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Traditional math operators, such as the inner product and curl, have been available in Diderot for a while and are not documented here. This directory provides examples with the newer features available in the Diderot-Dev branch. The features are organized in three different categories: Field and function operators "fn_", tools "tool_", and field definitions "dfn_". New operators that can applied in Diderot-Dev include composition, concatenation,inverse, min, max, clerp, clamp, field selection, and find cell. Tools that have been developed include Diderot's automated testing tool and printIR, which prints the intermediate representation. There are also new ways to define fields with closed-form expressions and by data created by solving a PDE (FEM).

Overview of Repo

A.Functions and Operators

A1. Function composition

Given two fields

field#k(d1)[α]F0;
field#k(d0)[d1]F1;

A user might want to use the result of one field to probe the other, also known as field composition

tennsor[α] t = F0(F1(x));

The problem with ^ is that the result is a tensor (and no longer differentiable). Instead, added a field composition operator that can be applied to differentiable fields.

field#k(d0)[α]G = F0 ∘ F1;

There is a type restriction on this type of operation:

field#k(d1)[α]× field#k(d0)[d1] → field#k(d1)[α] The second argument is a vector field, whose length is the dimension of the first argument.

The function name "compose" can also be used

field#k(d0)[α]G = compose(F0,F1);

The field arguments can be created as the result of another operation.

field#k(d0)[α]G = compose(F0, A∙B);

Note: The result of (A∙B) must still fit the type requirements mentioned earlier

The user might want to chain multiple composition operators:

field#k(d0)[α]G = F0 ∘ F1∘ F2;

Note: The new field variable F2 will have to meet the same type requirements mentioned earlier

Details

  • Branch: Diderot-Dev
  • Syntax: "compose" and "◦"
    • field#k(d1)[α]× field#k(d0)[d1] → field#k(d1)[α]
  • Text: EIN IR design, rewriting rules, and resolved bugs listed in Doc
  • Examples Directory : fn_composition

A2. Function concatenation

Given multiple fields

field#k(d)[α] A;  field#k(d)[α] B; field#k(d)[α] C;

A user might want to put two fields together to create a new field

field#k(d)[3,α] M = [A,B,C]

Note: The field arguments must have the same type

Details

  • Branch: Diderot-Dev

  • Syntax: [,]

    • field#k(d)[α] × field#k(d)[α]× ....field#k(d)[α] → field#k(d)[n,α] where n is the number of arguments

  • Syntax:concat

    • field#k(d)[α] × field#k(d)[α] → field#k(d)[2,α]

  • Text: concat Mentioned in dissertation-FW.

  • Examples Directory: fn_concatenation

A3. Matrix Inverse

Given matrix M and a second order tensor field F

tensor[d,d] M;
field#k(d)[d,d] F;

A user can take the matrix inverse

tensor[d,d] MI = inv(M);
field#k(d)[d,d] FI = inv(F)

Note: The two fields have the same type

Details

  • Branch: Diderot-Dev
  • Syntax: “inv()”
    • field#k(d)[d,d] → field#k(d)[d,d]
    • tensor[n,n] → tensor[n,n]
  • Text: Mentioned in dissertation-Design

A4. Min and Max

Given two scalar fields

tensor[] M;  
field#k(d)[] A;
field#k(d)[] B;

A user can compare them by taking the maximum and minimum

field#k(d)[] F = Max(A,B);  
field#k(d)[] G = Min(A,B)

Then take the derivative

field#k-1(d)[d] F = ∇ Max(A,B);

Details

  • Branch: Diderot-Dev
  • Syntax: “Max” and "Min"
    • field#k(d)[] × field#k(d)[] → field#k(d)[]
  • Examples Directory fn_min-max

A5. Clerp and Clamp

It is typical for a user to use clamp and lerp sequentially and on scalars

vec2 a; vec2 b; real t;
real xx = clamp(a[0], b[0], lerp(a[0], b[0], t));
real yy = clamp(a[1], b[1], lerp(a[1], b[1], t));
vec2 out = [xx,yy];

We addressed this issued in two ways. The first was to allow non-scalar arguments to math functions clamp() and lerp()

vec2 out = clamp (a, b, lerp(a, b, t))

The second was to create the new clerp() function

vec2 out = clerp (a,b,t)

which will apply clamp and lerp as expected.

Details

  • Branch: Vis15 & Diderot-Dev
  • Syntax: “clerp()” New Clerp function
    • tensor[i] × tensor[i] × real → tensor[i]
    • tensor[i] × tensor[i] × real × real × real → tensor[i]
  • Syntax: “clamp()” Clamp function can be applied to general tensors
    • tty = tensor[α]
    • tty × tty × tty → tty
  • Examples directory: fn_clerp

A6.Field Selection

Inside a Diderot program there may be many different field definitions and computations on those fields

field#k(d)[α] A; field#k(d)[α] B; field#k(d)[α] C;  field#k(d)[α] D;...

Typically, this meant that the Diderot programmer would comment some lines in and out and recompile when trying different definitions. To make it easier we created a new function "swap()" that alternates between different field definitions based on the selection_id.

int selection_id;
field#k(d)[] F = swap(selection_id,A,B,C,D);

As a note, the selection_id can also be an input variable.

input int selection_id;

Details

  • Branch: Diderot-Dev
  • Syntax: “swap()"
    • fty = field#k(d)[α]
    • int × fty × fty → fty
    • int × fty × fty× .... → fty
    • int × fty × fty × fty × fty × fty × fty → fty
  • Arguments
    • Selection id The first argument is an integer that serves to select a field. i.e. id=2 chooses the second field argument
    • Field arguments The function accepts 2-6 field arguments.
  • Potential issue: The value of the Selection id is clamped. If the first argument is id=-7 the id is set to 1 instead of throwing an error
  • Examples directory: fn_selection

A7. Math functions: GetCell()

A user can define a field created as the result of an outside tool. The code to define this type of field is copied below

input fem#1(2)[] f;
field#1(2)[] F = FEM(f, "Diderot-Dev/fnspace_data/data.json")

but check out the relevant directory dfn_fem for more details. When this type of field is probed at a position then the right cell needs to be found. The surface level operator GetCell() allows the Diderot user to use that value in the Diderot program.

int currentcell = GetCell(F,pos);

An inside test will return a boolean

bool TF = insideF(pos,F);

The user probes the FEM field at a position with

tensor[] out = F(pos);

Run

  • Change path to Diderot-Dev compiler in data/makedefs.gmk and in the relevant diderot program
  • Install Firedrake and activate with

    source firedrake/bin/activate

Details

  • Branch: Diderot-Dev

  • Syntax:

    • Inside Check if a position is inside a field-insideF(): tensor[d]×field#k(d)[α] →boolean
    • GetCell Get the cell number the point is located in-GetCell(): field#k(d)[α] ×tensor[d]× → int

    When there is no Cell the function returns -1.

  • Notes: Defining a FEM field: dfn_fem.

  • Examples directory: fn_getCell

A8. Math functions: Abs()

The absolute function can be used on fields

#version 2.0
field#2(1)[] A  (x) = x*2;
field#2(1)[] F = abs(A);
field#1(1)[] G = ∇F;

The derivative is derived as the following

output real out = |x*2|/(x*2);

Details

  • Branch: Diderot-Dev
  • Syntax: “abs”: field#k(d)[] → field#k(d)[]
  • Examples directory fn_abs

B.Tools

B1. DATm: Diderot's Automated testing

Details

  • Branch: Diderot-Dev
  • Use: Test operators on and between tensors/image data based on correctness
  • Tool: DATm:Diderot’s Automated Testing tool
  • Text: ICSE-AST paper and Testing chapter in Dissertation

    Testing environment variables in Pg 102 Adding a new operator in Pg 113

User Guide

Getting Started

Quick instructions

  1. Checkout github directory for DATm

    git clone https://github.com/cchiw/DATm.git

  2. Change cpath in Frame to your absolute path to diderot branches. See Set Up about other variables you might want to change. 3. Starting Testing with command line arguments. See Section on Running DATm.

    python3 datm.py

Set Up: variables and testing frame settings

Settings in the testing process can be changed by commenting in/out variables in the Frame (input.py).

  • Change branch(s_branch) : Tell DATm the name of branch being tested. Some branch names are built in(vis15,Diderot-Dev, Chiw17). Comment in the right s_branch variable in Frame

     #s_branch  = branch_vis15
 s_branch = branch_dev 
 #s_branch = branch_chiw17
 #s_branch  = branch_other

    or comment in branch_other and set the variable to a string in shared/base_constants.py

  • Complexity (s_layer): The core computation in a Diderot test program can be simple or more complicated. s_layer indicates the number of operators to apply in a core computation. That number can be 1, 2, or 3.

  • type of field(c_pde_test) : DATm test tensors and fields. The fields can either be made by nrrd files or by Firedrake (outside tool to solve PDE solutions). For an original Diderot Field types created with nrrd, comment in s_field = field_conv in the Frame. For PDE solutions comment in s_field = field_pde and change the path in fem/makedefs.gmk.

    s_field = field_conv 
#s_field = field_pde

  • type of search (s_random_range) For an exhaustive testing approach, set variable s_random_range to 0 in Frame. For randomized testing set the variable to x where the probability of a single test case being generated is 1 in x+1.

  • order of coefficients(s_coeff_style) : The order of coefficients for the polynomial creating synthetic data. The data can either be linear, quadratic or cubic.

     #s_coeff_style = coeff_linear
 s_coeff_style = coeff_quadratic
 #s_coeff_style = coeff_cubic

  • testing environment: You can comment in and out variables in Frame. This includes variables to change the number of samples, type of arguments,..

    More details in Pg 102 in Dissertation,

Labels

  • Each test case has a testing label

    Of the form "p_o1...l2"

Running DATm: command-line commands and targetting testing

The testing environment is indicated by the Frame. The scope helps target a specific operator, test, or family of programs.

  • Run everything:

    python3 datm.py

  • Test a single operator

    python3 datm.py id # where id is a number Note Each operator has a unique id. List printed to screen and copies to rst/stash/results_ops.txt

  • Family of computations
    Rerun (group of) tests by using 1-4 integers from the testing label. For instance, the label

    “p_o27_o0_t0_tN_tN_l2”

    can rerun with command

    python datm.py 27 0 0

Results passes/fails

Great, everything is running now, but how do I look at the results? In the directory rst/stash are several text files that record the test cases (with labels)

  • results_final.txt:The results of each test case
  • results_terrible.txt: Reports test cases with errors
  • results_ty.txt:Test Types used
  • results_ops.txt:Operators with ids

Development

  • Adding a new operator to DATm:
    1. Add to operator constant: shared/obj_operator.py
    2. Add case to type-checker: shared/obj_typechecker.py
    3. Add way to evaluate that operator applied to polynomials: nc/nc_eval.py

    More details in Pg 113 in Dissertation

B2. Printing the intermediate representation

Translate our compiler's intermediate representation (EIN) into a readable latex or unicode format.

Derivations by hand can be tedious and error-prone. To address that issue we support a higher-order notation. For instance, consider the following Diderot code

image(2)[] a;				image(2)[] b;
field#4(3)[]A =k⊛ a; 			field#4(3)[]B =k⊛ b;f		field#4(3)[3]V =k⊛v;
field#3(3)[3]G =  (A*V)/B;
tensor[3,3] T =  (∇⊗G)(pos);

Note that that the image type is defined separately. These are the variable names that will be used when printing out the IR.

The differentiation operations ∇⊗∇ are distributed across the tensor operations defined in filed type G. Internally, Diderot's rewriting system applies tensor calculus based rewrites that will distribute the differentiation operation. There are two ways to print out the intermediate representation: a surface level operator and with a command line argument. In both cases the user can use command line arguments to make specific formatting and rewriting choices.

Specify a single computation to print

We can use the operation printIR() to print a specific computation on the command line

output int out =  printIR(T);

or save the output to a text file "tmpRead"

output int out =  printIR(T,"tmpRead");

By default the tool will save one version of the IR in two files: Unicode in "tmpRead.txt" and latex version in tmpRead.latex.

Print whatever we can

In lieu of using the printIR() operation the user can use command line argument "--readEin". It will print a larger portion of the computations in the program (every EIN operator in the high-to-mid stage of the compiler).

Command Line Arguments

  • Format
    • (default) : both latex and unicode output
    • --formatTex : latex output only
    • --formatUni : unicode output only
  • Representation
    • (default) : surface level representation (without indices)
    • --repEin : EIN notation only
    • --repMultiple : Four variations of the computations in surface level representation
  • Output
    • (default) : print to terminal
    • --savePDF : Save to file "output_tmp"

Details

  • Branch: Diderot-Dev
  • Syntax: printIR
  • Run: runs with command line flags and surface level operator
  • Issues/Future Work
    • pull out of strands
    • image types needs to be defined separately so we have a unique variable to refer to

C. Field Definitions

C1. Field Definition: Closed Form expression

Users can define closed form expressions. The expression can include tensor operators and variables. Differentiation is applied by differentiating in respect to one or more variables.

Single input variable

It is natural to define a function with a closed form expression: F(x) = 7٭x. We allow a user to define such a field in Diderot

field#k(d)[d] F(x) = 7٭x;
tensor[d] v =...;
tensor[d] out = F(v);
tensor[d,d] jacob = ∇⊗F(v)
tensor[] divergence = ∇•F(v)

Note that variable x is a vector of length d, where d is the dimension of the field.

The probed field is evaluated as

F(x) ⇨7٭x ⇨ 7*[xa , xb ,..xd ]] where x = [xa , x b,..xd ]

If v = [3,5] then F(v) = [21, 35]

The user can apply other tensor and field operators on the field term F including differentiation. The Jacobian of our field F creates the following matrix.

∇⊗F(x) ⇨∇⊗(7٭x) ⇨ [[7,0],[0,7]]

and the divergence creates the following

∇•F(x) ⇨∇•(7٭x) ⇨ 14

Differentiation is applied to the entire expression on the right hand side of the field definition F and in respect to the variable x. Internally, The tensor variable x is expanded into it's components and the differentiation operator is applied to the components.

∇⊗F(x)⇨ [[∇a 7xa , ∇a 7xb ], [∇b 7xa ,∇b 7xb ]]

Details

Field Definition: Closed Form expression - Multiple Input

Users can define closed form expressions. The expression can include tensor operators and variables. Differentiation is applied by differentiating in respect to some variable(s).

A function can be defined with multiple variables. F (a, b) = a + b and similarly a field can be defined with multiple input variables

type F (real a, real b) = a+b;

The type is an ofield Dideot type. The dimension is a list of vector sizes for the corresponding input variables.

>   ofield#k(d1,...dn)[α] (tensor[d1] ×...× tensor[dn])→ ofield#k(d1,...dn)[α]

Differentiation is applied in respect to all the variables indicated between "(" ")": ∇ F= ∇ a +∇ b
Alternatively, to indicate some input variables as constant variables (differentiation of it is 0) use a second pair parathesis.

type G (real a)(real b) = exp; // takes derivative of exp in respect to a 
type I (real b)(real a) = exp; // takes derivative of exp in respect to b
type F (real a, real b) = exp; // takes derivative of exp in respect to a and b

Here is an example: F (s, a, b) = s*(A²-B)

ofield#k(1,1,1)[] K = (real A)(real B, real S) = s*(A²-B);  //takes derivative in respect to A
ofield#k(1,1,1)[] L = (real B)(real A real S) =s*(A²-B)     //takes derivative in respect to B
ofield#k(1,1,1)[] M = (real A, real B)(real S) =s*(A²-B)    //takes derivative in respect to A and B

The code does the following computations:
∇K = ∇A s*(A²-B) = ∇A s*(A²-B) = s2a
∇L = ∇B s*(A²-B) = ∇B s*(A²-B)= -s
∇M = ∇AB s*(A²-B) = ∇AB s*(A²-B)= s*(2*A -1)

Details

  • Branch: Diderot-Dev
  • Syntax:
    • ofield#k(d1,...dn)[α] (tensor[d1] ×...× tensor[dn])→ ofield#k(d1,...dn)[α]
    • ofield#k(d1,...dn)[α] (tensor[d1] ×tensor[di]) (tensor[di+1]×...× tensor[dn])→ ofield#k(d1,d2,...dn)[α]
  • Issues:
    • Using ofield here instead of field. Need to generalize field type or add new basis-vars.
  • Examples in directory dfn_input_multi

C2. FEM

Field Definition: FEM

We support computations on fields defined by outside sources.

In Action

There are four steps to the implementation process:

  1. Diderot code (observ.diderot)
  2. C code that communicates to the generated Diderot code (observ_init.c)
  3. Python code that initiates the C code and creates FEM data (observ.py)
  4. Running the program (run.sh)

For the most part steps 2-4 are the same for each example and code can be easily reused.

1.Diderot Code (observ.diderot)

The user declares a FEM field with the functionFEM and two arguments. The first argument is an input variable and the second is a path to the relevant data file.

input fem#k(d)[α] F0;
field#k(d)[α] F = FEM(F0, "data.json");

Define a fem field- FEM(): fem#k(d)[α] × string →field#k(d)[α]

2. C code that communicates to the generated Diderot code (observ_init.c)

The C code is used to communicate with the generated Diderot code. The function callDiderot_observ() can be called by outside tools.

void callDiderot_observ(char *Outfile, void *valF)

The function takes the name of the output nrrd file and a pointer to the field data.

Otherwise, the code here is FEM independent and does not need augmentation. .....

4. Running the whole thing (run.sh)

  • Change path to Diderot-Dev compiler in data/makedefs.gmk and in the relevant diderot program

  • Install Firedrake and activate with

    source firedrake/bin/activate

  • Make and run

    python observ.py

  • Branch: Diderot-Dev

  • Read full readme file in dfn_fem

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