### optim_mesh

parent 198c0856
 ... ... @@ -4,7 +4,7 @@ ## Initialization We need to interpolate the data on the mesh. These data contains : We need to interpolate the data on the mesh. These data contain : - The bed topography, the thickness (both from BedMachine) - The surface mass balance (SMB from a the MAR model) ... ... @@ -24,34 +24,71 @@ The models is now able to compute others variables. The goal of the inversion is to find the viscosity and the friction parameter which will permit to approach as best as possible the observed surface velocity. ### Body Force ### SSA The SSA is compute with the first guess on the value to optimize. We solve the equation : ### Material Viscosity exponent : \$\$ \frac{\partial}{\partial x}(2\eta h (2 \frac{\partial u}{\partial x} + \eta \frac{\partial v}{\partial y})) + \frac{\partial}{\partial y}(\eta h (\frac{\partial u}{\partial y} + \eta \frac{\partial v}{\partial x})) - \beta^2u =\rho_igh\frac{\partial s}{\partial x} Viscosity\ exponent = \frac{1}{n}\ avec\ n= \$\$ Critical shear stress : \$\$ \frac{\partial}{\partial y}(2\eta h (2 \frac{\partial v}{\partial y} + \eta \frac{\partial u}{\partial x})) + \frac{\partial}{\partial x}(\eta h (\frac{\partial v}{\partial x} + \eta \frac{\partial u}{\partial y})) - \beta^2v =\rho_igh\frac{\partial s}{\partial y} \tau_c = 1.0\times10^{-12} \$\$ Mean density : \$\$ \rho_i = 917 kg/m^{-3} \$\$ Mean Viscosity : \$\$ \eta=prefactor ^2 \times \mu \$\$ Mean Viscosity Derivative : \$\$ \partial\eta=2\times prefactor \times\mu \$\$ Friction Parameter : \$\$ \beta = 10^\alpha \$\$ Friction parameter derivative : \$\$ \frac{\partial J}{\partial\alpha}=\frac{\partial J}{\partial\beta}\times 10 ^ {\alpha}\times ln 10 \$\$ Grounded area coefficient : give the value 1 if it's grounded or 0 if it the grounding line Flux: \$\$ \$\$ ### SSA The SSA is computed with the first guess on the value \$\eta\$ and \$\beta\$. We solve the equation : \$\$ \frac{\partial}{\partial x}(2\eta h (2 \frac{\partial u}{\partial x} + \eta \frac{\partial v}{\partial y})) + \frac{\partial}{\partial y}(\eta h (\frac{\partial u}{\partial y} + \eta \frac{\partial v}{\partial x})) - \beta^2u =\rho_igh\frac{\partial s}{\partial x} \$\$ The effective viscosity : \$\$ \eta = \frac{1}{2}A^{-1/n_{\epsilon_e}(1-n)/n} \frac{\partial}{\partial y}(2\eta h (2 \frac{\partial v}{\partial y} + \eta \frac{\partial u}{\partial x})) + \frac{\partial}{\partial x}(\eta h (\frac{\partial v}{\partial x} + \eta \frac{\partial u}{\partial y})) - \beta^2v =\rho_igh\frac{\partial s}{\partial y} \$\$ The value of the velocity u and v are exported to be used in the cost function. The value of the velocity \$u\$ and \$v\$ are exported to computed in the cost function. ### The cost function The cost function is the comparison of the compute surface velocity and the observed one. and the observed one. We compute here the integer of all the difference \$\$ J = \int{\frac{1}{2}((u_{SSA} - u_{Obs})^2+(v_{SSA} - v_{Obs})^2)} d\Omega \$\$ ... ... @@ -80,8 +117,8 @@ \$\$ We need to compute the gradient for each parameters we have to optimize. Theoretically, we have the right solution when the gradient is equal to zero but we never obtains this value. So we define a lower limit which is define a the gradient is enough low to have a good result. \$\$ \nabla J= \left\{ \begin{array}{ll} \frac{\partial J}{\partial Beta}=\\ \frac{\partial J}{\partial Eta}= \frac{\partial J}{\partial Beta}=\frac{J(i)-J(i-1)}{Beta(i)-Beta(i-1)}\\ \frac{\partial J}{\partial Eta}=\frac{J(i)-J(i-1)}{Eta(i)-Eta(i-1)} \end{array} \right\} \$\$ ... ... @@ -90,20 +127,13 @@ \$\$ ### The regularization terms The regularization can be done one the first derivative. We gave a weight \$\lambda\$ to this the value computed. This weight have to be choice according the The regularization can be done one the first derivative. We gave a weight \$\lambda\$ to this the value computed. This weight have to be choice according to the order of the optimized parameter. \$\$ J_{reg} = \int_{\Omega} 0.5 (|dV/dx|)^2 d\Omega \\with\ V \ the\ nodal\ variable \$\$ \$\$ J_{reg} = \int_{\Omega} 0.5 ((V-V^{prior})/s^2)^2 d\Omega \\with\ V \ the\ nodal\ variable, V^{prior}\ is \ the\ estimate\ and\ s^2\ the\ variance. \$\$ ### The optimization M1QN3 The optimization will take the gradient and try to found the best choice by doing simulations to find the best direction and will pass to the next iteration. \ No newline at end of file
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