CONCLUSIONS
In this thesis, a method is developed which reduces the mathematical order of a inequality constrained optimization problem. Such problems exist in many fields of computational engineering including contact modeling and computa- tional finance. As such, strong demand exists to reduce these so called High Fidelity Models (HFMs) or Full Order Models (FOMs) to Reduced Order Mod- els (ROMs) which can replicate the results of the HFMs/FOMs at a fraction of the computational cost. Such speedup could enable much more expeditious and therefore impactful information to computational modelers.
While ROMs for many systems exist at present, few models exist for inequality constrained optimization problems. The reason for this dearth is the difficulty in reducing the constraint after successfully reducing the objective function. Several methods exist to solve this problem, each with its own tradeoffs. Gener- alized Coordinate Bounding (GCB) is a novel approach which delivers a stronger enforcement of the variational linear inequality constraints contained in these problems, and scales to larger more complex systems with higher dimensional Reduced Order Bases (ROBs) while maintaining model fidelity. The construc- tion and application of GCB is the central result of this thesis.
The method functions by reducing the constraint to an abstraction on the true feasible region defined in selecting a ROB. This ROB can be constructed using traditional methods such as POD, but is not dependent on any one approach. GCB supplements the underlying primal variable reduction to ensure feasibility in the resulting ROM. The proposed method asks how can solutions to the ROM be chosen from the total space of solutions such that the constraints are satisfied? This question is formulated into a mathematical constraint and replaces the full order constraint with out the need to reproduce the dual variable solution. This new constraint provides strong feasibility assurances without requiring an increase in the dimension of the reduced solution space.
The proposed GCB method was successfully used to reproduce and extrapo- late away from solutions contained in the snapshot data used to form the ROB for the model. These preliminary results are promising and leave open pathways to see this work implemented for increasingly complex problems. Such advance- ments could include using a Support Vector Machine (SVM) classifier or other such more complicated learning technique to determine feasibility for higher di-
mensional systems. Additionally, an SVM would eliminate some of the context dependent implementation specifics present in this work, thereby automating the process and improving overall robustness. This method could be readily ap- plied to higher dimensional, multi-equation systems. Finally, a chief benefit of this work is scalability to large scale computational systems. Implementation of the proposed method might yield computational speedup over a FOM while still providing the same strong enforcement of the constraints and model accuracy. Dynamic, unaligned, inelastic contact remains a very challenging problem. Gen- eralized Coordinate Bounding may provide a means to generate ROMs capable of simulating this behavior at a significantly reduced computational cost.
Reduced Order Modeling is a key and emerging field of research as computa- tional engineers seek to harvest the high fidelity physical information collected in decades of physical and numerical experiments, and leverage that knowl- edge to rapidly address the challenges of the present. Constrained ROMs will continue to challenge engineers and resist techniques successfully applied to un- constrained ROMs, but solving these problems will generate substantial utility for modelers.
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