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Add some notes and clean up TODO.md a little bit
author Jonatan Werpers <jonatan@werpers.com>
date Wed, 23 Mar 2022 13:09:31 +0100
parents 6b94663f4376
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1 # Notes
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2
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3 ## Reading operators
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4
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5 Jonatan's suggestion is to add methods to `Laplace`, `SecondDerivative` and
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6 similar functions that take in a filename from which to read stencils. These
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7 methods encode how to use the structure in a file to build the particular
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8 operator. The filename should be a keyword argument and could have a default
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9 value.
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10
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11 * This allows easy creation of operators without the user having to handle stencils.
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12 * The user can easily switch between sets of operators by changing the file stecils are read from.
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13
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14 Grids for optimized operators could be created by reading from a .toml file in
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15 a similar fashion as for the operators. The grid can then be used in a
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16 `Laplace` method which dispatches on the grid type and knows how to read the
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17 optimized operators. The method would also make sure the operators match the
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18 grid.
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19
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20 Idea: Make the current upper case methods lower case. Add types with the upper
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21 case names. These types are tensor mappings for the operator but also contain
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22 the associated operators as fields. For example:
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23
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24 ```julia
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25 L = Laplace(grid)
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26 L.H
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27 L.Hi
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28 L.e
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29 L.d
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30 L.M
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31
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32 wave = L - L.Hi∘L.e'∘L.d
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33 ```
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34
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35 These types could also contain things like borrowing and such.
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36
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37 ## Storage of operators
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38 We need to change the toml format so that it is easier to store several
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39 operator with different kinds of differentiations. For example there could be
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40 several operators of the same order but with different number of boundary
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41 points or different choice of boundary stencils.
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42
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43 Properties that differentiate operators should for this reason be stored in
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44 variables and not be section or table names.
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45
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46 Operators/sets of stencils should be stored in an [array of tables](https://toml.io/en/v1.0.0-rc.3#array-of-tables).
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47
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48 We should formalize the format and write simple and general access methods for
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49 getting operators/sets of stencils from the file. They should support a simple
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50 way to filter based on values of variables. There filters could possibly be
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51 implemented through keyword arguments that are sent through all the layers of
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52 operator creation.
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53
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54 * Remove order as a table name and put it as a variable.
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55
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56 ### Parsing of stencil sets
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57 At the moment the only parsing that can be done at the top level is conversion
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58 from the toml file to a dict of strings. This forces the user to dig through
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59 the dictionary and apply the correct parsing methods for the different parts,
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60 e.g. `parse_stencil` or `parse_tuple`. While very flexible there is a tight
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61 coupling between what is written in the file and what code is run to make data
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62 in the file usable. While this coupling is hard to avoid it should be made
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63 explicit. This could be done by putting a reference to a parsing function in
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64 the operator-storage format or somehow specifying the type of each object.
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65 This mechanism should be extensible without changing the package. Perhaps
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66 there could be a way to register parsing functions or object types for the
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67 toml.
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68
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69 If possible the goal should be for the parsing to get all the way to the
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70 stencils so that a user calls `read_stencil_set` and gets a
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71 dictionary-structure containing stencils, tuples, scalars and other types
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72 ready for input to the methods creating the operators.
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74 ## Variable second derivative
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75
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76 2020-12-08 after discussion with Vidar:
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77 We will have to handle the variable second derivative in a new variant of
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78 VolumeOperator, "SecondDerivativeVariable?". Somehow it needs to know about
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79 the coefficients. They should be provided as an AbstractVector. Where they are
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80 provided is another question. It could be that you provide a reference to the
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81 array to the constructor of SecondDerivativeVariable. If that array is mutable
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82 you are free to change it whenever and the changes should propagate
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83 accordingly. Another option is that the counter part to "Laplace" for this
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84 variable second derivate returns a function or acts like a functions that
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85 takes an Abstract array and returns a SecondDerivativeVariable with the
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86 appropriate array. This would allow syntax like `D2(a)*v`. Can this be made
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87 performant?
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88
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89 For the 1d case we can have a constructor
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90 `SecondDerivativeVariable(D2::SecondDerivativeVariable, a)` that just creates
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91 a copy with a different `a`.
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92
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93 Apart from just the second derivative in 1D we need operators for higher
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94 dimensions. What happens if a=a(x,y)? Maybe this can be solved orthogonally to
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95 the `D2(a)*v` issue, meaning that if a constant nD version of
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96 SecondDerivativeVariable is available then maybe it can be wrapped to support
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97 function like syntax. We might have to implement `SecondDerivativeVariable`
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98 for N dimensions which takes a N dimensional a. If this could be easily
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99 closured to allow D(a) syntax we would have come a long way.
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100
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101 For `Laplace` which might use a variable D2 if it is on a curvilinear grid we
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102 might want to choose how to calculate the metric coefficients. They could be
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103 known on closed form, they could be calculated from the grid coordinates or
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104 they could be provided as a vector. Which way you want to do it might change
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105 depending on for example if you are memory bound or compute bound. This choice
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106 cannot be done on the grid since the grid shouldn't care about the computer
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107 architecture. The most sensible option seems to be to have an argument to the
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108 `Laplace` function which controls how the coefficients are gotten from the
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109 grid. The argument could for example be a function which is to be applied to
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110 the grid.
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111
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112 What happens if the grid or the varible coefficient is dependent on time?
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113 Maybe it becomes important to support `D(a)` or even `D(t,a)` syntax in a more
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114 general way.
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115
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116 ```
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117 g = TimeDependentGrid()
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118 L = Laplace(g)
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119 function Laplace(g::TimeDependentGrid)
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120 g_logical = logical(g) # g_logical is time independent
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121 ... Build a L(a) assuming we can do that ...
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122 a(t) = metric_coeffs(g,t)
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123 return t->L(a(t))
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124 end
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125 ```
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126
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127 ## Known size of range and domain?
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128 Is there any reason to use a trait to differentiate between fixed size and unknown size?
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129
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130 When do we need to know the size of the range and domain?
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131 * When indexing to provide boundschecking?
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132 * When doing specialised computations for different parts of the range/domain?
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133 * More?
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134
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135 Maybe if we should have dynamic sizing it could be only for the range. `domain_size` would not be implemented. And the `range_size` would be a function of a vector that the LazyTensor is applied to.
332
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136
319
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137 ## Reasearch and thinking
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138 - [ ] Use a trait to indicate that a LazyTensor har the same range and domain?
319
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139 - [ ] Check how the native julia doc generator works
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140 - [ ] Check if Vidars design docs fit in there
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141 - [ ] Create a macro @lazy which replaces a binary op (+,-) by its lazy equivalent? Would be a neat way to indicate which evaluations are lazy without cluttering/confusing with special characters.
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142 - [ ] Dispatch on Lower() instead of the type Lower so `::Lower` instead of `::Type{Lower}` ???
332
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143 Seems better unless there is some specific reason to use the type instead of the value.
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144 - [ ] How do we handle mixes of periodic and non-periodic grids? Seems it should be supported on the grid level and on the 1d operator level. Between there it should be transparent.
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145 - [ ] Can we have a trait to tell if a LazyTensor is transposable?
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146 - [ ] Is it ok to have "Constructors" for abstract types which create subtypes? For example a Grids() functions that gives different kind of grids based on input?
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147 - [ ] Figure out how to treat the borrowing parameters of operators. Include in into the struct? Expose via function dispatched on the operator type and grid?
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148
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149 ## Identifiers for regions
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150 The identifiers (`Upper`, `Lower`, `Interior`) used for region indecies should probabily be included in the grid module. This allows new grid types to come with their own regions.
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151
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152 ## Regions and tensormappings
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153 - [ ] Use a trait to indicate if a LazyTensor uses indices with regions.
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154 The default should be that they do NOT.
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155 - [ ] What to name this trait? Can we call it IndexStyle but not export it to avoid conflicts with Base.IndexStyle?
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156 - [ ] Figure out repeated application of regioned LazyTensors. Maybe an instance of a tensor mapping needs to know the exact size of the range and domain for this to work?
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157
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158 ### Ideas for information sharing functions
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159 ```julia
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160 using StaticArrays
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161
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162 function regions(op::SecondDerivativeVariable)
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163 t = ntuple(i->(Interior(),),range_dim(op))
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164 return Base.setindex(t, (Lower(), Interior(), Upper()), derivative_direction(op))
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165 end
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166
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167 function regionsizes(op::SecondDerivativeVariable)
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168 sz = tuple.(range_size(op))
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169
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170 cl = closuresize(op)
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171 return Base.setindex(sz, (cl, n-2cl, cl), derivative_direction(op))
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172 end
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173
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174
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175 g = EquidistantGrid((11,9), (0.,0.), (10.,8.)) # h = 1
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176 c = evalOn(g, (x,y)->x+y)
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177
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178 D₂ᶜ = SecondDerivativeVariable(g, c, interior_stencil, closure_stencils,1)
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179 @test regions(D₂ᶜ) == (
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180 (Lower(), Interior(), Upper()),
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181 (Interior(),),
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182 )
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183 @test regionsizes(D₂ᶜ) == ((1,9,1),(9,))
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184
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185
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186 D₂ᶜ = SecondDerivativeVariable(g, c, interior_stencil, closure_stencils,2)
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187 @test regions(D₂ᶜ) == (
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188 (Interior(),),
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189 (Lower(), Interior(), Upper()),
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190 )
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191 @test regionsizes(D₂ᶜ) == ((11,),(1,7,1))
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192 ```
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193
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194
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195 ## Boundschecking and dimension checking
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196 Does it make sense to have boundschecking only in getindex methods?
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197 This would mean no bounds checking in applys, however any indexing that they do would be boundschecked. The only loss would be readability of errors. But users aren't really supposed to call apply directly anyway.
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198
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199 Preferably dimensions and sizes should be checked when lazy objects are created, for example TensorApplication, TensorComposition and so on. If dimension checks decreases performance we can make them skippable later.
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200
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201 ## Vector valued grid functions
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202 Från slack konversation:
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203
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204 Jonatan Werpers:
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205 Med vektorvärda gridfunktioner vill vi ju fortfarande att grid funktionen ska vara till exempel AbstractArray{LitenVektor,2}
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206 Och att man ska kunna göra allt man vill med LitenVektor
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207 typ addera, jämföra osv
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208 Och då borde points returnera AbstractArray{LitenVektor{Float,2},2} för ett 2d nät
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209 Men det kanske bara ska vara Static arrays?
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210
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211 Vidar Stiernström:
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212 Ja, jag vet inte riktigt vad som är en rimlig representation
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213 Du menar en vektor av static arrays då?
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214
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215 Jonatan Werpers:
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216 Ja, att LitenVektor är en StaticArray
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217
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218 Vidar Stiernström:
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219 Tuplar känns typ rätt inuitivt för att representera värdet i en punkt
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220 men
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221 det suger att man inte har + och - för dem
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222
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223 Jonatan Werpers:
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224 Ja precis
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225
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226 Vidar Stiernström:
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227 så kanske är bra med static arrays i detta fall
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228
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229 Jonatan Werpers:
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230 Man vill ju kunna köra en Operator rakt på och vara klar eller?
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231
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232 Vidar Stiernström:
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233 Har inte alls tänkt på hur det vi gör funkar mot vektorvärda funktioner
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234 men känns som staticarrays är hur man vill göra det
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235 tuplar är ju immutable också
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236 blir jobbigt om man bara agerar på en komponent då
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237
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238 Jonatan Werpers:
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239 Hm…
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240 Tål att tänkas på
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241 Men det lär ju bli mer indirektion med mutables eller?
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242 Hur fungerar det?
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243 Det finns ju hur som helst både SVector och MVector i StaticArrays
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244
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245 Vidar Stiernström:
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246 När vi jobbat i c/c++ och kollat runt lite hur man brukar göra så lagrar man i princip alla sina obekanta i en lång vektor och så får man specificera i funktioerna vilken komponent man agerar på och till vilken man skriver
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247 så man lagrar grejer enl: w = [u1, v1, u2, v2, …] i 1D.
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248 Men alltså har ingen aning hur julia hanterar detta
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249
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250 Jonatan Werpers:
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251 Det vi är ute efter kanske är att en grid funcktion är en AbstractArray{T,2} where T<:NågotSomViKanRäknaMed
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252 Och så får den typen var lite vad som helst.
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253
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254 Vidar Stiernström:
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255 Tror det kan vara farligt att ha nåt som är AbstractArray{LitenArray{NDof},Dim}
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256 Jag gissar att det kompilatorn vill ha är en stor array med doubles
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257
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258 Jonatan Werpers:
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259 Och sen är det upp till den som använder grejerna att vara smart
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260 Vill man vara trixig kan man väl då imlementera SuperHaxxorGridFunction <: AbstractArray{Array{…},2} som lagrar allt linjärt eller något sånt
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261 Det kommer väl lösa sig när man börjar implementera vektorvärda saker
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262 Euler nästa!
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263 New
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264 Vidar Stiernström:
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265 Det vore skönt att inte behöva skriva såhär varje gång man testar mot en tupel :smile: @test [gp[i]...] ≈ [p[i]...] atol=5e-13
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266
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267 Jonatan Werpers:
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268 https://github.com/JuliaArrays/ArraysOfArrays.jl
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269 https://github.com/jw3126/Setfield.jl
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270
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271 ### Test-applikationer
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272 div och grad operationer
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273
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274 Enligt Wikipedia verkar det som att `∇⋅` agerar på första dimensionen av ett tensor fält och `div()` på sista.
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275 Om man generaliserar kanske `∇` i så fall bara lägger till en dimension i början.
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276
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277 Kan vi implementera `⋅`(\cdot) så att de fungerar som man vill för både tensor-fält och tensor-operatorer?
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278
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279 Är `∇` ett tensor-fält av tensor-operatorer? Vad är ett tensor-fält i vår kod? Är det en special-fall av en tensor-mapping?
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280
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281 ### Grid-funktionen
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282 Grid-funktionon har typen `AbstractArray{T,2} where T`.
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283 `T` kan vara lite vad som helst, tillexemel en SVector eller Array, eller tuple. TensorOperatorerna bryr sig inte om exakt vad det är, mer än att typen måste stödja de operationer som operatorn använder.
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284
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285 En nackdel kan vara hur man ska få ut gridfunktionen för tex andra komponenten.
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286
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287 Syntax:
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288 ```
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289 f(x̄) = x̄
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290 gf = evalOn(g, f)
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291 gf[2,3] # x̄ för en viss gridpunkt
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292 gf[2,3][2] # x̄[2] för en viss gridpunkt
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293 ```
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294
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295 Note: Behöver bestämma om eval on skickar in `x̄` eller `x̄...` till `f`. Eller om man kan stödja båda.
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296
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297 ### Tensor operatorer
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298 Vi kan ha tensor-operatorer som agerar på ett skalärt fält och ger ett vektorfält eller tensorfält.
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299 Vi kan också ha tensor-operatorer som agerar på ett vektorfält eller tensorfält och ger ett skalärt fält.
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300
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301 TBD: Just nu gör `apply_transpose` antagandet att domän-typen är samma som range-typen. Det behöver vi på något sätt bryta. Ett alternativ är låta en LazyTensor ha `T_domain` och `T_range` istället för bara `T`. Känns dock lite grötigt. Ett annat alternativ skulle vara någon typ av trait för transpose? Den skulle kunna innehålla typen som transponatet agerar på? Vet inte om det fungerar dock.
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302
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303 TBD: Vad är målet med `T`-parametern för en LazyTensor? Om vi vill kunna applicera en difference operator på vad som helst kan man inte anta att en `LazyTensor{T}` bara agerar på instanser av `T`.
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304
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305 Man kan implementera `∇` som en tensormapping som agerar på T och returnerar `StaticVector{N,T} where N`.
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306 (Man skulle eventuellt också kunna låta den agera på `StaticMatrix{N,T,D} where N` och returnera `StaticMatrix{M,T,D+1}`. Frågan är om man vinner något på det...)
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307
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308 Skulle kunna ha en funktion `range_type(::LazyTensor, ::Type{domain_type})`
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309
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310 Kanske kan man implementera `⋅(tm::LazyTensor{R,D}, v::AbstractArray{T,D})` där T är en AbstractArray, tm på något sätt har komponenter, lika många som T har element.
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311
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312 ### Ratade alternativ:
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313
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314
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315 #### 2.AbstractArray{T,2+1} where T (NOPE!)
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316 Blir inte den här. Bryter mot alla tankar om hur grid funktioner ska fungera. Om de tillåts ha en annan dimension än nätet blir allt hemskt.
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317
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318 Man låter helt enkelt arrayen ha en extra dimension. En fördel är att man har en väldigt "native" typ. En nackdel kan vara att det eventuellt blir rörigt vilken dimension olika operatorer ska agera på. I värsta fall behöver vi "kroneckra in" de tillagda dimensionerna. Vektorfältets index kommer också att bli det första eftersom vi vill att de ska lagras kontinuerligt i minnet pga chachen. (Går kanske att lösa med en custom typ men då krånglar man till det för sig). En fördel skulle vara att man enkelt får ut olika komponenter.
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319
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320 Syntax:
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321 ```
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322 gf = eval_on_grid(g,f)
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323 gf[:,2,3] # Hela vektorn för en gridpunkt
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324 gf[2,2,3] # Andra komponenten av vektor fältet i en punkt.
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325 gf[2,:,:] #
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326 ```
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327
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328 ### Evaluering av funktioner på nät
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329 Hur ska man skriva funktioner som evalueras på nätet? `f(x,y) = ...` eller `f(x̄) = ...`? Eller båda? Kan eval_on_grid se skillnad eller får användaren specificera?
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330
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331 ```
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332 f(x,y) = [x^2, y^2]
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333 f(x̄) = [x̄[1]^2, x̄[2]^2]
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334 ```
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335
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336 Påverkas detta av hur vi förväntar oss kunna skapa lata gridfunktioner?
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337
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338 ### Komponenter som gridfunktioner
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339 En viktig operation för vektor fält är att kunna få ut komponenter som grid-funktioner. Detta behöver antagligen kunna ske lazy.
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340 Det finns ett par olika lösningar:
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341 * Implementera en egen typ av view som tar hand om detta. Eller Accessors.jl?
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342 * Använda en LazyTensor
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343 * Någon typ av lazy-broadcast
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344 * En lazy array som applicerar en funktion för varje element.
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345
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346 Skulle vara en fördel om det är hyffsat generiskt så att en eventuell användare kan utöka det enkelt om de har någon egen exotisk typ. Eller ska man vila helt på
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347
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348 Syntax:
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349 ```
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350 gf = eval(...)
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351 component(gf,2) # Andra komponenten av en vektor
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352 component(gf,2,3) # (2,3) elementet av en matris
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353 component(gf,:,2) # Andra kolumnen av en matris
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354 @ourview gf[:,:][2]
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355 ```
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356
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357 ## Grids embedded in higher dimensions
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358
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359 For grids generated by asking for boundary grids for a regular grid, it would
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360 make sense if these grids knew they were embedded in a higher dimension. They
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361 would return coordinates in the full room. This would make sense when
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362 drawing points for example, or when evaluating functions on the boundary.
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363
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364 Implementation of this is an issue that requires some thought. Adding an extra
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365 "Embedded" type for each grid would make it easy to understand each type but
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366 contribute to "type bloat". On the other hand adapting existing types to
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367 handle embeddedness would complicate the now very simple grid types. Are there
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368 other ways of doing the implentation?
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369
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370 ## Performance measuring
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371 We should be measuring performance early. How does our effective cpu and memory bandwidth utilization compare to peak performance?
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372
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373 We should make these test simple to run for any solver.
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374
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375 See [this talk](https://www.youtube.com/watch?v=vPsfZUqI4_0) for some simple ideas for defining effecive memory usage and some comparison with peak performance.
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376
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377
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378 ## Adjoint as a trait on the sbp_operator level?
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379
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380 It would be nice to have a way of refering to adjoints with resepct to the sbp-inner-product.
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381 If it was possible you could reduce the number of times you have to deal with the inner product matrix.
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382
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383 Since the LazyOperators package is sort of implementing matrix-free matrices there is no concept of inner products there at the moment. It seems to complicate large parts of the package if this was included there.
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384
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385 A different approach would be to include it as a trait for operators so that you can specify what the adjoint for that operator is.
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386
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387
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388 ## Name of the `VolumeOperator` type for constant stencils
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389 It seems that the name is too general. The name of the method `volume_operator` makes sense. It should return different types of `LazyTensor` specialized for the grid. A suggetion for a better name is `ConstantStencilVolumeOperator`