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#BBFMM2D

This is the first public release of the BBFMM2D library.
Date: July 24th, 2013

Version 3.1 - First external release.

%% Copyleft 2013: Sivaram Ambikasaran, Ruoxi Wang and Eric Darve
%% Developed by Sivaram Ambikasaran, Ruoxi Wang
%% Contact: [email protected] (Sivaram), [email protected] (Ruoxi)
%%
%% This program is free software; you can redistribute it and/or modify it under the terms of MPL2 license.
%% The Source Code Form is subject to the terms of the Mozilla Public License, v. 2.0. If a copy of the MPL was not %% distributed with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

###1. INTRODUCTION

BBFMM2D is an open source package of the Black-box Fast Multipole Method in 2 dimensions.
The Black-box Fast Multipole Method is an O(N) fast multipole method, which is a technique to calculate sums of the form

where is kernel function, are observation points, are locations of sources, and are charges at corresponding locations. BBFMM3D provides an O(N) solution to matrix-vector products of the type Ax. In that case the relation between A and K is:

This implementation of the FMM differs from other methods by the fact that it is applicable to all smooth kernels K. [Give examples of RBF kernels, 1/r, log r, Stokes, etc.] The precomputation time is also minimal.

The approximation scheme used in the FMM relies on Chebyshev interplation to construct low-rank approximations for well-separated clusters. In addition the use of Singular Value Decomposition ensures that the computational cost is minimal. In particular the rank is optimally chosen for a given error.

Please cite the following paper if you use this code:

Fong, William, and Eric Darve. "The black-box fast multipole method." Journal of Computational Physics 228, no. 23 (2009): 8712-8725. You can see details here.

###2. DIRECTORIES AND FILES

./examples/		:	Example input C++ codes; Needed to read input from user or from input file.  
./src/			:	Source code in C++  
./header/		:	Relevant header files  
./exec/			:	Executables for BBFMM2D  
./input/		:	The input file.  
./README.md		:	This file  
./License.md	:	License file  
./Makefile		:	Makefile

###3. TUTORIAL ####3.1 To Get Started

1. To use BBFMM2D, you need to have Eigen library.

2. Download Eigen from here: http://eigen.tuxfamily.org/index.php?title=Main_Page.

3. Create a directory named Codes/ inside the main Eigen folder and copy the directory BBFMM2D/ into the directory Codes/.

4. Open the Makefile, which is inside the folder BBFMM2D. Ensure that you have included the path to Eigen in the line containing CFLAGS. For instance, in the above setting, the path "-I ./../../" should be included in the Makefile.

5. To check whether things are set up correctly, you can perform the following action: Go to the directory where Makefile is in, then key in the following three commands in the terminal:

    make binary_file_mykernel
    cd exec/
    ./H2_2D_MVP_binary_file_mykernel
   

####3.2 Basic usage

#####3.2.1 BBFMM2D with standard kernel

The basic usage of BBFMM2D with standard kernel is as follows:

#include"BBFMM2D.hpp"  
...
{
unsigned long N;        // Number of charges;
unsigned m;             // Number of sets of charges;
vector<Point> location; // Locations of the charges;
double* charges;        // All the different sets of charges;
unsigned short nChebNodes	=	9; // Number of Chebyshev nodes per dimension; 
…
H2_2D_Tree Atree(nChebNodes, charges, location, N, m);// Build the fmm tree;

/* The following can be repeated with different kernels */
double* potential;
potential = new double[N*m];

kernel_Gaussian A;
A.calculate_Potential(Atree,potential);
...
}

This example first build a fmm tree with this line:
H2_2D_Tree Atree(nChebNodes, charges, location, N, m);
where H2_2D_Tree is a class of fmm tree, the constructor takes 5 arguments:

• nChebNodes(unsigned short):
  Number of Chebyshev nodes per dimension. The value should be at least 3, and we recommend to take value from 3 to 10. (Larger number of Chebyshev nodes would give better result but with much more time)
• charges(double*):
  All the different sets of charges. This pointer should point to an array with size N x m, and the data should be stored in column-wise. ( i.e. first set of charges, followd by second set of charges, etc)
• location(vector):
  Locations of the charges in 2D domain. In BBFMM2D, we assume here that observation points are the same as locations of sources. Here Point is a structure type with x and y coordinate defined.
• N(unsigned long):
  Number of charges.
• m(unsigned):
  Number of sets of charges.

Once the tree is created, you can compute the matrix-matrix product with as many kernels as you want.(see 3.2.4) The code shows an example using Gaussian kernel:

kernel_Gaussian A;
A.calculate_Potential(Atree,potential);

The result is computed via calculate_Potential(), which is a method of class kernel_Gaussian. The first argument of calculate_Potential() is the fmm tree that we just created; and the second argument potential is a pointer to the result, and the result is stored column-wise in potential.

#####3.2.2 Options of provided kernels

Below are the details of the kernel functions K we have provided:
( For all the kernel functions, we denote r to be Euclidean distance between x and y. )

Options of kernels:

• LOGARITHM kernel:
  usage: kernel_Logarithm
  kernel function:
  

• ONEOVERR2 kernel:
  usage: kernel_OneOverR2
  kernel function:
  

• GAUSSIAN kernel:
  usage: kernel_Gaussian
  kernel function:
  

• QUADRIC kernel:
  usage: kernel_Quadric
  kernel function:
  

• INVERSEQUADRIC kernel:
  usage: kernel_InverseQuadric
  kernel function:
  

• THINPLATESPLINE kernel:
  usage: kernel_ThinPlateSpline
  kernel function:
  If you want to define your own kernel, please see 3.2.3.

#####3.2.3 BBFMM2D with user defined kernel

The basic usage is almost the same as 3.2.1 except that you have to define your own routine of computing kernel. One example code is as follows:

#include"BBFMM2D.hpp"  
class myKernel: public kernel_Base {
public:
virtual double kernel_Func(Point r0, Point r1){
    //implement your own kernel here
    double rSquare	=	(r0.x-r1.x)*(r0.x-r1.x) + (r0.y-r1.y)*(r0.y-r1.y);
    return exp(-pow(pow(rSquare,0.5)/900.0;8,0.5));
	}
};
...
{
…
H2_2D_Tree Atree(nChebNodes, charges, location, N, m);// Build the fmm tree;

/* The following can be repeated with different kernels */
...
myKernel A;
A.calculate_Potential(Atree,potential);
...
}

You can define your own kernel inside kernel_Func(Point r0, Point r1), it takes two Points as input and returns a double value ().

#####3.2.4 Usage of multiple kernels

You can also use multiple kernels (user defined kernels and standard kernels) in one file, but make sure to have different class names. e.g.

class myKernelA: public kernel_Base{
	public:
	virtual double kernel_Func(Point r0, Point r1) {
	...
	}
}
class myKernelB: public kernel_Base{
	public:
	virtual double kernel_Func(Point r0, Point r1) {
	...
	}
}
…

{
…
H2_2D_Tree Atree(nChebNodes, charges, location, N, m);// Build the fmm tree;

/* The following can be repeated with different kernels */
...
myKernelA A;
A.calculate_Potential(Atree,potentialA);
...
myKernelB B;
B.calculate_Potential(Atree,potentialB);
…  
kernel_Gaussian C;
C.calculate_Potential(Atree,potentialC);
...
kernel_Quadric D;
D.calculate_Potential(Atree,potentialD);  
…
}

The basic usage is already domonstrated in 3.2.1 and 3.2.3. Once you have built the FMM tree, you can use different kernels to compute the matrix-matrix multiplication without rebuilding the tree. You can choose kernel type from standard kernels given by us ( see 3.2.2 ), or you can define your own kernel ( see 3.2.3 )

###4. ROUTINES FOR INPUTING AND OUTPUTING DATA We have provided several routines for reading data from text file and binary file, and writing data into binary file.

####4.1 Reading meta data from text file

void read_Metadata_BBFMM2D (const string& filenameMetadata, unsigned long& N, unsigned& m);

The first argument, filenameMetadata is the filename for your meta data (the number of locations, number of sets of charges). The number of locations is stored in N, and number of sets of charges is stored in m.

File format:

Number of charges, Number of sets of charges

For example:

5000, 10

####4.2 Reading from text file The prototype of function to read input from text file is:

void read_Location_Charges (const string& filename, unsigned long N, vector<Point>& location, unsigned m, double*& charges);

The first argument is the filename of your text file, the second argument N and forth argument m are the number of locations and number of sets of charges respectively. This function stores location in location and stores charges column-wise in charges.

File format:
For each row, it should start with locations, and followed by a row in B ( if we do AB multiplication, B is the R.H.S.). Here note that elements should be separated using ','. If some element is 0, you can leave it as empty instead of 0. If all the elements in a row is 0, nothing need to be typed after the location.(spaces are allowed)

The row should look like this:

(location[0],location[1]) (elem1,elem2,elem3,elem4,…,elemn)

For example:

(-0.999984,-0.676221)   	(0.480685,0.869803,-0.188232,0.548587,-0.771039,0.73709,0.126494)    
(0.869386,-0.5408)  
(0.0655345,0.891162) (-0.193033,,-0.0287383,,-0.520512,0.33891,)  
(0.342299,-0.246828) (0.0732668,,,,,,0.0951028)  
(-0.984604,-0.44417) (,0.782447,-0.867924,0.485731,-0.729282,-0.481031,0.541473)  

####4.3 Reading from binary file

void read_Location_Charges_binary(const string& filenameLocation, unsigned long N, vector<Point>& location, const string& filenameHtranspose,unsigned m, double* charges);

The first argument filenameLocation and the forth argument filenameHtranspose are binary file names for location and H' respectively (Here H'is the right part of AH'). N is the number of locations and m is the number of sets of charges. The data of locations is stored in location and the data of charges is stored in charges column-wise.

File format:

1. Binary file for charges:

   Elements of charges is stored this way:
   It should be stored column-wise, i.e.
   first set of charges, followed by second set of charges, etc.

2. Binary file for Location:

   Elements are stored column-wise:

    loc0.x
    loc1.x  
    …
    loc0.y
    loc1.y
    ...
   

####4.4 Writing into binary file

void write_Into_Binary_File(const string& filename, double* outdata, int numOfElems);  

This first argument is the filename for your output data. The second argument is a pointer to the output data, and the last argument is the number of elements in the array of your output data.

###5. EXAMPLES

We have provided several examples for BBFMM2D. Go to examples/, read through the files both must be self explanatory for the most part. You can use our examples with your own input. ####5.1 Making changes to the examples for your own application

1. If you want to generate input through your own routine, and use the standard kernels:

   Go to /examples/, open "H2_2D_MVP_get_input_through_routine_standard_kernel.cpp".

   • To generate input through routine:

     Change void get_Location() and void get_Charges()

   • To use standard kernels:

     Choose the kernel type in main(), options of kernels are in 3.2.2

2. If you want to read input from text file, and use standard kernels:

   Go to the folder /input/, put your input file inside of this folder.

   Go to the folder /examples/, open "H2_2D_MVP_textfile_standard_kernel.cpp".

   • To change input filename:

     string filenameInput = "../input/test_input.txt";

   • To use standard kernels:

     The same step as described in 1.

3. If you want to generate input through your own routine, and use your own kernel:

   Go to /examples/, open "H2_2D_MVP_get_input_through_routine_myKernel.cpp".

   • To define your own kernel:

     Modify class myKernel.

   • To generate your input:

     The same step as described in 1.

4. If you want to read input from text file, and use your own kernel:

   Go to /examples/, open "H2_2D_MVP_textfile_myKernel.cpp".

   • To define your own kernel:

     Modify class myKernel.

   • To change input filename:

     The same step as described in 2.

5. If you want to read input from binary file, and use standard kernel:

   Go to /examples/, open "H2_2D_MVP_binary_file_standard_kernel.cpp".

   • To change input filename:

     string filenameLocation = "../input/test_Location.bin";
string filenameCharges = "../input/test_Charges.bin";
string filenameMetadata = "../input/metadata.txt";
     change them into your input filenames.

   • To use standard kernels:

     The same step as described in 1. ;

6. If you want to read input from binary file, and use your own kernel:

   Go to /examples/, open "H2_2D_MVP_binary_file_mykernel.cpp".

   • To change the input filename:

     The same step as described in 5.

   • To define your own kernel:

     Modify class myKernel.

When using our examples, make sure that the input file format are the same as described in 4.

####5.2 Run examples

Here we give an example:
If you want to use "H2_2D_MVP_textfile_standard_kernel.cpp"

1. As stated earlier to run the code, go to the appropriate directory and key in the following:

    make textfile_standard_kernel
   

2. Make sure you have changed or cleaned the .o files from previous compilation. To clean the irrelevant files, key in:

    make clean
   

3. To tar the file, key in

    make tar
   

4. Read through the makefile for other options.

To run other .cpp files:

1. H2_2D_MVP_get_input_through_routine_myKernel.cpp
   key in:

    make get_input_through_routine_myKernel   
   

2. H2_2D_MVP_get_input_through_routine_standard_kernel.cpp
   key in:

    make get_input_through_routine_standard_kernel
   

3. H2_2D_MVP_textfile_myKernel.cpp
   key in:

    make textfile_myKernel 
   

4. H2_2D_MVP_binary_file_mykernel.cpp
   key in:

    make binary_file_mykernel
   

5. H2_2D_MVP_binary_file_standard_kernel.cpp
   key in:

    make binary_file_standard_kernel