manual.org

pochoir user manual

Introduction

This document describes how to use pochoir. In this section, high level concepts are introduced. Subsequent sections provide concrete guides to get started using pochoir, followed by a tutorial and finally a developer guide.

Overview of calculating response functions

• describe the steps and data products

Domains

A domain of a problem is the space on which it is defined. Driven by the finite-difference method (FDM), a pochoir domain is a finite, uniform, rectilinear (Cartesian) grid of points. In general, the domain may be N-dimensional though for practical problems we are restricted to 2D or 3D.

FDM for Laplace boundary value problem

• characteristics of problem where FDM is applicable (grid vs geometry feature sizes)
• define terms
  • domain
  • boundary value problem
    • boundary and initial values and boundary condition
  • initial value problem
    • paths

RK4/5 for paths initial value problem

Map S-R dot product across matrix of (path (x) weighting)

Getting started

Installation

• get package
• venv options (python -m venv venv ; source venv/bin/activate and echo layout python3 > .envrc; direnv allow)
• install (pip install -e . or pip install [email protected]:wirecell/pochoir.git or whatever)
• testing (pytest)

General usage

• CLI help
• general CLI vs command-level options
• environment variables (POCHOIR_STORE)

Using CPU vs GPU

• devices: “best” vs “numpy” vs torch “cpu” and “gpu” (still needs actual implementation to pick “best” or otherwise globally set)
• selection via CLI options
• when to care what device is used

Data and its storage

• main data objects
  • domain
  • scalar fields
  • vector fields
  • path start points
  • full paths
  • responses
• separate input and output vs input+output store
• HDF5 vs NPZ+JSON+directory
  • latter can work with snakemake or similar
  • converting between the two
• export formats
  • vtk
• input formats
  • invent something for electrode shape description, probably JSON based
    • maybe jsonnet
  • do we allow import of data object from, say, external NPZ?

Tutorial

Overview

This tutorial walks through calculating field response for a detector very similar to real detectors, if idealized for simplicity. It contains parallel ground plane and a cathode plane separated along the Z-axis which provides the nominal drift field. Near the ground plane are two “anode” planes each with a series of strips parallel in their plane and mutually perpendicular across planes. The anode planes have each a “bias voltage” applied and share through-holes so that electron drift paths may pass by the first “induction anode” plane and terminate on the second “collection anode” plane.

The tutorial will make a compromise of using a low-resolution grid in order to perform the solution quickly. After an initial solution, finer resolution will be explored in the context of comparing the use of CPU and GPU.

A number of “tricks” are used to further reduce computation. The drift field and paths are solved on a small domain exploiting periodic boundary conditions. The weighting fields far from a strip of interest do not vary in the dimension parallel to the strip which allows 2D “far fields” to be calculated on a broad 2D domain and then connected to 3D “near fields” calculated on a smaller 3D domain.

The tutorial then combines paths and weighting fields to produce current waveforms (the field response functions) and converts them to a format suitable for use by the Wire-Cell Toolkit detector simulation and signal processing. Along the way, the tutorial will show how to visualize and export intermediate results from the pochoir data store.

The tutorial will first walk through each step manually. The last step will show how to put these steps together so they may be repeated automatically and how this can facilitate exploring “hyper parameters” such as grid density. As the automation mechanism requires datasets as individual files we will use the NPZ+JSON store instead of a monolithic HDF5 file.

Initial Setup

Install pochoir as described elsewhere. To avoid repeating the pochoi -s/--store option we also set:

export POCHOIR_STORE=$(pwd)/tutorial-store

Define the problem domains

A domain can be created simply with the pochoir domain command:

pochoir domain --help

Usage: pochoir domain [OPTIONS] NAME

  Produce a "domain" and store it to the output dataset.

  A domain describes a finite, uniform grid in N-D space in these terms:

      - shape :: an N-D integer vector giving the number of grid
      points in each dimension.  Required.

      - origin :: an N-D spatial vector identifying the location of
      the grid point with all indices zero.

      - spacing :: a scalar or N-D vector in same distance units as
      used in origin and which gives a common or a per-dimension
      spacing between neighboring grid points.

  A vector is given as a comma-separated list of numbers.

  Note: this description corresponds to vtk/paraview uniform rectilinear
  grid, aka an "image".

Options:
  -s, --shape TEXT    The number of grid points in each dimension
  -o, --origin TEXT   The spatial location of zero index grid point (def=0's)
  -s, --spacing TEXT  The grid spacing as scalar or vector (def=1's)
  --help              Show this message and exit.

The origin vector is optional but allows one to adopt a convention other than identifying it with the “zero index” grid point. The shape describes the number of grid points on each dimension and spacing gives their common separation.

We will store our domains in a common group named domain. We need three:

drift

a 3D domain for drift field and paths

wfar

common 2D domain for all “far” weighting fields

wnear

common 3D domain for all “near” weighting field

Next we describe electrode shapes to be defined in the domain. To do this properly we must already know our models for the electrode shapes and how we wish to stitch wfar and wnear. For now, we just show the commands do create the domains and later will explain how the numbers are chosen.

pochoir domain --shape 51,51,201 --spacing 0.1 drift
pochoir domain --shape 151,151,201 --spacing 0.1 wnear
pochoir domain --shape 1051,1051 --spacing 0.1 wfar
ls -l $POCHOIR_STORE
ls -l $POCHOIR_STORE/drift

#+RESULTS[63028b25cb770b7d413ef3a72610e8ad4f9ae97e]:

total 12
drwxrwxr-x 2 bv bv 4096 Mar 22 14:12 drift
drwxrwxr-x 2 bv bv 4096 Mar 22 14:12 wfar
drwxrwxr-x 2 bv bv 4096 Mar 22 14:12 wnear
total 12
-rw-rw-r-- 1 bv bv 290 Mar 22 14:12 origin.npz
-rw-rw-r-- 1 bv bv 288 Mar 22 14:12 shape.npz
-rw-rw-r-- 1 bv bv 292 Mar 22 14:12 spacing.npz

Once defined, a domain is later referenced by its full path name for the group, eg domain/drift. The datasets are named according to their interpretation by pochoir.

Define initial and boundary value arrays

An initial value array (IVA) provides a scalar field from which the FDM solution begins. Each element holds either a known, applied potential or an initial guess of its solution. A boundary value array (BVA) asserts which of these two interpretations hold. A True element of a BVA implies that the corresponding element of an IVA is fixed, else it may be updated by FDM.

In general, the user must provide an IVA and BVA for drift and each weighting field calculation, though typically a common IVA is needed. To provide these arrays one must map a model of the real electrode geometry to the arrays via the constraints of the domain.

A user is free to do that any way that is convenient. pochoir provides two features that may help with the task.

First is support for “shape array painting” whereby the user provides descriptions of 2D or 3D shapes and a set of potential values and pochoir will apply them to produce IVA and BVA. See details in Shape Painting.

Second is pochoir provides high-level IVA/BVA generators for particular electrode configurations with are governed by higher level parameters. For this tutorial we will use the sandh (strips and holes) high level generator.

Solve Laplace equation

The Laplace equation can be solved by specifying initial and boundary value arrays, boundary conditions and convergence requirements.

pochoir fdm --help

Usage: pochoir fdm [OPTIONS] SOLUTION ERROR

  Solve a Laplace boundary value problem with finite difference method
  storing the result as named solution.  The error names an output array to
  hold difference in last two iterations.

Options:
  -i, --initial TEXT     Name initial value array, elements include boundary
                         values

  -b, --boundary TEXT    Name the boundary array, zero value elemnts subject
                         to solving

  -e, --edges TEXT       Comma separated list of 'fixed' or 'periodic' giving
                         domain edge conditions

  --precision FLOAT      Finish when no changes larger than precision
  --epoch INTEGER        Number of iterations before any check
  -n, --nepochs INTEGER  Limit number of epochs (def: one epoch)
  --help                 Show this message and exit.

We may make a trial solution which we save it and its error to caps-solution1 and caps-error1 arrays, respectively

pochoir fdm -e periodic,periodic \
        -i caps-initial -b caps-boundary \
        --epoch 10 -n 1 \
        caps-solution1 caps-error1  

maxerr: 43.046966552734375

The maximum difference between the solution at the penultimate and last iteration is the printed maxerr.

We can visualize solution and error:

pochoir plot-image caps-solution1 docs/caps-solution1.png
pochoir plot-image caps-error1 docs/caps-error1.png

The error is rather high and although this domain is small which makes the solution fast, we may reuse this first solution as the initial value array for continued solution:

pochoir fdm -e periodic,periodic \
        -i caps-solution1 -b caps-boundary \
        --epoch 10 -n 1 \
        caps-solution2 caps-error2  

maxerr: 21.263214111328125

pochoir plot-image caps-solution2 docs/caps-solution2.png
pochoir plot-image caps-error2 docs/caps-error2.png

We can continue this manual, high-level iteration or take a guess for the total number of internal iterations to reach the desired error level. Or, we may tell pochoir fdm to continue until either the requested number of epochs are reached or the maxerr falls below a requested precision. When using a precision, it is checked only after each epoch is complete and so the result will typically be over-precise.

pochoir fdm -e periodic,periodic \
        -i caps-solution2 -b caps-boundary \
        --epoch 10 -n 100 --precision 0.1 \
        caps-solution3 caps-error3  

maxerr: 13.02484130859375
maxerr: 8.48162841796875
maxerr: 5.6507568359375
maxerr: 3.921722412109375
maxerr: 2.8282470703125
maxerr: 2.056884765625
maxerr: 1.50787353515625
maxerr: 1.12823486328125
maxerr: 0.855926513671875
maxerr: 0.65093994140625
maxerr: 0.496307373046875
maxerr: 0.37939453125
maxerr: 0.2906494140625
maxerr: 0.2232666015625
maxerr: 0.17201995849609375
maxerr: 0.13336181640625
maxerr: 0.10352325439453125
maxerr: 0.08056640625

The solution is reached prior to 100 epochs. Again, let’s see it:

pochoir plot-image caps-solution3 docs/caps-solution3.png
pochoir plot-image caps-error3 docs/caps-error3.png

3D Laplace

• change in args w.r.t. 2D
• understand time/resource scaling with 2D
• visualize (matplotlib and paraview)

Use 2D as boundary condition for 3D

• derive 3D boundary values to 2D and merge with 3D boundary values
• understand precision of 2D as a function of 3D domain size

Weighting fields

The fantasy example of caps sets boundary values applicable for calculating the “real”, applied potential. The overall field response is tabulated for each sensitive electrode by calculating that electrode’s weighting potential. Thus we must apply the pochoir fdm command as above to a boundary value which sets the grid points on the sensitive electrode to unity and all others to zero.

Calculate and visualize gradient fields

The gradient of a scalar field gives a vector field. The E-field is the gradient of the applied potential scalar field and is needed for the next step of calculating paths. The W-fields, one per sensitive electrode are needed for the step after, calculating responses to paths.

The pochoir grad command will calculate and store the gradient allowing for visualization and later use.

pochoir grad --help

Usage: pochoir grad [OPTIONS] SCALAR VECTOR

  Calculate the gradient of a scalar field.

Options:
  --help  Show this message and exit.

pochoir grad \
        --domain adomain \
        caps-solution3 caps-efield3

We may visualize this field with:

pochoir plot-quiver \
        --domain adomain \
        caps-efield3 docs/caps-efield3.png

• 2D and 3D matplotlib
• paraview

Path initial value problem

• specify problem to solve
• specify initial value
• apply solver
• store result
• visualize (matplotlib and paraview)

Calculate responses

• combine path and fields for schockley-ramo
• exploit symmetry and equivalences
• visualize

Convert to Wire-Cell

Automation with Snakemake

• full chain, repeatable, performant processing
• what knobs to tune

Developer guide

todo

Some major features still wanted:

• [ ] a way for a 2D potential to serve as boundary conditions for a 3D problem by defining a 2D boundary and projecting it along one of the 3D dimensions.
• [ ] normalize the collection of metadata from each command which is stored on the result datasets. Name input datasets in a uniform manner, collect CLI parameters by name, collect and store time for body of command. Goal would be to produce post-hoc visualization showing how any given result was produced or determine which rules a given result influenced.
• [ ] form “cross product” of paths and weighting fields to give responses.