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Linear Inverse Problems

Linear Inverse Problems. A MATLAB Tutorial Presented by Johnny Samuels. What do we want to do?. We want to develop a method to determine the best fit to a set of data: e.g. The Plan. Review pertinent linear algebra topics Forward/inverse problem for linear systems Discuss well-posedness

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Linear Inverse Problems

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  1. Linear Inverse Problems A MATLAB Tutorial Presented by Johnny Samuels

  2. What do we want to do? • We want to develop a method to determine the best fit to a set of data: e.g.

  3. The Plan • Review pertinent linear algebra topics • Forward/inverse problem for linear systems • Discuss well-posedness • Formulate a least squares solution for an overdetermined system

  4. Linear Algebra Review • Represent m linear equations with n variables: • A = m x n matrix, y = nx1 vector, b = mx1 vector • If A = m x n and B = n x p then AB = m x p (number of columns of A = number of rows of B)

  5. Linear Algebra Review:Example • Solve system using MATLAB’s backslash operator “\”: A = [1 -1 3 1;3 -3 1 0;1 1 0 -2]; b=[2;-1;3]; y = A\b

  6. Linear Algebra Review:What does it mean? • Graphical Representation:

  7. Linear Algebra Review:Square Matrices • A = square matrix if A has n rows and n columns • The n x n identity matrix = • If there exists s.t. then A is invertible

  8. Linear Algebra ReviewSquare Matrices cont. • If then • Compute (by hand) and verify (with MATLAB’s “*” command) the inverse of

  9. Linear Algebra Review:One last thing… • The transpose of a matrix A with entries Aij is defined as Aji and is denoted as AT– that is, the columns of AT are the rows of A • Ex: implies • Use MATLAB’s “ ‘ “ to compute transpose

  10. Forward Problem:An Introduction • We will work with the linear system Ay = b where (for now) A = n x n matrix, y = n x 1 vector, b = n x 1 vector • The forward problem consists of finding b given a particular y

  11. Forward Problem:Example • g = 2y : Forward problem consists of finding g for a given y • If y = 2 then g = 4 • What if and ? • What is the forward problem for vibrating beam?

  12. Inverse Problem • For the vibrating beam, we are given data (done in lab) and we must determine m, c and k. • In the case of linear system Ay=b, we are provided with A and b, but must determine y

  13. Inverse Problem:Example • g = 2y : Inverse problem consists of finding y for a given g • If g = 10 then • Use A\b to determine y

  14. Well-posedness • The solution technique produces the correct answer when Ay=b is well-posed • Ay=b is well-posed when • Existence – For each b there exists an y such that Ay=b • Uniqueness – • Stability – is continuous • Ay=b is ill-posed if it is not well-posed

  15. Well-posedness:Example • In command window type y=well_posed_ex(4,0) • y is the solution to • K = condition number; the closer K is to 1 the more trusted the solution is

  16. Ill-posedness:Example • In command window type y=ill_posed_ex(4,0) • y is the solution to where • Examine error of y=ill_posed_ex(8,0) • Error is present because H is ill-conditioned

  17. What is an ill conditioned system? • A system is ill conditioned if some small perturbation in the system causes a relatively large change in the exact solution • Ill-conditioned system:

  18. Ill-Conditioned System:Example II

  19. What is the effect of noisy data? • Data from vibrating beam will be corrupted by noise (e.g. measurement error) • Compare: • y=well_posed_ex(4,0) and z=well_posed(4,.1) • y=well_posed_ex(10,0) and z=well_posed(10,.2) • y=ill_posed_ex(4,0) and z=ill_posed(4,.1) • y=ill_posed_ex(10,0) and z=ill_posed(10,.2) • How to deal with error? Stay tuned for next talk

  20. Are we done? • What if A is not a square matrix? In this case does not exist • Focus on an overdetermined system (i.e. A is mxn where m > n) • Usually there exists no exact solution to Ay=b when A is overdetermined • In vibrating beam example, # of data points will be much larger than # of parameters to solve (i.e. m > n)

  21. Overdetermined System:Example • Minimize

  22. Obtaining the Normal Equations • We want to minimize : • is minimized when y solves • provides the least squares solution

  23. Least Squares:Example • Approximate the spring constant k for Hooke’s Law: l is measured lengths of spring, E is equilibrium position, and F is the resisting force • least_squares_ex.m determines the least squares solution to the above equation for a given data set

  24. What did we learn? • Harmonic oscillator is a nonlinear system, so the normal equations are not directly applicable • Many numerical methods approximate the nonlinear system with a linear system, and then apply the types of results we obtained here

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