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Inverse Problems for Dummies Inverse Problems for Dummies Scott Ziegler (a very stable genius) Colorado State University September 13, 2018 Scott Ziegler (CSU) Green Slopes September 13, 2018 1 / 27 Geology, medicine, radar, astronomy, etc. The task of using the mathematical model of the situation to simulate data is called a forward problem. The task of using the gathered data to recreate the mathematical model of the situation is called an inverse problem. In practice, we typically look for a specific parameter of the mathematical model. What is an inverse problem? There are many physical situations in which we acquire data from some phenomenon or object which we cannot see. Scott Ziegler (CSU) Green Slopes September 13, 2018 2 / 27 The task of using the mathematical model of the situation to simulate data is called a forward problem. The task of using the gathered data to recreate the mathematical model of the situation is called an inverse problem. In practice, we typically look for a specific parameter of the mathematical model. What is an inverse problem? There are many physical situations in which we acquire data from some phenomenon or object which we cannot see. Geology, medicine, radar, astronomy, etc. Scott Ziegler (CSU) Green Slopes September 13, 2018 2 / 27 The task of using the gathered data to recreate the mathematical model of the situation is called an inverse problem. In practice, we typically look for a specific parameter of the mathematical model. What is an inverse problem? There are many physical situations in which we acquire data from some phenomenon or object which we cannot see. Geology, medicine, radar, astronomy, etc. The task of using the mathematical model of the situation to simulate data is called a forward problem. Scott Ziegler (CSU) Green Slopes September 13, 2018 2 / 27 In practice, we typically look for a specific parameter of the mathematical model. What is an inverse problem? There are many physical situations in which we acquire data from some phenomenon or object which we cannot see. Geology, medicine, radar, astronomy, etc. The task of using the mathematical model of the situation to simulate data is called a forward problem. The task of using the gathered data to recreate the mathematical model of the situation is called an inverse problem. Scott Ziegler (CSU) Green Slopes September 13, 2018 2 / 27 What is an inverse problem? There are many physical situations in which we acquire data from some phenomenon or object which we cannot see. Geology, medicine, radar, astronomy, etc. The task of using the mathematical model of the situation to simulate data is called a forward problem. The task of using the gathered data to recreate the mathematical model of the situation is called an inverse problem. In practice, we typically look for a specific parameter of the mathematical model. Scott Ziegler (CSU) Green Slopes September 13, 2018 2 / 27 Some simple examples Deblurring 1 1Jennifer Mueller and Samuli Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. 2012. Scott Ziegler (CSU) Green Slopes September 13, 2018 3 / 27 Some simple examples X-ray imaging 2 2Jennifer Mueller and Samuli Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. 2012. Scott Ziegler (CSU) Green Slopes September 13, 2018 4 / 27 When the operator A = A is a linear operator, we call the problem linear. The problem could still be infinite dimensional, but we will n m m×n typically discretize to get b 2 R , x 2 R and A = R . If the operator A is nonlinear, we cleverly call the inverse problem nonlinear. In this case we are typically dealing with A : H ! H where H is a Hilbert space. Classifying inverse problems Inverse problems can be broadly summarized by the following equation: b = Ax + where b represents the observation (or result of the forward problem), x represents the model parameters, A represents the operator governing the model, and is a random noise vector (typically normally distributed). Scott Ziegler (CSU) Green Slopes September 13, 2018 5 / 27 Classifying inverse problems Inverse problems can be broadly summarized by the following equation: b = Ax + where b represents the observation (or result of the forward problem), x represents the model parameters, A represents the operator governing the model, and is a random noise vector (typically normally distributed). When the operator A = A is a linear operator, we call the problem linear. The problem could still be infinite dimensional, but we will n m m×n typically discretize to get b 2 R , x 2 R and A = R . If the operator A is nonlinear, we cleverly call the inverse problem nonlinear. In this case we are typically dealing with A : H ! H where H is a Hilbert space. Scott Ziegler (CSU) Green Slopes September 13, 2018 5 / 27 We would like for our inverse problem to be well-posed, which means it satisfies the following three conditions: Existence: there should be a solution. Uniqueness: there should be at most one solution Stability: the solution must continuously depend on the data. These conditions are equivalent to saying that our map A (which in the finite dimensional case is just a matrix) should have a continuous inverse. If an inverse problem is not well-posed, it is ill-posed. Linear inverse problems We'll begin (and essentially end) by studying linear inverse problems since they are much easier to work with. Scott Ziegler (CSU) Green Slopes September 13, 2018 6 / 27 Existence: there should be a solution. Uniqueness: there should be at most one solution Stability: the solution must continuously depend on the data. These conditions are equivalent to saying that our map A (which in the finite dimensional case is just a matrix) should have a continuous inverse. If an inverse problem is not well-posed, it is ill-posed. Linear inverse problems We'll begin (and essentially end) by studying linear inverse problems since they are much easier to work with. We would like for our inverse problem to be well-posed, which means it satisfies the following three conditions: Scott Ziegler (CSU) Green Slopes September 13, 2018 6 / 27 Uniqueness: there should be at most one solution Stability: the solution must continuously depend on the data. These conditions are equivalent to saying that our map A (which in the finite dimensional case is just a matrix) should have a continuous inverse. If an inverse problem is not well-posed, it is ill-posed. Linear inverse problems We'll begin (and essentially end) by studying linear inverse problems since they are much easier to work with. We would like for our inverse problem to be well-posed, which means it satisfies the following three conditions: Existence: there should be a solution. Scott Ziegler (CSU) Green Slopes September 13, 2018 6 / 27 Stability: the solution must continuously depend on the data. These conditions are equivalent to saying that our map A (which in the finite dimensional case is just a matrix) should have a continuous inverse. If an inverse problem is not well-posed, it is ill-posed. Linear inverse problems We'll begin (and essentially end) by studying linear inverse problems since they are much easier to work with. We would like for our inverse problem to be well-posed, which means it satisfies the following three conditions: Existence: there should be a solution. Uniqueness: there should be at most one solution Scott Ziegler (CSU) Green Slopes September 13, 2018 6 / 27 These conditions are equivalent to saying that our map A (which in the finite dimensional case is just a matrix) should have a continuous inverse. If an inverse problem is not well-posed, it is ill-posed. Linear inverse problems We'll begin (and essentially end) by studying linear inverse problems since they are much easier to work with. We would like for our inverse problem to be well-posed, which means it satisfies the following three conditions: Existence: there should be a solution. Uniqueness: there should be at most one solution Stability: the solution must continuously depend on the data. Scott Ziegler (CSU) Green Slopes September 13, 2018 6 / 27 If an inverse problem is not well-posed, it is ill-posed. Linear inverse problems We'll begin (and essentially end) by studying linear inverse problems since they are much easier to work with. We would like for our inverse problem to be well-posed, which means it satisfies the following three conditions: Existence: there should be a solution. Uniqueness: there should be at most one solution Stability: the solution must continuously depend on the data. These conditions are equivalent to saying that our map A (which in the finite dimensional case is just a matrix) should have a continuous inverse. Scott Ziegler (CSU) Green Slopes September 13, 2018 6 / 27 Linear inverse problems We'll begin (and essentially end) by studying linear inverse problems since they are much easier to work with. We would like for our inverse problem to be well-posed, which means it satisfies the following three conditions: Existence: there should be a solution. Uniqueness: there should be at most one solution Stability: the solution must continuously depend on the data. These conditions are equivalent to saying that our map A (which in the finite dimensional case is just a matrix) should have a continuous inverse. If an inverse problem is not well-posed, it is ill-posed. Scott Ziegler (CSU) Green Slopes September 13, 2018 6 / 27 In practice every inverse problem is ill-posed, and it turns out that solving a least squares problem for ill-posed inverse problems goes very badly. Linear inverse problems If our inverse problem is well-posed, then our work is essentially done.
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