Greedy for Poisson equation
The main code for this tutorial can be found at poisson_local_rom_greedy.cpp, which demonstrates how to use libROM to execute greedy procedure for the ROM of the Poisson problem introduced in the Poisson equation tutorial. For the recap, the general procedure of the greedy algorithm follows:
- Define a parameter space
- Pick an initial point in the parameter space to build a ROM there (a good cancidiate initla point is either the centroid or one of end points)
- Evaluate error indicator of the current ROM (either global or local ROM) at $N$ random points within the parameter space
- Check if the maximum error indicator value is less than the desirable accuracy threshold
- If the answer to Step 4 is yes, then terminate the greedy process.
- If the answer to Step 4 is no, then collect the full order model simulation data at the maximum error indicator point and add them to update the ROM
- Go to Step 3.
As in the Poisson equation tutorial, we choose to vary the frequency $\kappa$ as parameter. First, we define the parameter space to be from $0.5$ to $3$, which is larger than the one used in Poisson equation tutorial. You can try the following command line options to run the greedy procedure:
./poisson_local_rom_greedy -build_database -greedy-param-min 0.5 -greedy-param-max 3 -greedy-param-size 40 -greedysubsize 10 -greedyconvsize 20 -greedyrelerrortol 0.01
The lower and upper bounds of the parameter are determined by the options, -greedy-param-min and -greedy-param-max, respectively. The option -greedy-param-size specifies the total number of random sample points within the parameter space for the whole greedy procedure. The option -greedysubsize sets the number of sub-sample points, where the error indicator is evaluated at every greedy iterations. The option -greedyconvsize sets the number of convergence-sample points on which the error indicator is checked and the termination of the greedy algorithm is determined if the error indicators are below the convergence threshold after the error indicator tests on the sub-sample points have been passed. Finally, -greedyrelerrortol sets the desirable relative error for the greedy algorithm to achieve.
The core class of the libROM for the greedy procedure is GreedySampler, which is defined on Line 126 of poisson_local_rom_greedy.cpp The GreedySampler generates sampling points within a given parameter space. The class has two sub-classes, i.e., GreedyCustomSampler and GreedyRandomSampler. The GreedyCustomSampler generates pre-defined sampling points, e.g., a tensor product sampling points, while the GreedyRandomSampler generates random sampling points, e.g., Latin hyper-cube sampling points. The GreedySampler also does book-keeping job of at which sampling point to evaluate the error indicator, when to move onto the next greedy step, and which sampling point has the maximum error indicator value.
Just to be clear, the libROM does not do everything for you. For example, the error indicator must be defined in the physics solver. For the Poisson example, the residual-based error indicator is defined at Lines 529-535 of poisson_loca_rom_greedy.cpp.
Once the greedy procedure completes, the log file, called poisson_greedy_algorithm_log.txt, is generated to show the progress of the greedy procedure and the final parameter points on which the global ROM was built. For example, you will see the following block at the end of the file, indicating the final parameter points on which the global ROM is built.
Sampled Parameter Points [ 0.591166 ] [ 0.844964 ] [ 1.066678 ] [ 1.353370 ] [ 1.710334 ] [ 1.992794 ] [ 2.568171 ] [ 2.985638 ]
Because we set the -greedyrelerrortol to be 0.01, the global ROM built through the greedy procedure must be able to predict a solution with a relative error less than 1$\%$ for any points in the parameter space. Indeed, let's try to predict the solution at $\kappa = 2.2$, which was not one of the final parameter points. Let's first generate the full order model solution with the following command line option:
./poisson_local_rom_greedy -offline -f 2.2
This full order model solution will be used to compute the relative error for the ROM solution. The ROM solution can be obtained by the following command line option:
./poisson_local_rom_greedy -use_database -online -f 2.2
Indeed, the relative error of 0.00167671 is achieved