# MWFunctions¶

Everything that is discussed in the following chapter is available to the application program by including:

#include "MRCPP/MWFunctions"


Multiwavelet (MW) representations of real-valued scalar functions are in MRCPP called FunctionTrees. These are in principle available in any dimension using the template parameter D (in practice D=1,2,3). There are several different ways of constructing MW functions (computing the expansion coefficients in the MW basis):

• Projection of analytic function

• Arithmetic operations

• Application of MW operator

The first two will be discribed in this chapter, while the last one regarding operators will be the topic of the next chapter.

The interface for constructing MW representations in MRCPP has a dual focus: on the one hand we want a simple, intuitive way of producing adaptive numerical approximations with guaranteed precision that does not require detailed knowledge of the internals of the MW code and with a minimal number of parameters that have to be set. On the other hand we want the possibility for more detailed control of the construction and refinement of the numerical grid where such control is possible and even necessary. In the latter case it is important to be able to reuse the existing grids in e.g. iterative algorithms without excessive allocation/deallocation of memory.

## MultiResolution Analysis (MRA)¶

template<int D>
class mrcpp::MultiResolutionAnalysis

Class collecting computational domain and MW basis.

In order to combine different functions and operators in mathematical operations, they need to be compatible. That is, they must be defined on the same computational domain and constructed using the same polynomial basis (order and type). This information constitutes an MRA, which needs to be defined and passed as argument to all function and operator constructors, and only functions and operators with compatible MRAs can be combined in subsequent calculations.

Public Functions

MultiResolutionAnalysis(const BoundingBox<D> &bb, const ScalingBasis &sb, int depth = MaxDepth)

Return

New MultiResolutionAnalysis object

Parameters
• [in] bb: Computational domain

• [in] sb: Polynomial basis

• [in] depth: Maximum allowed resolution depth, relative to root scale

template<int D>
class mrcpp::BoundingBox

Class defining the computational domain.

The computational domain is made up of a collection of D-dimensional boxes on a particular length scale $$n$$. The size of each box is then $$[2^{-n}]^D$$, i.e. higher scale means smaller boxes, and the scale may be negative. The number of boxes can be different in each dimension $$[n_x, n_y, \dots]$$, but they must all be on the same scale (size). Box translations relative to the world origin must be an integer multiple of the given scale size $$2^{-n}$$.

Subclassed by mrcpp::NodeBox< D >

Public Functions

BoundingBox(int n = 0, const std::array<int, D> &l = {}, const std::array<int, D> &nb = {}, const std::array<double, D> &sf = {})

Return

New BoundingBox object

Parameters
• [in] n: Length scale, default 0

• [in] l: Corner translation, default [0, 0, …]

• [in] nb: Number of boxes, default [1, 1, …]

• [in] sf: Scaling factor, default [1.0, 1.0, …]

class mrcpp::LegendreBasis : public mrcpp::ScalingBasis

Legendre scaling functions as defined by Alpert, SIAM J Math Anal 24 (1), 246 (1993).

Public Functions

LegendreBasis(int k)

Return

New LegendreBasis object

Parameters
• [in] k: Polynomial order of basis, 1 < k < 40

class mrcpp::InterpolatingBasis : public mrcpp::ScalingBasis

Interpolating scaling functions as defined by Alpert etal, J Comp Phys 182, 149-190 (2002).

Public Functions

InterpolatingBasis(int k)

Return

New InterpolatingBasis object

Parameters
• [in] k: Polynomial order of basis, 1 < k < 40

## FunctionTree¶

template<int D>
class mrcpp::FunctionTree : public mrcpp::MWTree<D>, public mrcpp::RepresentableFunction<D>

Function representation in MW basis.

Constructing a full grown FunctionTree involves a number of steps, including setting up a memory allocator, constructing root nodes according to the given MRA, building an adaptive tree structure and computing MW coefficients. The FunctionTree constructor does only half of these steps: It takes an MRA argument, which defines the computational domain and scaling basis (these are fixed parameters that cannot be changed after construction). The tree is initialized with a memory allocator and a set of root nodes, but it does not compute any coefficients and the function is initially undefined. An undefined FunctionTree will have a well defined tree structure (at the very least the root nodes of the given MRA, but possibly with additional refinement) and its MW coefficient will be allocated but uninitialized, and its square norm will be negative (minus one).

Public Functions

FunctionTree(const MultiResolutionAnalysis<D> &mra, SharedMemory *sh_mem = nullptr)

Constructs an uninitialized tree, containing only empty root nodes. If a shared memory pointer is provided the tree will be allocated in this shared memory window, otherwise it will be local to each MPI process.

Return

New FunctionTree object

Parameters
• [in] mra: Which MRA the function is defined

• [in] sh_mem: Pointer to MPI shared memory block

### Creating defined FunctionTrees¶

The following functions will define MW coefficients where there are none, and thus require that the output FunctionTree is in an undefined state. All functions marked with ‘adaptive grid’ will use the same building algorithm:

2. Compute the MW coefficients for the output function on the current grid

3. Refine the grid where necessary based on the local wavelet norm

4. Iterate points 2 and 3 until the grid is converged

With a negative precision argument, the grid will be fixed, e.i. it will not be refined beyond the initial grid. There is also an argument to limit the number of extra refinement levels beyond the initial grid, in which the adaptive refinement will stop, even if the local precision requirement is not met.

void mrcpp::MWTree::setZero()

Set the MW coefficients to zero, fixed grid.

Keeps the node structure of the tree, even though the zero function is representable at depth zero. Use cropTree to remove unnecessary nodes.

template<int D>
void mrcpp::project(double prec, FunctionTree<D> &out, RepresentableFunction<D> &inp, int maxIter, bool absPrec)

Project an analytic function onto the MW basis, adaptive grid.

The output function will be computed using the general algorithm:

• Compute MW coefs on current grid

• Refine grid where necessary based on prec

• Repeat until convergence or maxIter is reached

• prec < 0 or maxIter = 0 means NO refinement

• maxIter < 0 means no bound

Parameters
• [in] prec: Build precision of output function

• [out] out: Output function to be built

• [in] inp: Input function

• [in] maxIter: Maximum number of refinement iterations in output tree

• [in] absPrec: Build output tree based on absolute precision

Note

This algorithm will start at whatever grid is present in the out tree when the function is called (this grid should however be EMPTY, e.i. no coefs).

template<int D>
void mrcpp::copy_func(FunctionTree<D> &out, FunctionTree<D> &inp)

Copy function from one tree onto the grid of another tree, fixed grid.

The output function will be computed using the general algorithm:

• Loop through current leaf nodes of the output tree

• Copy MW coefs from the corresponding input node

Parameters
• [out] out: Output function

• [in] inp: Input function

Note

This algorithm will start at whatever grid is present in the out tree when the function is called and will overwrite any existing coefs.

template<int D>
void mrcpp::add(double prec, FunctionTree<D> &out, FunctionTreeVector<D> &inp, int maxIter, bool absPrec)

The output function will be computed as the sum of all input functions in the vector (including their numerical coefficients), using the general algorithm:

• Compute MW coefs on current grid

• Refine grid where necessary based on prec

• Repeat until convergence or maxIter is reached

• prec < 0 or maxIter = 0 means NO refinement

• maxIter < 0 means no bound

Parameters
• [in] prec: Build precision of output function

• [out] out: Output function to be built

• [in] inp: Vector of input function

• [in] maxIter: Maximum number of refinement iterations in output tree

• [in] absPrec: Build output tree based on absolute precision

Note

This algorithm will start at whatever grid is present in the out tree when the function is called (this grid should however be EMPTY, e.i. no coefs).

template<int D>
void mrcpp::add(double prec, FunctionTree<D> &out, double a, FunctionTree<D> &inp_a, double b, FunctionTree<D> &inp_b, int maxIter, bool absPrec)

The output function will be computed as the sum of the two input functions (including the numerical coefficient), using the general algorithm:

• Compute MW coefs on current grid

• Refine grid where necessary based on prec

• Repeat until convergence or maxIter is reached

• prec < 0 or maxIter = 0 means NO refinement

• maxIter < 0 means no bound

Parameters
• [in] prec: Build precision of output function

• [out] out: Output function to be built

• [in] a: Numerical coefficient of function a

• [in] inp_a: Input function a

• [in] b: Numerical coefficient of function b

• [in] inp_b: Input function b

• [in] maxIter: Maximum number of refinement iterations in output tree

• [in] absPrec: Build output tree based on absolute precision

Note

This algorithm will start at whatever grid is present in the out tree when the function is called (this grid should however be EMPTY, e.i. no coefs).

template<int D>
void mrcpp::multiply(double prec, FunctionTree<D> &out, FunctionTreeVector<D> &inp, int maxIter, bool absPrec, bool useMaxNorms)

Multiplication of several MW function representations, adaptive grid.

The output function will be computed as the product of all input functions in the vector (including their numerical coefficients), using the general algorithm:

• Compute MW coefs on current grid

• Refine grid where necessary based on prec

• Repeat until convergence or maxIter is reached

• prec < 0 or maxIter = 0 means NO refinement

• maxIter < 0 means no bound

Parameters
• [in] prec: Build precision of output function

• [out] out: Output function to be built

• [in] inp: Vector of input function

• [in] maxIter: Maximum number of refinement iterations in output tree

• [in] absPrec: Build output tree based on absolute precision

• [in] useMaxNorms: Build output tree based on norm estimates from input

Note

This algorithm will start at whatever grid is present in the out tree when the function is called (this grid should however be EMPTY, e.i. no coefs).

template<int D>
void mrcpp::multiply(double prec, FunctionTree<D> &out, double c, FunctionTree<D> &inp_a, FunctionTree<D> &inp_b, int maxIter, bool absPrec, bool useMaxNorms)

Multiplication of two MW function representations, adaptive grid.

The output function will be computed as the product of the two input functions (including the numerical coefficient), using the general algorithm:

• Compute MW coefs on current grid

• Refine grid where necessary based on prec

• Repeat until convergence or maxIter is reached

• prec < 0 or maxIter = 0 means NO refinement

• maxIter < 0 means no bound

Parameters
• [in] prec: Build precision of output function

• [out] out: Output function to be built

• [in] c: Numerical coefficient

• [in] inp_a: Input function a

• [in] inp_b: Input function b

• [in] maxIter: Maximum number of refinement iterations in output tree

• [in] absPrec: Build output tree based on absolute precision

• [in] useMaxNorms: Build output tree based on norm estimates from input

Note

This algorithm will start at whatever grid is present in the out tree when the function is called (this grid should however be EMPTY, e.i. no coefs).

template<int D>
void mrcpp::square(double prec, FunctionTree<D> &out, FunctionTree<D> &inp, int maxIter, bool absPrec)

Out-of-place square of MW function representations, adaptive grid.

The output function will be computed as the square of the input function, using the general algorithm:

• Compute MW coefs on current grid

• Refine grid where necessary based on prec

• Repeat until convergence or maxIter is reached

• prec < 0 or maxIter = 0 means NO refinement

• maxIter < 0 means no bound

Parameters
• [in] prec: Build precision of output function

• [out] out: Output function to be built

• [in] inp: Input function to square

• [in] maxIter: Maximum number of refinement iterations in output tree

• [in] absPrec: Build output tree based on absolute precision

Note

This algorithm will start at whatever grid is present in the out tree when the function is called (this grid should however be EMPTY, e.i. no coefs).

template<int D>
void mrcpp::power(double prec, FunctionTree<D> &out, FunctionTree<D> &inp, double p, int maxIter, bool absPrec)

Out-of-place power of MW function representations, adaptive grid.

The output function will be computed as the input function raised to the given power, using the general algorithm:

• Compute MW coefs on current grid

• Refine grid where necessary based on prec

• Repeat until convergence or maxIter is reached

• prec < 0 or maxIter = 0 means NO refinement

• maxIter < 0 means no bound

Parameters
• [in] prec: Build precision of output function

• [out] out: Output function to be built

• [in] inp: Input function to square

• [in] p: Numerical power

• [in] maxIter: Maximum number of refinement iterations in output tree

• [in] absPrec: Build output tree based on absolute precision

Note

This algorithm will start at whatever grid is present in the out tree when the function is called (this grid should however be EMPTY, e.i. no coefs).

template<int D>
void mrcpp::dot(double prec, FunctionTree<D> &out, FunctionTreeVector<D> &inp_a, FunctionTreeVector<D> &inp_b, int maxIter, bool absPrec)

Dot product of two MW function vectors, adaptive grid.

The output function will be computed as the dot product of the two input vectors (including their numerical coefficients). The precision parameter is used only in the multiplication part, the final addition will be on the fixed union grid of the components.

Parameters
• [in] prec: Build precision of output function

• [out] out: Output function to be built

• [in] inp_a: Input function vector

• [in] inp_b: Input function vector

• [in] maxIter: Maximum number of refinement iterations in output tree

• [in] absPrec: Build output tree based on absolute precision

Note

The length of the input vectors must be the same.

template<int D>
void mrcpp::map(double prec, FunctionTree<D> &out, FunctionTree<D> &inp, FMap fmap, int maxIter, bool absPrec)

map a MW function onto another representations, adaptive grid

The output function tree will be computed by mapping the input tree values through the fmap function, using the general algorithm:

• Compute MW coefs on current grid

• Refine grid where necessary based on prec

• Repeat until convergence or maxIter is reached

• prec < 0 or maxIter = 0 means NO refinement

• maxIter < 0 means no bound

Parameters
• [in] prec: Build precision of output function

• [out] out: Output function to be built

• [in] inp: Input function

• [in] fmap: mapping function

• [in] maxIter: Maximum number of refinement iterations in output tree

• [in] absPrec: Build output tree based on absolute precision

No assumption is made for how the mapping function looks. It is left to the end-user to guarantee that the mapping function does not lead to numerically unstable/inaccurate situations (e.g. divide by zero, overflow, etc…)

Note

This algorithm will start at whatever grid is present in the out tree when the function is called (this grid should however be EMPTY, e.i. no coefs).

### Creating undefined FunctionTrees¶

The grid of a FunctionTree can also be constructed without computing any MW coefficients:

template<int D>
void mrcpp::build_grid(FunctionTree<D> &out, const RepresentableFunction<D> &inp, int maxIter)

Build empty grid based on info from analytic function.

The grid of the output function will be EXTENDED using the general algorithm:

• Loop through current leaf nodes of the output tree

• Refine node based on custom split check from the function

• Repeat until convergence or maxIter is reached

• maxIter < 0 means no bound

Parameters
• [out] out: Output tree to be built

• [in] inp: Input function

• [in] maxIter: Maximum number of refinement iterations in output tree

Note

This algorithm will start at whatever grid is present in the out tree when the function is called. It requires that the functions isVisibleAtScale() and isZeroOnInterval() is implemented in the particular RepresentableFunction.

template<int D>
void mrcpp::build_grid(FunctionTree<D> &out, const GaussExp<D> &inp, int maxIter)

Build empty grid based on info from Gaussian expansion.

The grid of the output function will be EXTENDED using the general algorithm:

• Loop through current leaf nodes of the output tree

• Refine node based on custom split check from the function

• Repeat until convergence or maxIter is reached

• maxIter < 0 means no bound

Parameters
• [out] out: Output tree to be built

• [in] inp: Input Gaussian expansion

• [in] maxIter: Maximum number of refinement iterations in output tree

Note

This algorithm will start at whatever grid is present in the out tree when the function is called. It will loop through the Gaussians in the expansion and extend the grid based on the position and exponent of each term. Higher exponent means finer resolution.

template<int D>
void mrcpp::build_grid(FunctionTree<D> &out, FunctionTree<D> &inp, int maxIter)

Build empty grid based on another MW function representation.

The grid of the output function will be EXTENDED with all existing nodes in corresponding input function, using the general algorithm:

• Loop through current leaf nodes of the output tree

• Refine node if the corresponding node in the input has children

• Repeat until all input nodes are covered or maxIter is reached

• maxIter < 0 means no bound

Parameters
• [out] out: Output tree to be built

• [in] inp: Input tree

• [in] maxIter: Maximum number of refinement iterations in output tree

Note

This algorithm will start at whatever grid is present in the out tree when the function is called. This means that all nodes on the input tree will also be in the final output tree (unless maxIter is reached, but NOT vice versa.

template<int D>
void mrcpp::build_grid(FunctionTree<D> &out, FunctionTreeVector<D> &inp, int maxIter)

Build empty grid based on several MW function representation.

The grid of the output function will be EXTENDED with all existing nodes in all corresponding input functions, using the general algorithm:

• Loop through current leaf nodes of the output tree

• Refine node if the corresponding node in one of the inputs has children

• Repeat until all input nodes are covered or maxIter is reached

• maxIter < 0 means no bound

Parameters
• [out] out: Output tree to be built

• [in] inp: Input tree vector

• [in] maxIter: Maximum number of refinement iterations in output tree

Note

This algorithm will start at whatever grid is present in the out tree when the function is called. This means that the final output grid will contain (at least) the union of the nodes of all input trees (unless maxIter is reached).

template<int D>
void mrcpp::copy_grid(FunctionTree<D> &out, FunctionTree<D> &inp)

Build empty grid that is identical to another MW grid.

Note

The difference from the corresponding build_grid function is that this will first clear the grid of the out function, while build_grid will extend the existing grid.

Parameters
• [out] out: Output tree to be built

• [in] inp: Input tree

template<int D>
void mrcpp::clear_grid(FunctionTree<D> &out)

Clear the MW coefficients of a function representation.

Note

This will only clear the MW coefs in the existing nodes, it will not change the grid of the function. Use FunctionTree::clear() to remove all grid refinement as well.

Parameters
• [inout] out: Output function to be cleared

void mrcpp::FunctionTree::clear()

Remove all nodes in the tree.

Leaves the tree inn the same state as after construction, i.e. undefined function containing only root nodes without coefficients. The assigned memory (nodeChunks in SerialTree) is NOT released, but is immediately available to the new function.

### Changing FunctionTrees¶

There are also a number of in-place operations that change the MW coefficients of a given defined FunctionTree. All changing operations require that the FunctionTree is in a defined state.

void mrcpp::FunctionTree::rescale(double c)

In-place multiplication by a scalar, fixed grid.

The leaf node point values of the function will be in-place multiplied by the given coefficient, no grid refinement.

Parameters
• [in] c: Scalar coefficient

void mrcpp::FunctionTree::normalize()

In-place rescaling by a function norm $$||f||^{-1}$$, fixed grid.

void mrcpp::FunctionTree::add(double c, FunctionTree<D> &inp)

In-place addition with MW function representations, fixed grid.

The input function will be added in-place on the current grid of the function, i.e. no further grid refinement.

Parameters
• [in] c: Numerical coefficient of input function

• [in] inp: Input function to add

void mrcpp::FunctionTree::multiply(double c, FunctionTree<D> &inp)

In-place multiplication with MW function representations, fixed grid.

The input function will be multiplied in-place on the current grid of the function, i.e. no further grid refinement.

Parameters
• [in] c: Numerical coefficient of input function

• [in] inp: Input function to multiply

void mrcpp::FunctionTree::square()

In-place square of MW function representations, fixed grid.

The leaf node point values of the function will be in-place squared, no grid refinement.

void mrcpp::FunctionTree::power(double p)

In-place power of MW function representations, fixed grid.

The leaf node point values of the function will be in-place raised to the given power, no grid refinement.

Parameters
• [in] p: Numerical power

void mrcpp::FunctionTree::map(FMap fmap)

In-place mapping with a predefined function f(x), fixed grid.

The input function will be mapped in-place on the current grid of the function, i.e. no further grid refinement.

Parameters
• [in] fmap: mapping function

int mrcpp::FunctionTree::crop(double prec, double splitFac = 1.0, bool absPrec = true)

Reduce the precision of the tree by deleting nodes.

This will run the tree building algorithm in “reverse”, starting from the leaf nodes, and perform split checks on each node based on the given precision and the local wavelet norm.

Parameters
• prec: New precision criterion

• splitFac: Splitting factor: 1, 2 or 3

• absPrec: Use absolute precision

Note

The splitting factor appears in the threshold for the wavelet norm as $$||w|| < 2^{-sn/2} ||f|| \epsilon$$. In principal, s should be equal to the dimension; in practice, it is set to s=1.

template<int D>
int mrcpp::refine_grid(FunctionTree<D> &out, int scales)

Refine the grid of a MW function representation.

This will split ALL leaf nodes in the tree the given number of times, then it will compute scaling coefs of the new nodes, thus leaving the function representation unchanged, but on a larger grid.

Return

The number of nodes that were split

Parameters
• [inout] out: Output tree to be refined

• [in] scales: Number of refinement levels

template<int D>
int mrcpp::refine_grid(FunctionTree<D> &out, double prec, bool absPrec)

Refine the grid of a MW function representation.

This will first perform a split check on the existing leaf nodes in the tree based on the provided precision parameter, then it will compute scaling coefs of the new nodes, thus leaving the function representation unchanged, but (possibly) on a larger grid.

Return

The number of nodes that were split

Parameters
• [inout] out: Output tree to be refined

• [in] prec: Precision for initial split check

• [in] absPrec: Build output tree based on absolute precision

template<int D>
int mrcpp::refine_grid(FunctionTree<D> &out, FunctionTree<D> &inp)

Refine the grid of a MW function representation.

This will first perform a split check on the existing leaf nodes in the output tree based on the structure of the input tree (same as build_grid), then it will compute scaling coefs of the new nodes, thus leaving the function representation unchanged, but on a larger grid.

Return

The number of nodes that were split

Parameters
• [inout] out: Output tree to be refined

• [in] inp: Input tree that defines the new grid

### File I/O¶

void mrcpp::FunctionTree::saveTree(const std::string &file) override

Write the tree structure to disk, for later use.

Parameters
• [in] file: File name, will get “.tree” extension

void mrcpp::FunctionTree::loadTree(const std::string &file) override

Read a previously stored tree structure from disk.

Note

This tree must have the exact same MRA the one that was saved

Parameters
• [in] file: File name, will get “.tree” extension

### Extracting data¶

Given a FunctionTree that is a well defined function representation, the following data can be extracted:

double mrcpp::FunctionTree::integrate() const

Return

Integral of the function over the entire computational domain

double mrcpp::FunctionTree::evalf(const Coord<D> &r) const override

Return

Function value in a point, out of bounds returns zero

Note

This will only evaluate the scaling part of the leaf nodes in the tree, which means that the function values will not be fully accurate. If you want to include also the final wavelet part you’ll have to manually extend the MW grid by one level before evaluating, using mrcpp::refine_grid(tree, 1) This is done to allow a fast and const function evaluation that can be done in OMP parallel.

Parameters
• [in] r: Cartesian coordinate

double mrcpp::MWTree::getSquareNorm() const

Return

Squared L2 norm of the function

int mrcpp::MWTree::getNNodes(int depth = -1) const

Return

Total number of nodes in the tree, at given depth

Parameters
• [in] depth: Tree depth to count, negative means count all nodes

int mrcpp::MWTree::getSizeNodes() const

Return

Size of all MW coefs in the tree, in kB

template<int D>
double mrcpp::dot(FunctionTree<D> &bra, FunctionTree<D> &ket)

The dot product is computed with the trees in compressed form, i.e. scaling coefs only on root nodes, wavelet coefs on all nodes. Since wavelet functions are orthonormal through ALL scales and the root scaling functions are orthonormal to all finer level wavelet functions, this becomes a rather efficient procedure as you only need to compute the dot product where the grids overlap.

Return

Dot product <bra|ket> of two MW function representations

Parameters
• [in] bra: Bra side input function

• [in] ket: Ket side input function

## FunctionTreeVector¶

The FunctionTreeVector is simply an alias for a std::vector of std::tuple containing a numerical coefficient and a FunctionTree pointer.

template<int D>
void mrcpp::clear(FunctionTreeVector<D> &fs, bool dealloc = false)

Remove all entries in the vector.

Parameters
• [in] fs: Vector to clear

• [in] dealloc: Option to free FunctionTree pointer before clearing

template<int D>
double mrcpp::get_coef(const FunctionTreeVector<D> &fs, int i)

Return

Numerical coefficient at given position in vector

Parameters
• [in] fs: Vector to fetch from

• [in] i: Position in vector

template<int D>
FunctionTree<D> &mrcpp::get_func(FunctionTreeVector<D> &fs, int i)

Return

FunctionTree at given position in vector

Parameters
• [in] fs: Vector to fetch from

• [in] i: Position in vector

template<int D>
int mrcpp::get_n_nodes(const FunctionTreeVector<D> &fs)

Return

Total number of nodes of all trees in the vector

Parameters
• [in] fs: Vector to fetch from

template<int D>
int mrcpp::get_size_nodes(const FunctionTreeVector<D> &fs)

Return

Total size of all trees in the vector, in kB

Parameters
• [in] fs: Vector to fetch from

## Examples¶

### Constructing an MRA¶

An MRA is defined in two steps, first the computational domain is given by a BoundingBox (D is the dimension), e.g. for a total domain of $$[-32,32]^3$$ in three dimensions (eight root boxes of size $$[16]^3$$ each):

int n = -4;                                        // Root scale defines box size 2^{-n}
std::array<int, 3> l{-1, -1, -1};                  // Translation of first box [l_x,l_y,l_z]
std::array<int, 3> nb{2, 2, 2};                    // Number of boxes [n_x,n_y,n_z]
mrcpp::BoundingBox<3> world(n, l, nb);


which is combined with a ScalingBasis to give an MRA, e.g. interpolating scaling functions of order $$k=9$$:

int N = 20;                                        // Maximum refinement 2^{-(n+N)}
int k = 9;                                         // Polynomial order
mrcpp::InterpolatingBasis basis(k);                // Legendre or Interpolating basis
mrcpp::MultiResolutionAnalysis<D> MRA(world, basis, N);


Two types of ScalingBasis are supported (LegendreBasis and InterpolatingBasis), and they are both available at orders $$k=1,2,\dots,40$$ (note that some operators are constructed using intermediates of order $$2k$$, so in that case the maximum order available is $$k=20$$).

### Working withFunctionTreeVectors¶

Elements can be appended to the vector using the std::make_tuple, elements are obtained with the get_func and get_coef functions:

mrcpp::FunctionTreeVector<D> tree_vec;             // Initialize empty vector
tree_vec.push_back(std::make_tuple(2.0, &tree_a)); // Push back pointer to FunctionTree
auto coef = mrcpp::get_coef(tree_vec, 0);          // Get coefficient of first entry
auto &tree = mrcpp::get_func(tree_vec, 0);         // Get function of first entry
mrcpp::clear(tree_vec, false);                     // Bool argument for tree destruction


### Building empty grids¶

Sometimes it is useful to construct an empty grid based on some available information of the function that is about to be represented. This can be e.g. that you want to copy the grid of an existing FunctionTree or that an analytic function has more or less known grid requirements (like Gaussians). Sometimes it is even necessary to force the grid refinement beyond the coarsest scales in order for the adaptive refining algorithm to detect a wavelet “signal” that allows it to do its job properly (this happens for narrow Gaussians where none of the initial quadrature points hits a function value significantly different from zero).

The simplest way to build an empty grid is to copy the grid from an existing tree (assume that f_tree has been properly built so that it contains more than just root nodes)

mrcpp::FunctionTree<D> f_tree(MRA);   // Input tree
mrcpp::FunctionTree<D> g_tree(MRA);   // Output tree

mrcpp::project(prec, f_tree, f_func); // Build adaptive grid for f_tree
mrcpp::copy_grid(g_tree, f_tree);     // Copy grid from f_tree to g_tree


Passing an analytic function as argument to the generator will build a grid based on some predefined information of the function (if there is any, otherwise it will do nothing)

mrcpp::RepresentableFunction<D> func; // Analytic function
mrcpp::FunctionTree<D> tree(MRA);     // Output tree
mrcpp::build_grid(tree, func);        // Build grid based on f_func


The lambda analytic functions do not provide such information, this must be explicitly implemented as a RepresentableFunction sub-class (see MRCPP programmer’s guide for details).

Actually, the effect of the build_grid is to extend the existing grid with any missing nodes relative to the input. There is also a version of build_grid taking a FunctionTree argument. Its effect is very similar to the copy_grid above, with the only difference that now the output grid is extended with the missing nodes (e.i. the nodes that are already there are not removed first). This means that we can build the union of two grids by successive applications of build_grid

mrcpp::FunctionTree<D> f_tree(MRA);   // Construct empty grid of root nodes
mrcpp::build_grid(f_tree, g_tree);    // Extend f with missing nodes relative to g
mrcpp::build_grid(f_tree, h_tree);    // Extend f with missing nodes relative to h


In contrast, doing the same with copy_grid would clear the f_tree grid in between, and you would only get a (identical) copy of the last h_tree grid, with no memory of the g_tree grid that was once there. One can also make the grids of two functions equal to their union

mrcpp::build_grid(f_tree, g_tree);    // Extend f with missing nodes relative to g
mrcpp::build_grid(g_tree, f_tree);    // Extend g with missing nodes relative to f


The union grid of several trees can be constructed in one go using a FunctionTreeVector

mrcpp::FunctionTreeVector<D> inp_vec;
inp_vec.push_back(std::make_tuple(1.0, tree_1));
inp_vec.push_back(std::make_tuple(1.0, tree_2));
inp_vec.push_back(std::make_tuple(1.0, tree_3));

mrcpp::FunctionTree<D> f_tree(MRA);
mrcpp::build_grid(f_tree, inp_vec);   // Extend f with missing nodes from all trees in inp_vec


### Projection¶

The project function takes an analytic D-dimensional scalar function (which can be defined as a lambda function or one of the explicitly implemented sub-classes of the RepresentableFunction base class in MRCPP) and projects it with the given precision onto the MRA defined by the FunctionTree. E.g. a unit charge Gaussian is projected in the following way (the MRA must be initialized as above)

// Defining an analytic function
double beta = 10.0;
double alpha = std::pow(beta/pi, 3.0/2.0);
auto func = [alpha, beta] (const mrcpp::Coord<3> &r) -> double {
double R = std::sqrt(r[0]*r[0] + r[1]*r[1] + r[2]*r[2]);
return alpha*std::exp(-beta*R*R);
};

double prec = 1.0e-5;
mrcpp::FunctionTree<3> tree(MRA);
mrcpp::project(prec, tree, func);


This projection will start at the default initial grid (only the root nodes of the given MRA), and adaptively build the full grid. Alternatively, the grid can be estimated a priori if the analytical function has some known features, such as for Gaussians:

double prec;                                       // Precision of the projection
int max_iter;                                      // Maximum levels of refinement

mrcpp::GaussFunc<D> func;                          // Analytic Gaussian function
mrcpp::FunctionTree<D> tree(MRA);                  // Output tree

mrcpp::build_grid(tree, func);                     // Empty grid from analytic function
mrcpp::project(prec, tree, func, max_iter);        // Starts projecting from given grid


This will first produce an empty grid suited for representing the analytic function func (this is meant as a way to make sure that the projection starts on a grid where the function is actually visible, as for very narrow Gaussians, it’s not meant to be a good approximation of the final grid) and then perform the projection on the given numerical grid. With a negative prec (or max_iter = 0) the projection will be performed strictly on the given initial grid, with no further refinements.

Arithmetic operations in the MW representation are performed using the FunctionTreeVector, and the general sum $$f = \sum_i c_i f_i(x)$$ is done in the following way

double a, b, c;                                    // Addition parameters
mrcpp::FunctionTree<D> a_tree(MRA);                // Input function
mrcpp::FunctionTree<D> b_tree(MRA);                // Input function
mrcpp::FunctionTree<D> c_tree(MRA);                // Input function

mrcpp::FunctionTreeVector<D> inp_vec;              // Vector to hold input functions
inp_vec.push_back(std::make_tuple(a, &a_tree));    // Append to vector
inp_vec.push_back(std::make_tuple(b, &b_tree));    // Append to vector
inp_vec.push_back(std::make_tuple(c, &c_tree));    // Append to vector

mrcpp::FunctionTree<D> f_tree(MRA);                // Output function


The default initial grid is again only the root nodes, and a positive prec is required to build an adaptive tree structure for the result. The special case of adding two functions can be done directly without initializing a FunctionTreeVector

mrcpp::FunctionTree<D> f_tree(MRA);
mrcpp::add(prec, f_tree, a, a_tree, b, b_tree);


Addition of two functions is usually done on their (fixed) union grid

mrcpp::FunctionTree<D> f_tree(MRA);                // Construct empty root grid
mrcpp::build_grid(f_tree, a_tree);                 // Copy grid of g
mrcpp::build_grid(f_tree, b_tree);                 // Copy grid of h
mrcpp::add(-1.0, f_tree, a, a_tree, b, b_tree);    // Add functions on fixed grid


Note that in the case of addition there is no extra information to be gained by going beyond the finest refinement levels of the input functions, so the union grid summation is simply the best you can do, and adding a positive prec will not make a difference. There are situations where you want to use a smaller grid, though, e.g. when performing a unitary transformation among a set of FunctionTrees. In this case you usually don’t want to construct all the output functions on the union grid of all the input functions, and this can be done by adding the functions adaptively starting from root nodes.

If you have a summation over several functions but want to perform the addition on the grid given by the first input function, you first copy the wanted grid and then perform the operation on that grid

mrcpp::FunctionTreeVector<D> inp_vec;
inp_vec.push_back(std::make_tuple(a, a_tree));
inp_vec.push_back(std::make_tuple(b, b_tree));
inp_vec.push_back(std::make_tuple(c, c_tree));

mrcpp::FunctionTree<D> f_tree(MRA);                // Construct empty root grid
mrcpp::copy_grid(f_tree, get_func(inp_vec, 0));    // Copy grid of first input function


Here you can of course also add a positive prec to the addition and the resulting function will be built adaptively starting from the given initial grid.

### Multiplication¶

The multiplication follows the exact same syntax as the addition, where the product $$f = \prod_i c_i f_i(x)$$ is done in the following way

double a, b, c;                                    // Multiplication parameters
mrcpp::FunctionTree<D> a_tree(MRA);                // Input function
mrcpp::FunctionTree<D> b_tree(MRA);                // Input function
mrcpp::FunctionTree<D> c_tree(MRA);                // Input function

mrcpp::FunctionTreeVector<D> inp_vec;              // Vector to hold input functions
inp_vec.push_back(std::make_tuple(a, &a_tree));    // Append to vector
inp_vec.push_back(std::make_tuple(b, &b_tree));    // Append to vector
inp_vec.push_back(std::make_tuple(c, &c_tree));    // Append to vector

mrcpp::FunctionTree<D> f_tree(MRA);                // Output function
mrcpp::multipy(prec, f_tree, inp_vec);             // Adaptive multiplication


In the special case of multiplying two functions the coefficients are collected into one argument

mrcpp::FunctionTree<D> f_tree(MRA);
mrcpp::multiply(prec, f_tree, a*b, a_tree, b_tree);


For multiplications, there might be a loss of accuracy if the product is restricted to the union grid. The reason for this is that the product will contain signals of higher frequency than each of the input functions, which require a higher grid refinement for accurate representation. By specifying a positive prec you will allow the grid to adapt to the higher frequencies, but it is usually a good idea to restrict to one extra refinement level beyond the union grid (by setting max_iter=1) as the grids are not guaranteed to converge for such local operations (like arithmetics, derivatives and function mappings)

mrcpp::FunctionTree<D> f_tree(MRA);                    // Construct empty root grid
mrcpp::build_grid(f_tree, a_tree);                     // Copy grid of a
mrcpp::build_grid(f_tree, b_tree);                     // Copy grid of b
mrcpp::multiply(prec, f_tree, a*b, a_tree, b_tree, 1); // Allow 1 extra refinement


### Re-using grids¶

Given a FunctionTree that is a valid function representation, we can clear its MW expansion coefficients as well as its grid refinement

mrcpp::FunctionTree<D> tree(MRA);                      // tree is an undefined function
mrcpp::project(prec, tree, f_func);                    // tree represents analytic function f
tree.clear();                                          // tree is an undefined function
mrcpp::project(prec, tree, f_func);                    // tree represents analytic function g


This action will leave the FunctionTree in the same state as after construction (undefined function, only root nodes), and its coefficients can now be re-computed.

In certain situations it might be desireable to separate the actions of computing MW coefficients and refining the grid. For this we can use the refine_grid, which will adaptively refine the grid one level (based on the wavelet norm and the given precision) and project the existing function representation onto the new finer grid

mrcpp::refine_grid(tree, prec);


E.i., this will not change the function that is represented in tree, but it might increase its grid size. The same effect can be made using another FunctionTree argument instead of the precision parameter

mrcpp::refine_grid(tree_out, tree_in);


which will extend the grid of tree_out in the same way as build_grid as shown above, but it will keep the function representation in tree_out.

This functionality can be combined with clear_grid to make a “manual” adaptive building algorithm. One example where this might be useful is in iterative algorithms where you want to fix the grid size for all calculations within one cycle and then relax the grid in the end in preparation for the next iteration. The following is equivalent to the adaptive projection above (refine_grid returns the number of new nodes that were created in the process)

int n_nodes = 1;
while (n_nodes > 0) {
mrcpp::project(-1.0, tree, func);                  // Project f on fixed grid
n_nodes = mrcpp::refine_grid(tree, prec);          // Refine grid based on prec
if (n_nodes > 0) mrcpp::clear_grid(tree);          // Clear grid for next iteration
}