correctly the dynamics of the flow that is altered by the controller. To overcome the problem of unmodelled dynamics in the basis Afanasiev and Hinze =-=[2]-=-, Ravindran [84], Kunisch and Volkwein [65], Bergmann et al. [19], Ravindran [83], Yue and Meerbergen [99], Zahr and Farhat [100], Zahr et al. [101] proposed to update the basis according to the curre... |

30 |
Optimal rotary control of the cylinder wake using proper orthogonal decomposition reduced-order model
- Bergmann, Cordier, et al.
- 2005
(Show Context)
Citation Context ...dynamics of the flow that is altered by the controller. To overcome the problem of unmodelled dynamics in the basis Afanasiev and Hinze [2], Ravindran [84], Kunisch and Volkwein [65], Bergmann et al. =-=[19]-=-, Ravindran [83], Yue and Meerbergen [99], Zahr and Farhat [100], Zahr et al. [101] proposed to update the basis according to the current optimal control. In Arian et al. [9], Bergmann and Cordier [18... |

30 |
Bloemen Waanders. Goal-oriented, modelconstrained optimization for reduction of large-scale systems
- Bui-Thanh, Willcox, et al.
(Show Context)
Citation Context ...e pure Petrov-Galerkin projection does not guarantee the stability of the ARRA reduced order forward model (18a). The stability of the reduced pencil is not guaranteed, even when the pencil is stable =-=[22]-=-. In our numerical experiments using shallow water equation model, the ARRA approach exhibited large instabilities in the solution of the forward reduced ordered model when the initial conditions were... |

29 | Reduced order controllers for spatially distributed systems via proper orthogonal decomposition
- Atwell, King
- 2004
(Show Context)
Citation Context ... formulation of the cost function is employed in the “adjoint of reduced” case [39]. In the control literature these approaches are referred to as “design-then-reduce” and “reduce-thendesign” methods =-=[11, 12]-=-. The “reduced of adjoint” approach avoids the implementation of the adjoint of the tangent linear approximation of the original nonlinear model by replacing it with a reduced adjoint. This is obtaine... |

28 | A dual-weighted approach to order reduction in 4D-Var data assimilation. Monthly Weather Review
- Daescu, Navon
- 2007
(Show Context)
Citation Context ...ong constraint four dimensional variational (4D-Var) data assimilation problems for oceanic problems (Cao et al. [23], Fang et al. [43]) and atmospheric models (Chen et al. [30, 29], Daescu and Navon =-=[36, 37]-=-, Du et al. [41]). A strategy that formulates first order optimality conditions starting from POD models has been implemented in 4D-Var systems in Vermeulen and Heemink [95], Sava [90], while hybrid m... |

24 | 2009 Enablers for robust POD models
- Bergmann, Bruneau, et al.
(Show Context)
Citation Context ...ledge, the reduced order models with a pressure component use two strategies. In the decoupled approach [1, 77], the velocity and pressure snapshots are considered separately. In the coupled approach =-=[20, 97]-=-, each snapshot and POD mode have both velocity and the corresponding pressure component. In our manuscript we followed the decoupled approach to build our reduced order models. Moreover, we also deco... |

17 |
A state space error estimate for POD-DEIM nonlinear model reduction
- Chaturantabut, Sorensen
(Show Context)
Citation Context ...polation points, V ∈ RNstate×m gathers the first m POD basis modes of the nonlinear term while P ∈ RNstate×m is the DEIM interpolation selection matrix (Chaturantabut [26], Chaturantabut and Sorensen =-=[27, 28]-=-). The systematic application of these techniques in the Petrov-Galerkin projection framework to (1b) leads to the following reduced order forward model x̃i+1 = M̃i,i+1 (x̃i) , M̃i,i+1 (x̃i) = W TMi,i... |

14 | Mesh refinement and numerical sensitivity analysis for parameter calibration of partial differential equations
- Becker, Vexler
- 2005
(Show Context)
Citation Context ...y justify how the projection errors in the three optimality equations impact the optimal solution. A more rigorous argument can be made using the aposteriori error estimation methodology developed in =-=[3, 4, 16, 81]-=-. The full order KKT equations (5) form a large system of nonlinear equations, written abstractly as F(ζa) = 0, (20) where ζa = (xa,λa) ∈ RNstate×(N+1) × RNstate×(N+1) is obtained by running the forwa... |

14 |
Dimension Reduction for Unsteady Nonlinear Partial Differential Equations via Empirical Interpolation Methods
- Chaturantabut
- 2008
(Show Context)
Citation Context ...8) where m is the number of interpolation points, V ∈ RNstate×m gathers the first m POD basis modes of the nonlinear term while P ∈ RNstate×m is the DEIM interpolation selection matrix (Chaturantabut =-=[26]-=-, Chaturantabut and Sorensen [27, 28]). The systematic application of these techniques in the Petrov-Galerkin projection framework to (1b) leads to the following reduced order forward model x̃i+1 = M̃... |

13 | POD/DEIM Nonlinear model order reduction of an ADI implicit shallow water equations model
- Stefanescu, Navon
(Show Context)
Citation Context ...n this paper: standard POD (SPOD), tensorial POD (TPOD), and standard POD/Discrete Empirical Interpolation Method 5 (POD/DEIM), which were described in Ştefănescu et al. [35] and Ştefănescu and Navon =-=[32]-=-. The reduced Jacobians required by the ADI schemes are obtained analytically for all three ROMs via tensorial calculus and their computational complexity depends only on k, the dimension of POD basis... |

12 | A dual weighted trust-region adaptive POD 4D-Var applied to a Finite -Element Shallow water Equations
- Chen, Navon, et al.
- 2011
(Show Context)
Citation Context ...sfully applied to solve strong constraint four dimensional variational (4D-Var) data assimilation problems for oceanic problems (Cao et al. [23], Fang et al. [43]) and atmospheric models (Chen et al. =-=[30, 29]-=-, Daescu and Navon [36, 37], Du et al. [41]). A strategy that formulates first order optimality conditions starting from POD models has been implemented in 4D-Var systems in Vermeulen and Heemink [95]... |

12 |
A POD reduced order 4D-Var adaptive mesh ocean modelling approach
- Fang, Pain, et al.
(Show Context)
Citation Context ... in Lass and Volkwein [69]. POD was successfully applied to solve strong constraint four dimensional variational (4D-Var) data assimilation problems for oceanic problems (Cao et al. [23], Fang et al. =-=[43]-=-) and atmospheric models (Chen et al. [30, 29], Daescu and Navon [36, 37], Du et al. [41]). A strategy that formulates first order optimality conditions starting from POD models has been implemented i... |

12 |
Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition
- Hay, Borggaard, et al.
(Show Context)
Citation Context ... Other studies proposed to include time derivatives, nonlinear terms, and adjoint information (see Diwoky and Volkwein [40], Hinze and Volkwein [52], Hinze [51], Gubisch and Volkwein [48], Hay et al. =-=[50]-=-, Carlberg and Farhat [24], Zahr and Farhat [100]) into POD basis for reduced order optimization purposes. A-posteriori analysis for POD applied to optimal control problems governed by parabolic and e... |

10 |
A low-cost, goal-oriented compact proper orthogonal decomposition basis for model reduction of static systems
- Carlberg, Farhat
(Show Context)
Citation Context ... include time derivatives, nonlinear terms, and adjoint information (see Diwoky and Volkwein [40], Hinze and Volkwein [52], Hinze [51], Gubisch and Volkwein [48], Hay et al. [50], Carlberg and Farhat =-=[24]-=-, Zahr and Farhat [100]) into POD basis for reduced order optimization purposes. A-posteriori analysis for POD applied to optimal control problems governed by parabolic and elliptic PDEs were develope... |

10 |
A survey of finite difference schemes for the primitive equations for a barotropic fluid
- Grammeltvedt
- 1969
(Show Context)
Citation Context ...e computational performances of the three proposed reduced order SWE data assimilation systems. For all tests we derived the initial conditions from the initial height condition No. 1 of Grammeltvedt =-=[46]-=- i.e. h(x, y) = H0 +H1 + tanh ( 9 D/2− y 2D ) +H2sech2 ( 9 D/2− y 2D ) sin ( 2pix L ) , 0 ≤ x ≤ L, 0 ≤ y ≤ D. The initial velocity fields were derived from the initial height field using the geostroph... |

9 |
An alternating direction implicit method for solving the shallow water equations
- Gustafsson
- 1971
(Show Context)
Citation Context ...ation of Rossby and Kelvin waves in the atmosphere, rivers, lakes and oceans as well as gravity waves in a smaller domain. The alternating direction fully implicit finite difference scheme Gustafsson =-=[49]-=- was considered in this paper and it is stable for large CFL condition numbers (we tested the stability of the scheme for a CFL condition number equal up to 8.9301). We refer to Fairweather and Navon ... |

8 |
Stabilization of projection-based reduced-order models
- Amsallem, Farhat
(Show Context)
Citation Context ...y. Future work will consider stabilization approaches such as posing the problem of selecting the reduced bases as a goal-oriented optimization problem [22] or small-scale convex optimization problem =-=[7]-=-. In addition we can constrain the construction of the left basis UTa to minimize a norm of the residual arising at each Newton iteration to promote stability [25]. It will be worth checking how this ... |

7 |
A linear ADI method for the shallow water equations
- Fairweather, Navon
- 1980
(Show Context)
Citation Context ... was considered in this paper and it is stable for large CFL condition numbers (we tested the stability of the scheme for a CFL condition number equal up to 8.9301). We refer to Fairweather and Navon =-=[42]-=-, Navon and Villiers [75] for other research work on this topic. The SWE model using the β-plane approximation on a rectangular domain is introduced (see Gustafsson [49]) ∂w ∂t = A(w) ∂w ∂x +B(w) ∂w ∂... |

6 |
Space-time adaptive solution of inverse problems with the discrete adjoint method. Inverse Probl. 2011, submitted for publication
- Alexe, Sandu
(Show Context)
Citation Context ...y justify how the projection errors in the three optimality equations impact the optimal solution. A more rigorous argument can be made using the aposteriori error estimation methodology developed in =-=[3, 4, 16, 81]-=-. The full order KKT equations (5) form a large system of nonlinear equations, written abstractly as F(ζa) = 0, (20) where ζa = (xa,λa) ∈ RNstate×(N+1) × RNstate×(N+1) is obtained by running the forwa... |

6 | Numerical simulations with data assimilation using an adaptive POD procedure
- Dimitriu, Apreutesei, et al.
(Show Context)
Citation Context ...hes for the optimal solution by solving a collection of reduced optimization problems in the reduced space. Reduced order formulation of the cost function is employed in the “adjoint of reduced” case =-=[39]-=-. In the control literature these approaches are referred to as “design-then-reduce” and “reduce-thendesign” methods [11, 12]. The “reduced of adjoint” approach avoids the implementation of the adjoin... |

5 | Proper Orthogonal Decomposition for LinearQuadratic Optimal Control. Submitted
- Gubisch, Volkwein
- 2013
(Show Context)
Citation Context ...nd Volkwein [89]. Other studies proposed to include time derivatives, nonlinear terms, and adjoint information (see Diwoky and Volkwein [40], Hinze and Volkwein [52], Hinze [51], Gubisch and Volkwein =-=[48]-=-, Hay et al. [50], Carlberg and Farhat [24], Zahr and Farhat [100]) into POD basis for reduced order optimization purposes. A-posteriori analysis for POD applied to optimal control problems governed b... |

4 |
Efficiency of a POD-based reduced second order adjoint model in 4-D VAR data assimilation
- Daescu, Navon
- 2007
(Show Context)
Citation Context ...to solve strong constraint four dimensional variational (4D-Var) data assimilation problems for oceanic Cao et al. [13], Fang et al. [26] and atmospheric models Chen et al. [18, 17], Daescu and Navon =-=[19, 20]-=-, Du et al. [24]. A strategy that formulates first order optimality conditions starting from POD models has been implemented in 4D-Var systems in Vermeulen and Heemink [64], Sava [58], while hybrid me... |

4 |
Certified PDE-constrained parameter optimization using reduced basis surrogate models for evolution problems. Submitted to the
- Dihlmann, Haasdonk
(Show Context)
Citation Context ...llem et al. [8] used a two-step gappy POD procedure to decrease the computational complexity of the reduced nonlinear terms in the solution of shape optimization problems. Reduced basis approximation =-=[13, 47, 78, 86, 38]-=- is known to be very efficient for parameterized problems and has recently been applied in the context of reduced order optimization [48, 70, 74, 85]. This paper develops a systematic approach to POD ... |

3 |
Adjoint-based space-time adaptive solution algorithms for sensitivity analysis and inverse problems
- Alexe
- 2011
(Show Context)
Citation Context ...y justify how the projection errors in the three optimality equations impact the optimal solution. A more rigorous argument can be made using the aposteriori error estimation methodology developed in =-=[3, 4, 16, 81]-=-. The full order KKT equations (5) form a large system of nonlinear equations, written abstractly as F(ζa) = 0, (20) where ζa = (xa,λa) ∈ RNstate×(N+1) × RNstate×(N+1) is obtained by running the forwa... |

3 | Comparisons of POD reduced order strategies for the nonlinear 2D Shallow Water Equations Model
- Stefanescu, Sandu, et al.
(Show Context)
Citation Context ...e compare three reduced 4D-Var data assimilation systems using three different POD Galerkin based reduced order methods namely standard POD, tensorial POD and standard POD/DEIM (see Ştefănescu et al. =-=[35]-=-). For Petrov-Galerkin projection stabilization strategies have to be considered. To the best of our knowledge this is the first application of POD/DEIM to obtain suboptimal solutions of reduced data ... |

3 |
Nonlinear boundary control for the heat equation utilizing proper orthogonal decomposition
- Diwoky, Volkwein
(Show Context)
Citation Context ... Bergmann and Cordier [17], Leibfritz and Volkwein [71], Sachs and Volkwein [89]. Other studies proposed to include time derivatives, nonlinear terms, and adjoint information (see Diwoky and Volkwein =-=[40]-=-, Hinze and Volkwein [52], Hinze [51], Gubisch and Volkwein [48], Hay et al. [50], Carlberg and Farhat [24], Zahr and Farhat [100]) into POD basis for reduced order optimization purposes. A-posteriori... |

2 |
Design Optimization Using HyperReduced-Order Models
- Amsallem, Zahr, et al.
- 2013
(Show Context)
Citation Context ...g in full space 2 were introduced in Altaf et al. [5] and Ambrozic [6]. POD/DEIM has been employed to reduce the CPU complexity of a 1D Burgers 4D-Var system in Baumann [15]. Recently Amsallem et al. =-=[8]-=- used a two-step gappy POD procedure to decrease the computational complexity of the reduced nonlinear terms in the solution of shape optimization problems. Reduced basis approximation [13, 47, 78, 86... |

2 |
Nonlinear Model Order Reduction using POD/DEIM for Optimal Control of Burgersquation
- Baumann
- 2013
(Show Context)
Citation Context ...ed adjoint models but optimizing in full space 2 were introduced in Altaf et al. [5] and Ambrozic [6]. POD/DEIM has been employed to reduce the CPU complexity of a 1D Burgers 4D-Var system in Baumann =-=[15]-=-. Recently Amsallem et al. [8] used a two-step gappy POD procedure to decrease the computational complexity of the reduced nonlinear terms in the solution of shape optimization problems. Reduced basis... |

1 |
A numerical investigation of velocityâĂŞpressure reduced order models for incompressible flows
- Caiazzo, Iliescu, et al.
- 2014
(Show Context)
Citation Context ...ssure variable is replaced by the height of the surface topography [96]. To the best of our knowledge, the reduced order models with a pressure component use two strategies. In the decoupled approach =-=[1, 77]-=-, the velocity and pressure snapshots are considered separately. In the coupled approach [20, 97], each snapshot and POD mode have both velocity and the corresponding pressure component. In our manusc... |

1 |
A reduced adjoint approach to variational data assimilation
- Altaf, Gharamti, et al.
(Show Context)
Citation Context ...odels has been implemented in 4D-Var systems in Vermeulen and Heemink [95], Sava [90], while hybrid methods using reduced adjoint models but optimizing in full space 2 were introduced in Altaf et al. =-=[5]-=- and Ambrozic [6]. POD/DEIM has been employed to reduce the CPU complexity of a 1D Burgers 4D-Var system in Baumann [15]. Recently Amsallem et al. [8] used a two-step gappy POD procedure to decrease t... |

1 | A Study of Reduced Order 4D-VAR with a Finite Element Shallow Water Model
- Ambrozic
- 2013
(Show Context)
Citation Context ...plemented in 4D-Var systems in Vermeulen and Heemink [95], Sava [90], while hybrid methods using reduced adjoint models but optimizing in full space 2 were introduced in Altaf et al. [5] and Ambrozic =-=[6]-=-. POD/DEIM has been employed to reduce the CPU complexity of a 1D Burgers 4D-Var system in Baumann [15]. Recently Amsallem et al. [8] used a two-step gappy POD procedure to decrease the computational ... |

1 |
Drag minimization of the cylinder wake by trustregion proper orthogonal decomposition
- Bergmann, Cordier
(Show Context)
Citation Context ... a trust region (TR) strategy to determine whether after an optimization step an update of the POD-basis should be performed. Additional work on TR/POD proved its efficiency, see Bergmann and Cordier =-=[17]-=-, Leibfritz and Volkwein [71], Sachs and Volkwein [89]. Other studies proposed to include time derivatives, nonlinear terms, and adjoint information (see Diwoky and Volkwein [40], Hinze and Volkwein [... |

1 | Optimal Control in Chemotherapy of a Viral Infection - Ştefănescu, Pogan |

1 | Efficient Approximation of Sparse Jacobians for Time-Implicit Reduced Order Models
- Ştefănescu, Sandu
- 2014
(Show Context)
Citation Context ...is dimension in order to speed up the reduced optimization systems. Future work will consider Petrov-Galerkin stabilization approaches [22, 7]. Moreover, we will focus on a generalized DEIM framework =-=[34]-=- to approximate operators since faster reduced Jacobian computations will further decrease the computational complexity of POD/DEIM reduced data assimilation systems. We will also address the impact o... |

1 | Reduced order modeling based on POD of a parabolized Navier-Stokes equations model
- Du, Navon, et al.
(Show Context)
Citation Context ...dimensional variational (4D-Var) data assimilation problems for oceanic problems (Cao et al. [23], Fang et al. [43]) and atmospheric models (Chen et al. [30, 29], Daescu and Navon [36, 37], Du et al. =-=[41]-=-). A strategy that formulates first order optimality conditions starting from POD models has been implemented in 4D-Var systems in Vermeulen and Heemink [95], Sava [90], while hybrid methods using red... |

1 |
Adaptive concepts in reduced order modeling with emphasis on PDE constrained optimization
- Hinze
- 2011
(Show Context)
Citation Context ... and Volkwein [71], Sachs and Volkwein [89]. Other studies proposed to include time derivatives, nonlinear terms, and adjoint information (see Diwoky and Volkwein [40], Hinze and Volkwein [52], Hinze =-=[51]-=-, Gubisch and Volkwein [48], Hay et al. [50], Carlberg and Farhat [24], Zahr and Farhat [100]) into POD basis for reduced order optimization purposes. A-posteriori analysis for POD applied to optimal ... |