The Measurable Outcomes Index provides the links associated to each of the measurable outcomes for the course, arranged according to three units. Each unit corresponds to a set of outcomes that you will be able to demonstrate upon successfully completing that unit. The outcomes are stated in a manner such that they can (hopefully) be measured. The entire course is designed to help you achieve these outcomes. Further, the various assessment problems and exams are designed to address one or more of these outcomes. Throughout 16.90, as you consider your progress on learning a particular unit, you should always review these measurable outcomes and ask yourself: *Can I demonstrate each measurable outcome?*

By clicking on a measurable outcome below, you will see the content you can use to learn about that measurable outcome and the content that we use to assess your understanding of that measurable outcome, except for measurable outcomes 3.13 - 3.16 and 3.18 - 3.20. You can also visualize these outcomes and their relationships to one another at MIToces.

## Unit 1: Numerical Integration of Ordinary Differential Equations

+ Measurable Outcome 1.1 Define a first-order ODE. |
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+ Measurable Outcome 1.2 Use analytical solutions to validate numerical solutions of ODE. |
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+ Measurable Outcome 1.3 Distinguish nonlinear ODEs from linear ODEs. |
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+ Measurable Outcome 1.4 Approximate the behavior of a nonlinear equation with a linear one. |
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+ Measurable Outcome 1.5 Discretize a univariate function and its derivative, assess the truncation error using Taylor series analysis. |
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+ Measurable Outcome 1.6 Describe the Forward Euler methods and the Midpoint methods. |
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+ Measurable Outcome 1.7 Assess whether a numerical method converges, and calculate its global order of accuracy. |
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+ Measurable Outcome 1.8 Assess whether a numerical method is consistent, and calculate its local order of accuracy. |
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+ Measurable Outcome 1.9 Assess whether a numerical method is zero stable, and whether a numerical method is eigenvalue stable. |
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+ Measurable Outcome 1.10 Demonstrate an understanding of the Dahlquist Equivalence Theorem by describing the relationship between a convergent method, consistency, and stability. |
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+ Measurable Outcome 1.11 Explain the concept of stiffness of a system of equations. |
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+ Measurable Outcome 1.12 Describe how stiffness impacts the choice of numerical method for solving the equations. |
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+ Measurable Outcome 1.13 Explain the differences and relative advantages between explicit and implicit methods to integrate systems of ordinary differential equations. |
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+ Measurable Outcome 1.14 For nonlinear systems of equations, explain how a Newton-Raphson can be used in the solution of an implicit method. |
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+ Measurable Outcome 1.15 Describe multi-step methods, including the Adams-Bashforth, Adams-Moulton, and backwards differentiation families. |
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+ Measurable Outcome 1.16 Describe the form of the Runge-Kutta family of multi-stage methods. |
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+ Measurable Outcome 1.17 Explain the relative computational costs of multi-step versus multi-stage methods. |
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+ Measurable Outcome 1.18 Determine the stability boundary for a multi-step or multi-stage method applied to a linear system of ODEs. |
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+ Measurable Outcome 1.19 Recommend an appropriate ODE integration method based on the features of the problem being solved. |
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## Unit 2: Numerical Methods for Partial Differential Equations

+ Measurable Outcome 2.1 Identify whether a PDE is in the form of a conservation law, describe the characteristic of a conservation law and how the solution behaves along the characteristic. |
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+ Measurable Outcome 2.2 Qualitatively describe the solution to simple PDEs: convection equation, diffusion equation, convection-diffusion equation, Burgers equation. |
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+ Measurable Outcome 2.3 Implement a finite difference or finite volume discretization to solve a representative PDE (or set of PDEs) from an engineering application. |
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+ Measurable Outcome 2.4 Describe finite volume discretization of two-dimensional convection on an unstructured mesh. |
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+ Measurable Outcome 2.5 Define the physical domain of dependence for a problem. |
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+ Measurable Outcome 2.6 Define and determine the numerical domain of dependence for a discretization. |
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+ Measurable Outcome 2.7 Explain the CFL condition and determine the timestep constraints resulting from the CFL condition. |
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+ Measurable Outcome 2.8 Determine the local truncation error for a finite difference approximation of a PDE using a Taylor series analysis. |
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+ Measurable Outcome 2.9 Explain the difference between a centered and a one-sided (e.g., upwind) discretization. |
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+ Measurable Outcome 2.10 Define eigenvalue stability. |
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+ Measurable Outcome 2.11 Perform an eigenvalue stability analysis of a finite difference approximation of a PDE using either Von Neumann analysis or a semi-discrete (method of lines) analysis. |
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+ Measurable Outcome 2.12 Describe how the method of weighted residuals can be used to calculate an approximate solution to a PDE. |
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+ Measurable Outcome 2.13 Describe the differences between the method of weighted residuals, the collocation method, and the least-squares method for approximating a PDE. |
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+ Measurable Outcome 2.14 Describe the Galerkin method of weighted residuals. |
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+ Measurable Outcome 2.15 Describe the choice of approximate solutions (i.e., the test functions or interpolants) used in the finite element method. |
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+ Measurable Outcome 2.16 Give examples of a basis for the approximate solutions, in particular, including a nodal basis for at least linear and quadratic solutions. |
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+ Measurable Outcome 2.17 Describe how integrals are performed using a reference element. |
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+ Measurable Outcome 2.18 Explain how Gaussian quadrature rules are derived, Describe how Gaussian quadrature is used to approximate an integral in the reference element. |
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+ Measurable Outcome 2.19 Explain how Dirichlet and Neumann boundary conditions are implemented for Laplace's equation, discretized by the finite element method. |
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+ Measurable Outcome 2.20 Describe how the finite element method discretization results in a system of discrete equations and, for linear problems, gives rise to the stiffness matrix. Describe the meaning of the entries (rows and columns) of the stiffness matrix and of the right-hand side vector for linear problems. |
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## Unit 3: Probabilistic Methods and Optimization

+ Measurable Outcome 3.1 Define random variables and how they can be used in mathematical modeling. |
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+ Measurable Outcome 3.2 Define events and outcomes, list the axioms of probability. |
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+ Measurable Outcome 3.3 Describe the process of Monte Carlo sampling from uniform distributions. |
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+ Measurable Outcome 3.4 Describe how to generalize Monte Carlo sampling from uniform distributions to arbitrary univariate distributions. |
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+ Measurable Outcome 3.5 Use Monte Carlo simulation to propagate uncertainty through an ODE or PDE model. |
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+ Measurable Outcome 3.6 Describe what an estimator is. |
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+ Measurable Outcome 3.7 Define the bias and variance of an estimator. |
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+ Measurable Outcome 3.8 State unbiased estimators for mean and variance of a random variable, and for the probability of particular events. |
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+ Measurable Outcome 3.9 Describe the typical convergence rate of Monte Carlo methods. |
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+ Measurable Outcome 3.10 Define the standard error and sampling distribution of an estimator. |
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+ Measurable Outcome 3.11 Give standard errors for sample estimators of mean, variance, and event probability. |
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+ Measurable Outcome 3.12 Obtain confidence intervals for sample estimates of the mean, variance, and event probability. |
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+ Measurable Outcome 3.13 Describe how to apply design of experiments methods, including parameter study, one-at-a-time, Latin hypercube sampling, and orthogonal arrays. |
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+ Measurable Outcome 3.14 Describe the Response Surface Method. |
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+ Measurable Outcome 3.15 Describe the construction of a response surface through Taylor series, design of experiments with least-squares regression, and random sampling with least-squares regression. |
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+ Measurable Outcome 3.16 Describe the R2-metric, its use in measuring the quality of a response surface, and its potential problems. |
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+ Measurable Outcome 3.17 Describe the steepest descent, conjugate gradient, and the Newton method for optimization of multivariate functions, and apply these optimization techniques to simple unconstrained design problems. |
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+ Measurable Outcome 3.18 Describe methods to estimate gradients. |
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+ Measurable Outcome 3.19 Use finite difference approximations to estimate gradients. |
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+ Measurable Outcome 3.20 Interpret sensitivity information and explain its relevance to aerospace design examples. |
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