# Measurable Outcome Index

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.
### Learn 1.2.2 An example of first order ODE 1.6.1 Nonlinear Systems 1.6.2 Linear Constant Coefficient Systems ### Assess 1.2.1 First-order ODEs

+ Measurable Outcome 1.2 Use analytical solutions to validate numerical solutions of ODE.
### Learn 1.6.2 Linear Constant Coefficient Systems 1.6.3 Eigenvalue Stability for a Linear ODE 1.7.1 Stiffness ### Assess 1.2.1 First-order ODEs 1.6.4 Imaginary Eigenvalues

+ Measurable Outcome 1.3 Distinguish nonlinear ODEs from linear ODEs.
### Learn 1.2.2 An example of first order ODE 1.6.1 Nonlinear Systems 1.6.2 Linear Constant Coefficient Systems 1.6.3 Eigenvalue Stability for a Linear ODE 1.7.4 Newton-Raphson: Implement Implicit Methods on Nonlinear Problems ### Assess No specific content identified.

+ Measurable Outcome 1.4 Approximate the behavior of a nonlinear equation with a linear one.
### Learn 1.2.2 An example of first order ODE 1.6.2 Linear Constant Coefficient Systems 1.7.4 Newton-Raphson: Implement Implicit Methods on Nonlinear Problems ### Assess 1.7.5 Apply Newton-Rhapson

+ Measurable Outcome 1.5 Discretize a univariate function and its derivative, assess the truncation error using Taylor series analysis.
### Learn 1.2.3 Discretization 1.2.4 The Forward Euler Method 1.2.5 The Midpoint Method 1.3.1 Errors 1.3.2 Local Truncation Error ### Assess 1.3.3 Local Order of Accuracy 1.3.4 Definition of Multi-Step Methods 1.3.5 Example of Most Accurate Multi-Step Method 1.4.1 Types of Errors

+ Measurable Outcome 1.6 Describe the Forward Euler methods and the Midpoint methods.
### Learn 1.2.4 The Forward Euler Method 1.2.5 The Midpoint Method 1.6.1 Nonlinear Systems 1.6.3 Eigenvalue Stability for a Linear ODE ### Assess 1.3.5 Example of Most Accurate Multi-Step Method 1.4.1 Types of Errors

+ Measurable Outcome 1.7 Assess whether a numerical method converges, and calculate its global order of accuracy.
### Learn 1.4.2 Convergence of Numerical Methods ### Assess 1.4.3 Rate of Convergence (Global Order of Accuracy) 1.5.3 Dahlquist Equivalence Theorem

+ Measurable Outcome 1.8 Assess whether a numerical method is consistent, and calculate its local order of accuracy.
### Learn No specific content identified. ### Assess 1.3.3 Local Order of Accuracy 1.3.4 Definition of Multi-Step Methods 1.3.5 Example of Most Accurate Multi-Step Method 1.5.1 Consistency 1.5.3 Dahlquist Equivalence Theorem

+ Measurable Outcome 1.9 Assess whether a numerical method is zero stable, and whether a numerical method is eigenvalue stable.
### Learn 1.5.2 Stability 1.6.2 Linear Constant Coefficient Systems 1.6.3 Eigenvalue Stability for a Linear ODE 1.7.1 Stiffness 1.7.3 Implicit Methods for Linear Systems of ODE's ### Assess 1.5.3 Dahlquist Equivalence Theorem 1.6.4 Imaginary Eigenvalues 1.7.2 Spectral Condition Number

+ Measurable Outcome 1.10 Demonstrate an understanding of the Dahlquist Equivalence Theorem by describing the relationship between a convergent method, consistency, and stability.
### Learn 1.5.2 Stability ### Assess 1.3.3 Local Order of Accuracy 1.5.3 Dahlquist Equivalence Theorem

+ Measurable Outcome 1.11 Explain the concept of stiffness of a system of equations.
### Learn 1.7.1 Stiffness 1.7.3 Implicit Methods for Linear Systems of ODE's ### Assess 1.7.2 Spectral Condition Number

+ Measurable Outcome 1.12 Describe how stiffness impacts the choice of numerical method for solving the equations.
### Learn 1.7.1 Stiffness 1.7.3 Implicit Methods for Linear Systems of ODE's 1.7.4 Newton-Raphson: Implement Implicit Methods on Nonlinear Problems 1.8.2 Adams-Moulton Methods 1.8.3 Backwards Differentiation Methods ### Assess 1.7.2 Spectral Condition Number 1.8.4 Backwards Differentiation Excercise

+ Measurable Outcome 1.13 Explain the differences and relative advantages between explicit and implicit methods to integrate systems of ordinary differential equations.
### Learn 1.7.3 Implicit Methods for Linear Systems of ODE's 1.7.4 Newton-Raphson: Implement Implicit Methods on Nonlinear Problems ### Assess 1.3.4 Definition of Multi-Step Methods 1.7.5 Apply Newton-Rhapson

+ Measurable Outcome 1.14 For nonlinear systems of equations, explain how a Newton-Raphson can be used in the solution of an implicit method.
### Learn 1.7.4 Newton-Raphson: Implement Implicit Methods on Nonlinear Problems ### Assess 1.7.5 Apply Newton-Rhapson

+ Measurable Outcome 1.15 Describe multi-step methods, including the Adams-Bashforth, Adams-Moulton, and backwards differentiation families.
### Learn 1.8.1 Adams-Bashforth Methods 1.8.2 Adams-Moulton Methods 1.8.3 Backwards Differentiation Methods ### Assess 1.3.4 Definition of Multi-Step Methods 1.3.5 Example of Most Accurate Multi-Step Method 1.8.4 Backwards Differentiation Excercise

+ Measurable Outcome 1.16 Describe the form of the Runge-Kutta family of multi-stage methods.
### Learn 1.9.1 Two-stage Runge-Kutta Methods 1.9.2 Four-stage Runge-Kutta Method ### Assess 1.9.3 Stability Regions

+ Measurable Outcome 1.17 Explain the relative computational costs of multi-step versus multi-stage methods.
### Learn 1.8.1 Adams-Bashforth Methods 1.8.2 Adams-Moulton Methods 1.8.3 Backwards Differentiation Methods 1.9.1 Two-stage Runge-Kutta Methods 1.9.2 Four-stage Runge-Kutta Method ### Assess No specific content identified.

+ Measurable Outcome 1.18 Determine the stability boundary for a multi-step or multi-stage method applied to a linear system of ODEs.
### Learn 1.6.3 Eigenvalue Stability for a Linear ODE 1.8.1 Adams-Bashforth Methods 1.8.2 Adams-Moulton Methods 1.8.3 Backwards Differentiation Methods ### Assess 1.8.4 Backwards Differentiation Excercise 1.9.3 Stability Regions

+ Measurable Outcome 1.19 Recommend an appropriate ODE integration method based on the features of the problem being solved.
### Learn 1.7.4 Newton-Raphson: Implement Implicit Methods on Nonlinear Problems 1.8.1 Adams-Bashforth Methods 1.8.2 Adams-Moulton Methods 1.8.3 Backwards Differentiation Methods 1.9.1 Two-stage Runge-Kutta Methods 1.9.2 Four-stage Runge-Kutta Method ### Assess 1.7.5 Apply Newton-Rhapson 1.8.4 Backwards Differentiation Excercise 1.9.3 Stability Regions

## 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.
### Learn 2.2.1 Conservation Laws in Integral and Differential Form 2.2.3 Convection 2.2.6 Convection-Diffusion 2.5.1 Finite Volume Method in 1-D 2.5.3 Finite volume method in 2-D 2.5.5 Finite Volume Method for Nonlinear Systems ### Assess 2.2.2 One-Dimensional Burgers' Equation 2.2.4 Characteristics for One-Dimensional Burgers' Equation 2.5.2 Finite Volume Method applied to 1-D Convection 2.5.4 Finite Volume Method for 2-D Convection on a Rectangular Mesh

+ Measurable Outcome 2.2 Qualitatively describe the solution to simple PDEs: convection equation, diffusion equation, convection-diffusion equation, Burgers equation.
### Learn 2.2.3 Convection 2.2.5 Diffusion 2.2.6 Convection-Diffusion 2.2.7 Linear Elasticity 2.3.3 Finite Difference Method applied to 1-D Convection 2.7.2 Matrix Stability for Finite Difference Methods ### Assess 2.2.2 One-Dimensional Burgers' Equation 2.2.4 Characteristics for One-Dimensional Burgers' Equation 2.5.2 Finite Volume Method applied to 1-D Convection 2.5.4 Finite Volume Method for 2-D Convection on a Rectangular Mesh 2.6.1 Upwinding 2.6.2 The CFL Condition 2.7.1 Fourier Analysis of PDE's

+ 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.
### Learn 2.3.1 Finite Difference Approximations 2.3.2 Finite Difference Methods 2.3.3 Finite Difference Method applied to 1-D Convection 2.4.4 Finite Difference Methods in Matrix Form 2.4.6 Boundary Conditions for Finite Differences 2.5.1 Finite Volume Method in 1-D 2.5.3 Finite volume method in 2-D 2.5.5 Finite Volume Method for Nonlinear Systems ### Assess 2.3.4 Forward Time-Backward Space (FTBS) 2.4.5 General Finite Difference Approximations 2.5.2 Finite Volume Method applied to 1-D Convection 2.5.4 Finite Volume Method for 2-D Convection on a Rectangular Mesh

+ Measurable Outcome 2.4 Describe finite volume discretization of two-dimensional convection on an unstructured mesh.
### Learn 2.5.3 Finite volume method in 2-D 2.5.5 Finite Volume Method for Nonlinear Systems ### Assess 2.5.4 Finite Volume Method for 2-D Convection on a Rectangular Mesh

+ Measurable Outcome 2.5 Define the physical domain of dependence for a problem.
### Learn 2.2.3 Convection 2.2.5 Diffusion 2.2.6 Convection-Diffusion ### Assess 2.2.4 Characteristics for One-Dimensional Burgers' Equation 2.6.1 Upwinding 2.6.2 The CFL Condition

+ Measurable Outcome 2.6 Define and determine the numerical domain of dependence for a discretization.
### Learn 2.3.1 Finite Difference Approximations ### Assess 2.6.1 Upwinding 2.6.2 The CFL Condition

+ Measurable Outcome 2.7 Explain the CFL condition and determine the timestep constraints resulting from the CFL condition.
### Learn No specific content identified. ### Assess 2.6.2 The CFL Condition

+ Measurable Outcome 2.8 Determine the local truncation error for a finite difference approximation of a PDE using a Taylor series analysis.
### Learn 2.4.1 Local Truncation Error for a Derivative Approximation 2.4.4 Finite Difference Methods in Matrix Form 2.4.6 Boundary Conditions for Finite Differences ### Assess 2.4.2 Truncation Error of Central Difference Approximation 2.4.3 Truncation Error for a PDE 2.4.5 General Finite Difference Approximations

+ Measurable Outcome 2.9 Explain the difference between a centered and a one-sided (e.g., upwind) discretization.
### Learn 2.3.3 Finite Difference Method applied to 1-D Convection 2.4.1 Local Truncation Error for a Derivative Approximation 2.7.2 Matrix Stability for Finite Difference Methods ### Assess 2.3.4 Forward Time-Backward Space (FTBS) 2.6.1 Upwinding 2.7.4 Stability Exercises

+ Measurable Outcome 2.10 Define eigenvalue stability.
### Learn 2.3.3 Finite Difference Method applied to 1-D Convection 2.7.2 Matrix Stability for Finite Difference Methods 2.7.3 Circulant Matrices ### Assess 2.3.4 Forward Time-Backward Space (FTBS) 2.7.4 Stability Exercises

+ 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.
### Learn 2.7.2 Matrix Stability for Finite Difference Methods 2.7.3 Circulant Matrices ### Assess 2.7.1 Fourier Analysis of PDE's 2.7.4 Stability Exercises

+ Measurable Outcome 2.12 Describe how the method of weighted residuals can be used to calculate an approximate solution to a PDE.
### Learn 2.8.1 Functional Approximation of the Solution 2.8.2 The Collocation Method 2.8.3 The Method of Weighted Residuals ### Assess 2.8.4 Galerkin Method with New Basis 2.9.4 Weak Form of the Weighted Residual 2.9.5 Calculation of the Finite Element Weighted Residual 2.9.6 Calculation of the Stiffness Matrix

+ 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.
### Learn 2.8.2 The Collocation Method 2.8.3 The Method of Weighted Residuals ### Assess 2.8.4 Galerkin Method with New Basis

+ Measurable Outcome 2.14 Describe the Galerkin method of weighted residuals.
### Learn 2.8.3 The Method of Weighted Residuals 2.10.1 Gaussian Quadrature ### Assess 2.8.4 Galerkin Method with New Basis 2.9.4 Weak Form of the Weighted Residual 2.9.5 Calculation of the Finite Element Weighted Residual 2.9.6 Calculation of the Stiffness Matrix

+ Measurable Outcome 2.15 Describe the choice of approximate solutions (i.e., the test functions or interpolants) used in the finite element method.
### Learn 2.11.2 Reference Element and Linear Elements ### Assess 2.9.4 Weak Form of the Weighted Residual 2.9.5 Calculation of the Finite Element Weighted Residual 2.9.6 Calculation of the Stiffness Matrix

+ 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.
### Learn 2.9.2 1-D Finite Element Mesh and Notation 2.9.3 1-D Linear Elements and the Nodal Basis 2.11.2 Reference Element and Linear Elements ### Assess 2.9.5 Calculation of the Finite Element Weighted Residual 2.9.6 Calculation of the Stiffness Matrix

+ Measurable Outcome 2.17 Describe how integrals are performed using a reference element.
### Learn 2.10.1 Gaussian Quadrature 2.11.3 Differentiation using the Reference Element 2.11.4 Construction of the Stiffness Matrix ### Assess 2.9.5 Calculation of the Finite Element Weighted Residual 2.9.6 Calculation of the Stiffness Matrix

+ 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.
### Learn 2.10.1 Gaussian Quadrature ### Assess No specific content identified.

+ Measurable Outcome 2.19 Explain how Dirichlet and Neumann boundary conditions are implemented for Laplace's equation, discretized by the finite element method.
### Learn 2.10.2 Boundary Conditions for Finite Elements ### Assess No specific content identified.

+ 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.
### Learn 2.10.2 Boundary Conditions for Finite Elements 2.11.4 Construction of the Stiffness Matrix 2.11.5 Integration in the Reference Element ### Assess 2.9.5 Calculation of the Finite Element Weighted Residual 2.9.6 Calculation of the Stiffness Matrix

## Unit 3: Probabilistic Methods and Optimization

+ Measurable Outcome 3.1 Define random variables and how they can be used in mathematical modeling.
### Learn No specific content identified. ### Assess 3.2.1 Random Variables

+ Measurable Outcome 3.2 Define events and outcomes, list the axioms of probability.
### Learn 3.2.2 Outcomes and Events ### Assess No specific content identified.

+ Measurable Outcome 3.3 Describe the process of Monte Carlo sampling from uniform distributions.
### Learn 3.3.1 Introduction 3.3.2 Monte Carlo Analysis ### Assess 3.2.3 Distributions 3.3.3 Monte Carlo Example

+ Measurable Outcome 3.4 Describe how to generalize Monte Carlo sampling from uniform distributions to arbitrary univariate distributions.
### Learn 3.5.2 Importance Sampling ### Assess 3.2.3 Distributions 3.3.4 Inversion Method for Sampling

+ Measurable Outcome 3.5 Use Monte Carlo simulation to propagate uncertainty through an ODE or PDE model.
### Learn 3.3.2 Monte Carlo Analysis ### Assess 3.2.4 Expectations 3.3.3 Monte Carlo Example

+ Measurable Outcome 3.6 Describe what an estimator is.
### Learn No specific content identified. ### Assess 3.2.4 Expectations 3.4.1 The Error in Estimating the Mean

+ Measurable Outcome 3.7 Define the bias and variance of an estimator.
### Learn No specific content identified. ### Assess 3.2.5 The Central Limit Theorem 3.4.1 The Error in Estimating the Mean

+ Measurable Outcome 3.8 State unbiased estimators for mean and variance of a random variable, and for the probability of particular events.
### Learn 3.4.3 The Error in Estimating the Variance ### Assess 3.4.1 The Error in Estimating the Mean 3.4.2 The Error in Estimating Probabilities

+ Measurable Outcome 3.9 Describe the typical convergence rate of Monte Carlo methods.
### Learn 3.5.1 Monte Carlo Error as a Random Variable ### Assess 3.2.5 The Central Limit Theorem 3.4.1 The Error in Estimating the Mean

+ Measurable Outcome 3.10 Define the standard error and sampling distribution of an estimator.
### Learn No specific content identified. ### Assess 3.2.4 Expectations 3.4.1 The Error in Estimating the Mean

+ Measurable Outcome 3.11 Give standard errors for sample estimators of mean, variance, and event probability.
### Learn 3.4.3 The Error in Estimating the Variance ### Assess 3.4.1 The Error in Estimating the Mean 3.4.2 The Error in Estimating Probabilities

+ Measurable Outcome 3.12 Obtain confidence intervals for sample estimates of the mean, variance, and event probability.
### Learn 3.4.3 The Error in Estimating the Variance ### Assess 3.4.1 The Error in Estimating the Mean 3.4.2 The Error in Estimating Probabilities

+ 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.
### Learn No specific content identified. ### Assess No specific content identified.

+ Measurable Outcome 3.14 Describe the Response Surface Method.
### Learn No specific content identified. ### Assess No specific content identified.

+ 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.
### Learn No specific content identified. ### Assess No specific content identified.

+ Measurable Outcome 3.16 Describe the R2-metric, its use in measuring the quality of a response surface, and its potential problems.
### Learn No specific content identified. ### Assess No specific content identified.

+ 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.
### Learn 3.6.1 Design Optimization 3.6.3 Unconstrained Gradient-Based Optimization Methods ### Assess 3.6.2 Gradient Based Optimization 3.6.5 The 1D search

+ Measurable Outcome 3.18 Describe methods to estimate gradients.
### Learn No specific content identified. ### Assess No specific content identified.

+ Measurable Outcome 3.19 Use finite difference approximations to estimate gradients.
### Learn 2.3.2 Finite Difference Methods ### Assess No specific content identified.

+ Measurable Outcome 3.20 Interpret sensitivity information and explain its relevance to aerospace design examples.
### Learn No specific content identified. ### Assess No specific content identified.