MATH4064 2020-2021 Feedback Computational Applied Mathematics Open Book Exam (3 hours) Date released: 4 June 2021, 10am BST Exam result details (worth 60% of final mark) Numberofsittingstudents:40 Maximummarkpossible:100(=100%) Averagemark:63.5outof100(=63.5%) Maximummarkobtainedbystudent:91outof100(=91.0%) Numberofstudentswithmark70:17(=42.5%) Number of students with mark <50: 10 (=20.0%) Exam performance details per question Q1Approximation:average~56%(low-scoringquestion) Q2Nonlinearity;MachineLearning:average~77%(bestquestion) Q3NumericalODEs;MachineLearning:average~69% Q4NumericalPDEs(elliptic);Nonlinearity:average~60% Q5NumericalPDEs(parabolicandhyperbolic):average~56%(low- scoring question) Detailed feedback per question Q1 (a low-scoring question) a) i) Well done by almost all students. ii) Most students had some marks. Few students had full marks. Marks were received if a low-regular function (e.g., discontinuous) was mentioned, or a polynomial function. b) i) Well done by many students. Almost all students achieved some marks. But few students had full marks. ii) Some students did well on this. Many students skipped this question, likely because of time constraints. Q2 (best question) (a)i) Well done by almost all students. Some students had minor mistakes for which some marks were deducted. (a)ii) Most students that tried this question had many marks. Some students skipped this question. (b) i) Well done by almost all students. ii) Well done by almost all students. While marking the exam, it became clear that many students mis-interpreted what was requested: 2 x 2 = 4 matrices W, but instead provided 2 matrices W. It was easy to misinterpret this question, hence full marks could be obtained for 2 correct matrices only. Q3 (a)i) Well done by almost all students. ii) - iii) Many students did well and scored many marks or full marks. (b)i) Well done by almost all students. ii) Well done by many students. Almost all students got at least some marks. Q4 (a)i) Well done by almost all students. ii) Well done by approximately half of the students. Many students achieved many marks. Some skipped this question. (b)i) Well done by almost all students. ii) Skipped by many students. But those that attempted it did very well. Q5 (a low-scoring question) (a)i) Well done by many students. ii) Some students had full marks, but most students had some marks, due to some minor mistake. (b)i) Well done by almost all students. ii) Many students did not attempt this question. Approximately half of the students did, and got many marks. A LEVEL 4 MODULE, SPRING SEMESTER 2020-2021 COMPUTATIONAL APPLIED MATHEMATICS Time to complete THREE Hours plus THIRTY Minutes upload time Paper set: 04/06/2021 - 10:00 Paper due: 04/06/2021 - 13:30 Answer ALL questions Your solutions should be written on white paper using dark ink (not pencil), on a tablet, or typeset. Do not write close to the margins. Your solutions should include complete explanations and all intermediate derivations. Your solutions should be based on the material covered in the module and its prerequisites only. Any notation used should be consistent with that in the Lecture Notes. Submit your answers as a single PDF with each page in the correct orientation, to the appropriate dropbox on the modules Moodle page. Use the standard naming convention for your document: [StudentID]_[ModuleCode].pdf. A scan of handwritten notes is completely acceptable. Make sure your PDF is easily readable and does not require magnification. Text which is not in focus or is not legible for any other reason will be ignored. If your scan is larger than 20Mb, please see if it can easily be reduced in size (e.g. scan in black & white, use a lower dpi but not so low that readability is compromised). Staff are not permitted to answer assessment or teaching queries during the assessment period. If you spot what you think may be an error on the exam paper, note this in your submission but answer the question as written. Where necessary, minor clarifications or general guidance may be posted on Moodle for all students to access. If you submit your paper after the deadline, you will receive a mark of zero, unless you have an EC accepted. If you submit your paper less than thirty minutes after the deadline, please submit a fast track ECF. If you submit your paper more than thirty minutes after the deadline, please submit a normal ECF. MATH4064-E1 MATH4064-E1 Turn over 1. Approximation (a) i) Let and , for . Consider the cubic spline 1 MATH4064-E1 Referring to Study Material In your answers to the questions, you are allowed to refer to theorems, statements and other results in the Lecture Slides and Notes, as well as the Accompanying Reading Material. where , and is the standard B-spline of degree defined by Find a set of coefficients , , for which the spline interpolates the following data: and ii) Complete cubic splines that interpolate a function on an interval, using a uniform grid with grid-size , converge according to (1) where the constants and may depend on (but not on ). State three examples of (if possible): I. A function for which (1) holds with , but not with much larger than . II. A function for which (1) holds with some much larger than . III. A function for which (1) holds with some , but not with . (There is no need to provide an explanation for any of them.) (b) Consider the integral , and its quadrature approximation [6 marks] MATH4064-E1 i) Assume that and are the two roots of . Find possible weights and for which the quadrature approximation is exact for all polynomials of degree , with as large as possible. Also state the corresponding value of . Explain how you arrived at your answer. ii) Repeat the previous question, but now assuming that and . [5 marks] [4 marks] [5 marks] 2 MATH4064-E1 (a) Let be a function. To find such that , consider the following modified Newtons method: for (2) Note that, in contrast to the standard Newtons method, the Jacobian matrix in Eq. (2) is fixed for all . 2. Nonlinearity; Machine Learning i) Assuming that and , compute and when . ii) To analyse the method in Eq. (2), assume there is a and such that where the matrix is used as a short-hand notation for the initial Jacobian matrix, i.e., , and the matrix norm that is used is compatible with the vector norm . Prove that the approximations obtained by the method in (2) satisfy: Hint: You may use without proof the following corollary to the mean-value theorem: for any (b) Let denote the ReLU activator function. Consider the following artificial neural network: where , , , , ,and . i) Sketch the neural network architecture, that is, sketch the neurons, their connections, and indicate the input and output. [3 marks] ii) Suppose , , , ,and . Find two distinct for which classifies and as and , respectively. MATH4064-E1 Turn Over [5 marks] [6 marks] [6 marks] 3 (a) Consider the system of three ODEs and initial condition: MATH4064-E1 (3) 3. Numerical ODEs; Machine Learning where and. Itisgiventhat,with and . Let , denote the approximation obtained using Eulers method with step-size , i.e., where (4) i) Write the general solution of Eulers method in the following form: where , , , , and are to be determined, and where , and are constants independent of . Show that your answer is consistent with Eq. (4). Hint: You may use without proof that , where is any positive integer. [6 marks] ii) Assuming that , use your result from i) to state an initial condition for which does not go to as (in exact arithmetic). Provide a brief explanation to your answer. [2 marks] iii) Assuming again that , use your result from i) to now state all possible initial conditions for which as (in exact arithmetic). Provide a brief explanation to your answer. [2 marks] (5) (b) Let denote the ReLU activator function, and let a cost function be given by where and . i) Considering only the cases and , compute the partial derivatives and . [4 marks] MATH4064-E1 ii) To find the minimum of the cost function in (5), consider the gradient descent method. Suppose that , , and the initial approximation is . Find the values of the learning rate (step size) for which one step of the gradient descent method will ensure a decrease of the initial cost. [6 marks] 4 MATH4064-E1 (a) Consider the following elliptic differential equation subject to Dirichlet boundary conditions: 4. Numerical PDEs (elliptic); Nonlinearity and where . To approximate the solution, consider a standard, second-order, finite difference method using a grid with points for , and mesh-width . i) State the matrix in the system . where is the vector of interior grid-point approximations, and ii) Show that, for found in part i), the matrix 2-norm of its inverse remains bounded as . In other words, show that for all (sufficiently small) . Hint: You may use without proof that the eigenvalues of in the case of the differentialequation areequalto for (b) Consider now the following nonlinear differential equation subject to Dirichlet boundary conditions: and To approximate the solution, consider a standard, second-order, finite difference method usingagridwithpoints for,andmesh-width. i) Assuming (hence two interior grid points), obtain the system of nonlinear equations for the vector of unknowns . [4 marks] ii) Next, for the nonlinear system obtained in part (b)i), propose an iterative solution method that obtains successive approximations Clearly state the linear system that needs to be solved at each iteration. [4 marks] MATH4064-E1 Turn Over [7 marks] [5 marks] 5. Numerical PDEs (parabolic and hyperbolic) (a) Consider the following parabolic PDE, initial condition and boundary conditions: 5 MATH4064-E1 To approximate the solution, consider a finite difference method inspired by the theta method for systems of ODEs. In other words, this method has a parameter such that yields the standard explicit method, yields the standard implicit method, and yields the CrankNicolson method. Assume a time-step size and a spatial grid with grid-points , and . solved in this method for each time step . i) Using matrix and vector notation, state the system of equations that has to be ii) Denoting the error vector at time step by , it is given that [5 marks] where denotes the grid-function 2-norm, is a vector of truncation errors at time step , is a constant, and withand forall. Consideringseparately,, and , determine if (unconditionally). If not possible, find a condition on and to indeed obtain . (b) Consider the hyperbolic PDE and initial condition [5 marks] and (6) where . To approximate the solution, consider the following nonstandard scheme: where , with denoting the time-step size, and denoting the mesh-width. i) State the numerical flux so that the above scheme can be written in the form: ii) Define the truncation error for the above scheme and show that it is . MATH4064-E1 END [4 marks] [6 marks] 9/16 1/16 11/32 3/64 ()