Remarks: When Newton’s method is implemented has second order information while Gauss-Newton use only f

1. Numerical Comparison of PDE Constrained Optimization Schemes  for Solving Earthquake Modeling Problems L. Velazquez¹, C. Burstede ², A.A. Sosa ¹,  M. Argaez¹, O. Ghattas ² ¹ Computational Science Program, The University of Texas at El Paso (UTEP) ²Center for Computational Geosciences and Optimization, The University of Texas at Austin (UTA) Summary Logarithmic Barrier Method Reduced Nonlinear System We are implementing a Primal-Dual Interior-Point method to solve an inverse seismic wave problem. We compare the numerical behavior of the logarithmic barrier (LB), primal-dual interior-point (IPM), projected conjugate gradient (PRCG), and primal-dual active sets (PDAS) methods for solving one-dimensional partial differential equations (PDE) constrained optimization problems arising on earthquake applications. Our main interest is to identify an effective strategy that allows to incorporate efficiently physical bound constraints into the problem. We run several test cases and identify that the strategy IPM works best. The problem that we present here needs from the expertise of researchers from different areas (geologists, engineers, computational scientists) to develop the capability for estimating the geological structure and mechanical properties of the earth structure. Our contribution comes to help in the development of a new optimization approach to the new inversion methods based on PDE’s. We can reduce the nonlinear system (6) in terms of and then solve for We solve problem (2) through a sequence of log-barrier sub-problems as parameter goes to zero. The lagrangian function associated to (3) is We can solve (7) using Newton’s method where: or we can apply Gauss-Newton’s method having: Where y is the Lagrange multiplier associated to the equality constraints. Primal-Dual Interior-Point Method • Remarks: • When Newton’s method is implemented has second order information while Gauss-Newton use only first order information. • The only differences between the systems (6) and (7) are the terms C and B that depend only on the choice of the method LB or IPM. • does not change when either method is implemented The lagrangian associated to (2) is: Forward and Inverse Modeling Where is the Lagrange multiplier associated with the inequality constraints. The methodology is to keep the iterates positive except at the solution of the problem. At the heart of the site characterization problem using active (or passive) dynamic sources lies an inverse wave-based problem. Inverse problems are notoriously more difficult than the corresponding forward wave propagation problems. In the forward problem, one wishes to determine the soil response due to a prescribed excitation under the assumption that the source and the material properties are known. By contrast, in the inverse problem one wants to estimate the spatial distribution of the soil properties that results in a predicted response that most closely matches the observed records generated by either an active source or a seismic signal. The forward problems are, thus, but a mere “inner iteration” of the inverse problem, which entails multiple forward wave propagation simulations. It is thus important that forward wave simulations are carried out fast and accurately. Optimality Conditions For Algorithm Step 1 Select a method j=LB or IPM. Choose an initial point and according to the method selected. Step 2 For k=0,1,2,… until Step 3 Choose according to the method selected Step 4 Solve (7) for step and calculate Step 5 (Force positivity) Calculate according to the method selected Step 6 (Sufficient decrease) Find where t is the smallest nonnegative integer such that satisfies Step 7 (Update) Set Step 8 Return to Step 2 Seismic Wave Propagation Problem Where F is a square nonlinear function, The forward problem is based on the elastic wave equation for displacement (state) , shear-modulus , and force subject to additional boundary conditions: Nonlinear System We apply Newton’s method to (4) or (5), and reduce the system to obtain: Then the space-time discretized inverse PDE constrained problem is: References • Numerical Comparison of Constrained PDEs Optimization Schemes for Solving Earthquake Modeling Problems. L. Velazquez, C. Burstede, A.A. Sosa, M. Argaez, and O. Ghattas, Technical Report. • Numerical Optimization. J. Nocedal, and S. J. Wright, . • An Introduction to Seismology Earthquakes and Earth Structure, • S. Stein and M. Wysession, Blackwell Publishing, , 2006 Numerical Results Where and are the physical bounds. The regularization term varies depending on the method chosen. We can select Tikhonov regularization or Total Variation regularization We use four random initial points for solving two test problems (step, wild) with two different choices of level and grid coarse (6,3) and (7,5), respectively. The numerical results of PRCG and PDAS, and the Matlab code were provided by the UT team. The experimentation shows that Interior-Point Methods, without regularization, to be competitive in terms of robustness, accuracy , and total number of conjugate gradient iterations. Where Acknowledgements Contact Information This work is being funded by NSF Grants NEESR-SG CMMI-0619078 and Crest Cyber-ShARE HRD-0734825. Uram A. Sosa, Graduate Student Computational Science Program usosaaguirre@miners.utep.edu