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

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!

[Torsten]

My advice: Start simple and learn from step to step.

Answer 1

[Nipun]

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