Notebook

Engy-5310: Computational Continuum Transport Phenomena Spring 2021 UMass Lowell; Prof. V. F. de Almeida 07Feb21

Homework 04 (08Feb21)¶

$ \newcommand{\Amtrx}{\boldsymbol{\mathsf{A}}} \newcommand{\Bmtrx}{\boldsymbol{\mathsf{B}}} \newcommand{\Mmtrx}{\boldsymbol{\mathsf{M}}} \newcommand{\Imtrx}{\boldsymbol{\mathsf{I}}} \newcommand{\Pmtrx}{\boldsymbol{\mathsf{P}}} \newcommand{\Lmtrx}{\boldsymbol{\mathsf{L}}} \newcommand{\Umtrx}{\boldsymbol{\mathsf{U}}} \newcommand{\Smtrx}{\boldsymbol{\mathsf{S}}} \newcommand{\xvec}{\boldsymbol{\mathsf{x}}} \newcommand{\avec}{\boldsymbol{\mathsf{a}}} \newcommand{\bvec}{\boldsymbol{\mathsf{b}}} \newcommand{\cvec}{\boldsymbol{\mathsf{c}}} \newcommand{\rvec}{\boldsymbol{\mathsf{r}}} \newcommand{\fvec}{\boldsymbol{\mathsf{f}}} \newcommand{\mvec}{\boldsymbol{\mathsf{m}}} \newcommand{\gvec}{\boldsymbol{\mathsf{g}}} \newcommand{\zerovec}{\boldsymbol{\mathsf{0}}} \newcommand{\norm}[1]{\bigl\lVert{#1}\bigr\rVert} \newcommand{\transpose}[1]{{#1}^\top} \DeclareMathOperator{\rank}{rank} \newcommand{\Reals}{\mathbb{R}} \newcommand{\thetavec}{\boldsymbol{\theta}} $

Name: `your name`¶

Rubric for each assignment:¶

Context	Points
Precision of the answer	80%
Answer Markdown readability	10%
Code readability	10%

Guidance:¶

Save your work frequently to a file locally to your computer.
During your work and before submitting the final version do: Kernel -> Restart & Run All, to verify your notebook runs correctly.
Save your file again.
Upload the notebook file in the Blackboard submission portal as:
lastname-firstname-labwork-xx.ipynb

Table of Assignments ¶

Assignment 1 (95 pts) Function approximation with Lagrange polynomial basis functions.

1.1)(25 pts) Minimum 2-norm residual method proof.
1.2)(25 pts) Minimum 3-norm residual method proof.
1.3)(45 pts) Algorithm for the minimum 3-norm residual method.

Assignment 2 (5 pts) .

Assignment 1 (95 pts)¶

1.1)(25 pts) Prove the following about the least-squares method.¶

As described by the theory of linear combination of basis functions approximation in this course, one choice for optimal combination of coefficients is the minimization of the sum of the squares of the residual vector \begin{equation*} \phi(\tilde{\cvec}^*) = \min\limits_{\cvec\,\in\,\Reals^n} \, \norm{\tilde{\rvec}(\cvec)}^2_2. \end{equation*}

Demonstrate that the optimum $\tilde{\cvec}^*$ is also the optimum of the 2-norm of the residual vector, that is the conventional magnitude

\begin{equation*} \phi(\tilde{\cvec}^*) = \min\limits_{\cvec\,\in\,\Reals^n} \, \norm{\tilde{\rvec}(\cvec)}_2. \end{equation*}

Therefore the least-squares method delivers the residual with smallest 2-norm.

Analytical mathematical proof:

1.2)(25 pts) Prove the following about the least-cubes method:¶

Demonstrate that the optimum $\tilde{\cvec}^*$ in the least-cubes method \begin{equation*} \phi(\tilde{\cvec}^*) = \min\limits_{\cvec\,\in\,\Reals^n} \, \norm{\tilde{\rvec}(\cvec)}^3_3, \end{equation*}

is also the optimum of the minimum 3-norm of the residual vector, that is the conventional magnitude

\begin{equation*} \phi(\tilde{\cvec}^*) = \min\limits_{\cvec\,\in\,\Reals^n} \, \norm{\tilde{\rvec}(\cvec)}_3. \end{equation*}

Therefore the least-cubes method delivers the residual with smallest 3-norm.

Analytical mathematical proof:

1.3)(45 pts) Develop an algorithm for the least-cubes method:¶

Describe and algorithm to compute the optimum $\tilde{\cvec}^*$ of the least-cubes residual \begin{equation*} \phi(\tilde{\cvec}^*) = \min\limits_{\cvec\,\in\,\Reals^n} \, \norm{\tilde{\rvec}(\cvec)}^3_3. \end{equation*}

Analytical algorithm:

Assignment 2 (5 pts)¶

Describe the most unclear point in the content presented so far. Then, make an effort to clarify the point on your own.¶

Answer: