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Figure 3: Speedup analysis in logarithmic scale shown with respect to the basis number N

Figure 3 Speedup analysis in logarithmic scale shown with respect to the basis number N

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Then, the above system could be written in the following form: In order to solve the system in an all-at-once approach we can write:
Then, the above system could be written in the following form: In order to solve the system in an all-at-once approach we can write:
Figure 1: FE vs ROM relative errors with respect to the basis number N in logarithmic scale. Top left: state error; top right: control error; bottom left: adjoint error; bottom right: output error.
Figure 1: FE vs ROM relative errors with respect to the basis number N in logarithmic scale. Top left: state error; top right: control error; bottom left: adjoint error; bottom right: output error.
Figure 2: FE vs ROM simulations of state and control variables for w = [1/12,2,2.5} and a = 107?. First row: FE state for t = 0.17,1.5,3.5,5; second row: ROM state for ¢ = 0.17,1.5,3.5,5; third row: FE control for t = 0.17,1.5,3.5,5; fourth row: ROM control for t = 0.17,1.5,3.5,5. In order to better visualize the boundary control, we represent it in a third dimension.
Figure 2: FE vs ROM simulations of state and control variables for w = [1/12,2,2.5} and a = 107?. First row: FE state for t = 0.17,1.5,3.5,5; second row: ROM state for ¢ = 0.17,1.5,3.5,5; third row: FE control for t = 0.17,1.5,3.5,5; fourth row: ROM control for t = 0.17,1.5,3.5,5. In order to better visualize the boundary control, we represent it in a third dimension.
Figure 4: Considered reference domain and subdomains for 3 = 1. 5 Linear Quadratic Time Dependent OCP()s governed by Stokes Equations In this section, we introduce a distributed OCP() governed by time dependent Stokes equations. As we did for the parabolic case in section 2.1, first of all, we provide the proof of well-posedness of this specific linear quadratic OCP(u) in a saddle point framework. Then, we will adapt the space-time discretization and the aggregated POD-Galerkin technique of Sections 2.2 and 3.4, respectively, to this more complex problem.
Figure 4: Considered reference domain and subdomains for 3 = 1. 5 Linear Quadratic Time Dependent OCP()s governed by Stokes Equations In this section, we introduce a distributed OCP() governed by time dependent Stokes equations. As we did for the parabolic case in section 2.1, first of all, we provide the proof of well-posedness of this specific linear quadratic OCP(u) in a saddle point framework. Then, we will adapt the space-time discretization and the aggregated POD-Galerkin technique of Sections 2.2 and 3.4, respectively, to this more complex problem.
We are now able to define the functional of the form (9) and, then, the Lagrangian functional:
We are now able to define the functional of the form (9) and, then, the Lagrangian functional:
Figure 5: FE vs ROM simulations of state velocity and pressure variables for w = [1/2,10~?] and a = 107%. First row: FE state velocity for t = 0.05,0.5,0.75; second row: ROM state velocity for t = 0.05, 0.5,0.75; third row: FE state pressure for t = 0.05,0.5,0.75; fourth row: ROM state pressure for t = 0.05, 0.5, 0.75. (o manage several repeated simulations for different values of the parameter considered.
Figure 5: FE vs ROM simulations of state velocity and pressure variables for w = [1/2,10~?] and a = 107%. First row: FE state velocity for t = 0.05,0.5,0.75; second row: ROM state velocity for t = 0.05, 0.5,0.75; third row: FE state pressure for t = 0.05,0.5,0.75; fourth row: ROM state pressure for t = 0.05, 0.5, 0.75. (o manage several repeated simulations for different values of the parameter considered.
Figure 7: Speedup analysis in logarithmic scale shown with respect to the reduced basis dimension NV. Figure 6: Domain and boundary considered.
Figure 7: Speedup analysis in logarithmic scale shown with respect to the reduced basis dimension NV. Figure 6: Domain and boundary considered.
Table 2: Data for the OCP() governed by a Stokes equation.
Table 2: Data for the OCP() governed by a Stokes equation.
Figure 8: FE vs ROM errors in logarithmic scale. Top left: state error (velocity and pressure), top right: control error, bottom left: adjoint error (velocity and pressure), bottom right: output error.
Figure 8: FE vs ROM errors in logarithmic scale. Top left: state error (velocity and pressure), top right: control error, bottom left: adjoint error (velocity and pressure), bottom right: output error.
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