Sample Project Report, CMSC 878R/AMSC 698RA Student ...December 7, 2002AbstractAn abstract about your problem1 Format for pr oject submission1.1 Summary InformationTitle of the Project: Solution of the Laplace equationStudent’s Name: Nail GumerovDimensionality of the Problem: 2Fast computational method used: MLFMMMother Function: Φ(y, x)=ln|y − x|Size of the problem: N ∼ 104,M∼ 104Computational domain Square, [0, 10] × [0, 10]Max allowed error 10−6, absoluteYou can also provide other details such asWhether FMM is part of an iterative procedure No; Yes, for linear system; Yes for something elseWhat is the iterative method PCG, GMRES, Arnoldi, etc,How many iterations does it takeWhether preconditioning was usedYou can also include details on your MLFMM implementationType of E2 neighborhood 3-neighborhood, 5 neighborhood ...Scheme of translation reducedTranslation operators p × p matrixSingle S|R-translation complexity O(p2) for matrix evaluation and multiplicationLanguage of implementation mixed: C++/Matlab11.2 Project Description1.2.1 BackgroundSolution of 2D Laplace equation is important for many physical applications including fluid dy-namics, magneto and electrostatics, and computation of particle motion in potential fields.... (oneparagraph).Also mention if this project is of research interest for you, and if you intend continuing workingon it ...Some literature review ...1.2.2 Mathematical Background(at least one paragraph)2D Laplace equation can be written in the form∇2φ =0, ∇2=∂2∂x2+∂2∂y2.Fundamental solution of this equation (Green’s function) is... .1.2.3 Statement of the ProblemThe problem addressed in the project is to develop a fast method for evaluation of the potentialgenerated by a large number of sources of arbitrary intencities and spatial locations... (one para-graph).1.3 Method of Solution1.3.1 Description of the FMM version usedMiddleman, SL, ML1.3.2 R- and S- basesWhar are the function basis for the R and S expansions1.3.3 S|S-, S|R-, and R|R-operatorsHow do you translate? How do you compute the entries of the matrices?1.3.4 Error Bounds and Selection of the Truncation Number1.3.5 Selection of Grouping (Clustering Parameter)Analytical optimization, numerical optimization21.4 Comments on Implementation1.5 Results of Computations1.5.1 PerformanceCompare fast and slow methods.1.5.2 ErrorIllustrate error of computation.1.5.3 OptimizationTell what parameters of the FMM you varied to achieve better performance (e.g. variation of thegroping parameter in range showed, that the optimal parameter should be...)1.5.4 Computational ExamplesE.g. several random and regular point distributions. Some analysis and discussion.1.6 Future WorkDo you plan to continue working on this?1.7