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I am very interested in numerical simulation because it always provides new ideas to examine how the world works. I believe I will continue to work on numerical optimization in the future. My potential work should be to develop optimization algorithms, interpolation strategies, and interfaces between numerical simulation and optimization.
Numerical simulation is the basic tool that supports numerical optimization. The simulation results must represent real life in some way or to some extent. It is exciting to develop new methods to improve accuracy, efficiency and versatility. However, we should not just be commercial software operators.
Experiments are an important method to validate the numerical results. PhD student? Do your experiment!
All other fantasies
Sometimes you have fantastic ideas. It's really fun to implement them, even if you later discover that they are completely rubbish.