ZMCintegral

ZMCintegral (Numba backened) is an easy to use python package which uses Monte Carlo Evaluation Method to do numerical integrations on Multi-GPU devices. It supports integrations with up to 16 multi-variables, and it is capable of even more than 16 variables if time is not of the priori concern.

To understand how ZMCintegral works, please refer to

https://arxiv.org/pdf/1902.07916v2.pdf

This new version supports parameter grid search, for this new functionality please refer to

https://arxiv.org/pdf/1910.01965.pdf

ZMCintegral usually takes a few minutes to finish the task.

Newest Features

Full flexibility of user defined functions
Multi-dimension integration
Multi-GPU supports
Stratified sampling
Heuristic tree search
Parameter grid search
very many functions

Installation

To run ZMCintegral (Numba-Ray version), the following packages needs to be pre-installed:

Numba
Ray

cudatoolkit


# create a new environment
$: conda create -n zmcintegral

install relavant package sequentially

$: pip install ZMCintegral $: conda install python=3.6 $: conda install numba=0.45.1 $: conda install cudatoolkit (**notification) $: pip install -U ray[debug]==0.7.1


** notification

make sure it is compatable with your driver verison, other wise you will get an error:

numba.cuda.cudadrv.driver.LinkerError: [218] Call to cuLinkAddData results in UNKNOWN_CUDA_ERROR ptxas application ptx input, line 9; fatal : Unsupported .version 6.4; current version is '6.2' ptxas fatal : Ptx assembly aborted due to errors


#### Prepare for running ZMCintegral
First of all, prepare machines with Nvidia GPU devices. choose one of them as a head node:

for head node

$: ray start --head --redis-port=XXXX --num-cpus=10 --num-gpus=4

for other nodes, here the redis-address is the ip of head node.

$: ray start --redis-address=XXX.XXX.XX.XX:XXXX --num-cpus=5 --num-gpus=2


Remeber to use

for head node

$: ray stop

for other nodes

$: ray stop

after evaluation.

## ![#1589F0](https://placehold.it/15/1589F0/000000?text=+) Integration of 6-dimensionals

#### The integration is of the form

![An image](./pics/sin6d.PNG) <!-- .element height="20%" width="20%" -->

#### whose result can be obtained via

![Image of expression 1](./pics/sin6d_theoretical.PNG)

#### In ZMCintegral this is done via

this is for installation by pip install zmcintegral

from ZMCintegral.ZMC.ZMCintegral_normal import MCintegral_normal import math import numpy as np

import time start = time.time()

user defined function

fun = """ import math

define a device function that should be used by cuda kernel

@cuda.jit(device=True) def fun(x): return math.sin(x[0]+x[1]+x[2]+x[3]+x[4]+x[5]+x[6]) """

define arguments that MCintegral_normal requires

depth = 1 sigma_multiplier = 5 num_trials = 5 num_chunks_in_one_dimension = 12

call MCintegral_normal

MC = MCintegral_normal(my_func = fun, domain = [[0,10],[0,10],[0,10],[0,10],[0,10],[0,10]], head_node_address = "XXX.XXX.XX.XX:XXXX", depth = depth, sigma_multiplier = sigma_multiplier, num_trials = num_trials, num_chunks_in_one_dimension = num_chunks_in_one_dimension)

obtaining the result

result = MC.evaluate()

print the formatted result

print('result = %s std = %s' % (result[0], result[1])) print('evaluation time {}'.format(time.time()-start))

#### The output is

total number of GPUs: 1 140 hypercube(s) need(s) to be recalculated, to save time, try increasing sigma_multiplier. result = -49.47563512703137 std = 1.9873890591413763 evaluation time 37.8058066368103


## ![#1589F0](https://placehold.it/15/1589F0/000000?text=+) Integration of 4-dimensionals with parameters

#### The integration is of the form

![Image of expression 1](./pics/parameter%20integration.PNG)

#### wohse results can be obtained via

![Image of expression 1](./pics/parameter%20integration%20theoretical.PNG)

#### In ZMCintegral this is done via

this is for installation by pip install zmcintegral

from ZMCintegral.ZMC.ZMCintegral_functional import MCintegral_functional import math import numpy as np

import time start = time.time()

user defined function

fun = """ import math

define a device function that should be used by cuda kernel

@cuda.jit(device=True) def fun(x,para): return math.sin(x[0]+x[1]+x[2]+x[3]+para[0]+para[1]) """

para contains two parameters

para = [[1,2,3,4,5],[1.1,2.2,3.1]]

sample points is taken to 10**6

sample_points = 10**6

the parameter grid has totally 5*3=15 points

we choose 3 batches as an example

batch_size = 5

call MCintegral_functional

MC = MCintegral_functional(my_func = fun, domain = [[0,1],[0,1],[0,1],[0,1]], parameters = para, head_node_address = "XXX.XXX.XX.XX:XXXX", num_points = sample_points, batch_size = batch_size)

obtaining the result

result = MC.evaluate() result = print(np.reshape(result,[3,5])) print('evaluation time {}'.format(time.time()-start))

#### The output is (which can be compared with Mathematica's results)

<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< Detected total number of GPUs: 1 Total parameter grid size: 15, Each GPU cycle will cover 5 grid values, Total GPU cycles: 3 Evaluating, please wait... [[-0.64027657 -0.81054914 -0.23614549 0.55573641 0.83644859] [-0.8100885 -0.23610632 0.55550291 0.83625073 0.34960498] [-0.23645369 0.55572789 0.83633015 0.34781143 -0.45968675]] evaluation time 4.728282690048218


## ![#1589F0](https://placehold.it/15/1589F0/000000?text=+) Very many integrations

#### Suppose integration is of the form and results can be obtained via
![Image of expression 1](./pics/multi-function.PNG)

#### In ZMCintegral this is done via

this is for installation by pip install zmcintegral

from ZMCintegral.ZMC.ZMCintegral_multifunctions import MCintegral_multi_function import math import numpy as np

import time start = time.time()

user defined function and domains

funs = ["math.sin(x[0]+x[1]+x[2]+x[3])*math.tan(x[0])",\ "math.cos(x[0]+x[1]+x[2]+x[3])"]

domains = [[[0,1],[0,10],[0,1],[0,1]],\ [[0,1],[0,10],[0,1],[0,1]]]

sample points is taken to 10**6

sample_points = 10**6

call MCintegral_functional

MC = MCintegral_multi_function(my_funcs = funs, domains = domains, head_node_address = "XXX.XXX.XXX.XXX:XXXX", num_points = sample_points)

obtaining the result

result = MC.evaluate() print('result {}'.format(np.array(result).flatten())) print('evaluation time {:.4f} s'.format(time.time()-start))


#### The output is (which can be compared with Mathematica's results)

Detected total number of GPUs: 1 Evaluating, please wait... result [-0.425192 -1.64925066] evaluation time 0.9935 s


## ![#1589F0](https://placehold.it/15/1589F0/000000?text=+) - tuning parameters

The following four parameters can be tuned to fit special cases.

| parameter        | usage           | example           | default  |
|:-------------:|:-------------:|:-------------:|:-----:|
| num_trials     | Evaluate the integration for num_trials times. Better kept within 10. | 10 | 5 |
| depth | For importance sampling. A domain is magnified for depth times. Better kept within 3. |3|2|
| num_chunks_in_one_dimension     | The number of chunks users want to set along one dimension | 10 | 4 |
| sigma_multiplier | Only domains that have very large standardand deviations (hence, very unstable) should be magnified and re-evaluated. Domains which are beyond sigma_multiplication * &sigma; should be recalculated.|3|4|

#### Attention
The user defined function must be organized in string format as shown in the following example. And the function name in the string mutst be `fun`, something like:

user defined function

fun = """ import math

define a device function that should be used by cuda kernel

@cuda.jit(device=True) def fun(x): # here the function name must be set as fun return xxx """


## ![#1589F0](https://placehold.it/15/1589F0/000000?text=+) More Help

**One should read the [documentation](https://numba.pydata.org/numba-doc/dev/cuda/index.html) for the Numba package's CUDA capabilities when trying to use this package.** ZMCintegral is only compatible with device functions as Numba does not support dynamic parallelism. This is important when designing the integrated function.

Issues with CUDA should first be resolved by looking at the [CUDA documentation](https://docs.nvidia.com/cuda/index.html).

For further questions and technical issues, please contact us at

whz168@mail.ustc.edu.cn (Hong-Zhong Wu 伍宏忠)

zjacob@mail.ustc.edu.cn (Jun-Jie Zhang 张俊杰)

mosescao@mail.ustc.edu.cn (Xiao-Yan Cao 曹晓岩)

## ![#1589F0](https://placehold.it/15/1589F0/000000?text=+) License
----

The package is coded by Jun-Jie Zhang and checked by Hong-Zhong Wu of University of Science and Technology of China.

**This package is free**
you can redistribute it and/or modify it under the terms of 
the Apache License Version 2.0, January 2004 (http://www.apache.org/licenses/).

**File Structure**

ZMCintegral │ README.md │ LICENSE │ setup.py │ └───ZMC │ │ ZMCintegral_functional.py │ │ ZMCintegral_normal.py │ │ ZMCintegral_multifunctions.py │ │ init.py │ └───pics │ multi-function.PNG │ sin6d.PNG │ sin6d_theoretical.PNG │ parameter integration.PNG │ parameter integration theoretical.PNG



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Letianwu / ZMCintegral

readme

ZMCintegral

Newest Features

Installation

install relavant package sequentially

for head node

for other nodes, here the redis-address is the ip of head node.

for head node

for other nodes

this is for installation by pip install zmcintegral

user defined function

define a device function that should be used by cuda kernel

define arguments that MCintegral_normal requires

call MCintegral_normal

obtaining the result

print the formatted result

this is for installation by pip install zmcintegral

user defined function

define a device function that should be used by cuda kernel

para contains two parameters

sample points is taken to 10**6

the parameter grid has totally 5*3=15 points

we choose 3 batches as an example

call MCintegral_functional

obtaining the result

this is for installation by pip install zmcintegral

user defined function and domains

sample points is taken to 10**6

call MCintegral_functional

obtaining the result

user defined function

define a device function that should be used by cuda kernel