Levenberg Marquardt Algorithm

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In summary: Trust region is a method used in optimization algorithms to limit the search space to a region where the algorithm is confident it can find the minimum. This region is defined by a trust radius, within which the algorithm iteratively tries to improve the minimum value of the objective function. In summary, the LMA technique is used for non-linear fitting and works by minimizing the sum of the square of the residuals. It is important to set sensible initial starting parameters, as the method can only find a local minimum and not a global one. Trust region is a method used to limit the search space in optimization algorithms to a region where the algorithm is confident it can find the minimum.
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yellowputty
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Hello Physics Forums,

In my project work, I've had to use the LMA technique for some non linear fitting. I really want to understand what it's doing rather than just using it. I have a poor knowledge of non linear fitting so please bear with me!

When producing the line of best fit, it needs to follow the data points extremely closely, obviously. To do this it minimises the sum of the square of the residuals, correct? So when I'm reading this literature, why does it always talk about minimising the function? I understand the LMA is only able to find a local minimum and not a global one, and hence it's important to set sensible initial starting parameters.

But why does the method need to find the minimum? Is it that a local minimum that might not be the global minimum trick it into thinking it found it, and send the curve off in the wrong direction? (In my head I have the idea of an x^3 trend with an x^2 fit if that makes any sense).

When the literature is talking about minimisation, what is it referring to? The residuals or the curve itself?

Also, for bonus points, could anyone explain what a trust region is in simple terms?

Any insight at all would be greatly appreciated!

Thank you in advance.
 
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1. What is the Levenberg Marquardt algorithm?

The Levenberg Marquardt algorithm is a numerical method used for solving nonlinear least squares problems. It is commonly used in data fitting and optimization, particularly in the fields of statistics and engineering.

2. How does the Levenberg Marquardt algorithm work?

The Levenberg Marquardt algorithm combines the steepest descent method and the Gauss-Newton method to find the minimum value of a nonlinear function. It uses a trust region approach, meaning it iteratively adjusts the solution in small steps while taking into account the local curvature of the function.

3. What are the advantages of using the Levenberg Marquardt algorithm?

The Levenberg Marquardt algorithm is known for its speed and efficiency in solving nonlinear least squares problems. It is also less sensitive to the initial guess and is able to handle noisy data. Additionally, it has been shown to be more accurate and robust compared to other nonlinear optimization methods.

4. What are the limitations of the Levenberg Marquardt algorithm?

While the Levenberg Marquardt algorithm is a powerful tool for solving nonlinear least squares problems, it is not suitable for all types of optimization problems. It may struggle with highly non-convex problems or problems with a large number of parameters.

5. How is the Levenberg Marquardt algorithm used in practice?

The Levenberg Marquardt algorithm is commonly implemented in software packages for data analysis and optimization. It is also used in various scientific and engineering applications, such as in fitting mathematical models to experimental data and in parameter estimation for complex systems.

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