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author Jonatan Werpers <jonatan@werpers.com>
date Tue, 07 Mar 2023 09:21:27 +0100
parents 23f3b62388ba
children e352630a0309
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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
829
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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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73
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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?
320
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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.
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136
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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?
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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}` ???
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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 We implement this by refactoring RegionIndices to be agnostic to the region types and then moving the actual types to Grids.
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152
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153 ## Regions and tensormappings
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154 - [ ] Use a trait to indicate if a LazyTensor uses indices with regions.
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155 The default should be that they do NOT.
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156 - [ ] What to name this trait? Can we call it IndexStyle but not export it to avoid conflicts with Base.IndexStyle?
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157 - [ ] 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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158
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159 ### Ideas for information sharing functions
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160 ```julia
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161 using StaticArrays
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162
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163 function regions(op::SecondDerivativeVariable)
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164 t = ntuple(i->(Interior(),),range_dim(op))
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165 return Base.setindex(t, (Lower(), Interior(), Upper()), derivative_direction(op))
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166 end
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167
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168 function regionsizes(op::SecondDerivativeVariable)
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169 sz = tuple.(range_size(op))
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170
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171 cl = closuresize(op)
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172 return Base.setindex(sz, (cl, n-2cl, cl), derivative_direction(op))
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173 end
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174
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175
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176 g = EquidistantGrid((11,9), (0.,0.), (10.,8.)) # h = 1
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177 c = evalOn(g, (x,y)->x+y)
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178
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179 D₂ᶜ = SecondDerivativeVariable(g, c, interior_stencil, closure_stencils,1)
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180 @test regions(D₂ᶜ) == (
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181 (Lower(), Interior(), Upper()),
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182 (Interior(),),
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183 )
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184 @test regionsizes(D₂ᶜ) == ((1,9,1),(9,))
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185
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186
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187 D₂ᶜ = SecondDerivativeVariable(g, c, interior_stencil, closure_stencils,2)
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188 @test regions(D₂ᶜ) == (
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189 (Interior(),),
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190 (Lower(), Interior(), Upper()),
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191 )
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192 @test regionsizes(D₂ᶜ) == ((11,),(1,7,1))
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193 ```
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194
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195
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196 ## Boundschecking and dimension checking
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197 Does it make sense to have boundschecking only in getindex methods?
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198 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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199
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200 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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201
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202 ## Vector valued grid functions
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203
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204 ### Test-applikationer
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205 div och grad operationer
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206
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207 Enligt Wikipedia verkar det som att `∇⋅` agerar på första dimensionen av ett tensor fält och `div()` på sista.
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208 Om man generaliserar kanske `∇` i så fall bara lägger till en dimension i början.
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209
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210 Kan vi implementera `⋅`(\cdot) så att de fungerar som man vill för både tensor-fält och tensor-operatorer?
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211
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212 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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213
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214 ### Grid-funktionen
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215 Grid-funktionon har typen `AbstractArray{T,2} where T`.
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216 `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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217
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218 En nackdel kan vara hur man ska få ut gridfunktionen för tex andra komponenten.
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219
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220 Syntax:
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221 ```
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222 f(x̄) = x̄
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223 gf = evalOn(g, f)
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224 gf[2,3] # x̄ för en viss gridpunkt
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225 gf[2,3][2] # x̄[2] för en viss gridpunkt
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226 ```
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227
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228 Note: Behöver bestämma om eval on skickar in `x̄` eller `x̄...` till `f`. Eller om man kan stödja båda.
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229
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230 ### Tensor operatorer
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231 Vi kan ha tensor-operatorer som agerar på ett skalärt fält och ger ett vektorfält eller tensorfält.
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232 Vi kan också ha tensor-operatorer som agerar på ett vektorfält eller tensorfält och ger ett skalärt fält.
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233
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234 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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235
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236 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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237
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238 Man kan implementera `∇` som en tensormapping som agerar på T och returnerar `StaticVector{N,T} where N`.
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239 (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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240
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241 Skulle kunna ha en funktion `range_type(::LazyTensor, ::Type{domain_type})`
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242
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243 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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244
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245 ### Komponenter som gridfunktioner
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246 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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247 Det finns ett par olika lösningar:
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248 * Implementera en egen typ av view som tar hand om detta. Eller Accessors.jl?
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249 * Använda en LazyTensor
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250 * Någon typ av lazy-broadcast
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251 * En lazy array som applicerar en funktion för varje element.
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252
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253 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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254
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255 Syntax:
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256 ```
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257 gf = eval(...)
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258 component(gf,2) # Andra komponenten av en vektor
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259 component(gf,2,3) # (2,3) elementet av en matris
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260 component(gf,:,2) # Andra kolumnen av en matris
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261 @ourview gf[:,:][2]
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262 ```
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263
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264 ### Prestanda-aspekter
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265 [Vidar, Discord, 2023-03-03]
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266 Typiskt sett finns det två sätt att representera vektorvärda gridfunktioner AbstractArray{T,Dim} där T är en vektor över komponenterna. Man skulle alltså i 1D ha
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267 u = [ [u1[x1], u2[x1]] , [u1[x2], u2[x2]], ... [u1[xN], u2[xN]]]. Detta brukar kallas array of structs (AoS). Alternativet är struct of arrays (SoA), där man har alla gridpunkter för en given komponent u = [[u1[x1], u1[x2]],... u1[xN]], [u2[x1], u2[x2], ... u2[xN]]].
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268
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269 Personligen tycker jag att AoS känns som den mer naturliga representationen? Det skulle göra det enklarare att parallelisera en vektorvärd gridfunktion över gridpunkterna, och om man opererar på olika komponenter i samma funktion så är det också bra ur en minnesaccess-synpunkt då dessa kommer ligga nära vandra i minnet. Problemet är att AoS sabbar vektorisering på CPU då två gridpunkter i en komponent ligger långt bort från varandra. Efter lite eftersökningar (och efter att snackat lite med Ossian) så verkar det ändå som att AoS är dåligt på GPU, där man vill att trådar typiskt sett utföra samma operation på närliggande minne.
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270
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271 Vad tänker du kring detta ur ett interface-perspektiv? Jag hittade paketet https://github.com/JuliaArrays/StructArrays.jl som verkar erbjuda AoS-interface men SoA-minneslayout så det kanske kan vara något vi kan använda? Inte native-stödd på samma sätt som SVector, men verkar iaf utvecklas aktivt.
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272
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273 [Efter telefonsamtal] För optimal prestanda behöver vi antagligen se till att man kan räkna ut varje komponent i en punkt individuellt. Detta så att man har frihet att till exempel låta den innersta loopen hålla komponentindexet konstant för att underlätta intruktionsvektorisering.
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274
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275
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276 [Vidare tankar]
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277 * Det borde bara vara output-gridfunktionen som behöver special-indexeras? Det viktiga på inputsidan är att den är lagrad på rätt sätt i minnet.
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278 * Det borde inte vara några problem att behålla det "optimala" interfacet (gf[1,1,1][2]) till gridfunktionerna. Om man verkligen behöver kan skapa parallella indexeringsmetoder som gör det man behöver, i.e, "deep indexing".
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279 * Det är inte säkert att vi behöver göra något speciellt på outputsidan överhuvudtaget. Det känns inte orimligt att kompilatorn skulle kunna optimera bort den koden som räknar ut onödiga komponenter.
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280 * Om vi behöver special-indexering kommer till exempel LazyTensorApplication att behöva implementera det.
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281 * För att komma vidare med något mer avancerat behöver vi antagligen implementera några operatorer som ger och agerar på vektorvärda funktioner. Tex grad, elastiska operatorn, andra?
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282
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283
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284 ## Performance measuring
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285 We should be measuring performance early. How does our effective cpu and memory bandwidth utilization compare to peak performance?
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286
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287 We should make these test simple to run for any solver.
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288
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289 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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290
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291
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292 ## Adjoint as a trait on the sbp_operator level?
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293
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294 It would be nice to have a way of refering to adjoints with resepct to the sbp-inner-product.
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295 If it was possible you could reduce the number of times you have to deal with the inner product matrix.
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296
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297 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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298
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299 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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300
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301
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302 ## Name of the `VolumeOperator` type for constant stencils
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303 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`
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304
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305
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306 ## Implementation of LazyOuterProduct
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307 Could the implementation of LazyOuterProduct be simplified by making it a
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308 struct containing two or more LazyTensors? (using split_tuple in a similar way
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309 as TensorGrid)