Optimizing lsqnonlin for Nonlinear Inverse Problems: Handling Large Unknown Parameter Sizes and Measurement Datasets

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Sumithra.
Sumithra. 2024-4-30
回答： Nipun 2024-6-11
Can anyone provide insights or guidance on effectively utilizing the lsqnonlin function with the Levenberg-Marquardt algorithm and Tikhonov regularization for solving a nonlinear inverse problem with a large unknown parameter size of 109 and 380 measurements? Any tips on parameter initialization, optimization options, and dealing with computational challenges would be greatly appreciated!

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回答（1 个）

Nipun
Nipun 2024-6-11
Hi Sumithra,
I understand that you want to effectively utilize the lsqnonlin function with the Levenberg-Marquardt algorithm and Tikhonov regularization for a nonlinear inverse problem. Here are some tips:
1. Parameter Initialization:
  • Provide a good initial guess to ensure faster convergence.
  • Use domain knowledge or a preliminary analysis to set initial parameters.
2. Optimization Options:
  • Set optimoptions for lsqnonlin to use the Levenberg-Marquardt algorithm
options = optimoptions('lsqnonlin', 'Algorithm', 'levenberg-marquardt', 'Display', 'iter');
3. Tikhonov Regularization:
  • Modify the objective function to include a Tikhonov regularization term:
alpha = 0.01; % Regularization parameter
regularization = @(x) sqrt(sum(x.^2));
objective = @(x) [yourOriginalFunction(x); alpha * regularization(x)];
4. Dealing with Computational Challenges:
  • Use sparse matrices if applicable.
  • Ensure efficient computation by using vectorized operations.
  • If the problem is large, consider breaking it into smaller subproblems or using parallel computing.
options = optimoptions('lsqnonlin', 'Algorithm', 'levenberg-marquardt', 'Display', 'iter');
alpha = 0.01; % Regularization parameter
regularization = @(x) sqrt(sum(x.^2));
objective = @(x) [yourOriginalFunction(x); alpha * regularization(x)];
x0 = ...; % Initial guess
[x, resnorm, residual, exitflag, output] = lsqnonlin(objective, x0, [], [], options);
For more details, refer to MATLAB documentation:
https://www.mathworks.com/help/optim/ug/lsqnonlin.html
Hope this helps.
Regards,
Nipun

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