lowrankfunction.h

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//
// Created by Florian Bischoff on 8/10/23.
//
 
#ifndef MADNESS_LOWRANKFUNCTION_H
#define MADNESS_LOWRANKFUNCTION_H
 
 
#include<madness/mra/mra.h>
#include<madness/mra/vmra.h>
#include<madness/world/timing_utilities.h>
#include<madness/chem/electronic_correlation_factor.h>
#include<madness/chem/IntegratorXX.h>
#include <random>
 
 
 
namespace madness {
 
    struct LowRankFunctionParameters : QCCalculationParametersBase {
 
        LowRankFunctionParameters() : QCCalculationParametersBase() {
 
            // initialize with: key, value, comment (optional), allowed values (optional)
            initialize<double>("radius",2.0,"the radius");
            initialize<double>("gamma",1.0,"the exponent of the correlation factor");
            initialize<double>("volume_element",0.1,"volume covered by each grid point");
            initialize<double>("tol",1.e-8,"rank-reduced cholesky tolerance");
            initialize<std::string>("f12type","Slater","correlation factor",{"Slater","SlaterF12"});
            initialize<std::string>("orthomethod","cholesky","orthonormalization",{"cholesky","canonical","symmetric"});
            initialize<std::string>("transpose","slater2","transpose of the matrix",{"slater1","slater2"});
            initialize<std::string>("gridtype","random","the grid type",{"random","cartesian","dftgrid"});
            initialize<std::string>("rhsfunctiontype","exponential","the type of function",{"exponential"});
            initialize<int>("optimize",1,"number of optimization iterations");
        }
 
        void read_and_set_derived_values(World& world, const commandlineparser& parser, std::string tag) {
            read_input_and_commandline_options(world,parser,tag);
        }
 
        double radius() const {return get<double>("radius");}
        double gamma() const {return get<double>("gamma");}
        double volume_element() const {return get<double>("volume_element");}
        double tol() const {return get<double>("tol");}
        int optimize() const {return get<int>("optimize");}
        std::string gridtype() const {return get<std::string>("gridtype");}
        std::string orthomethod() const {return get<std::string>("orthomethod");}
        std::string rhsfunctiontype() const {return get<std::string>("rhsfunctiontype");}
        std::string f12type() const {return get<std::string>("f12type");}
    };
 
 
    class gridbase {
    public:
        double get_volume_element() const {return volume_element;}
        double get_radius() const {return radius;}
 
        virtual ~gridbase() = default;
        // visualize the grid in xyz format
        template<std::size_t NDIM>
        void visualize(const std::string filename, const std::vector<Vector<double,NDIM>>& grid) const {
            print("visualizing grid to file",filename);
            print("a total of",grid.size(),"grid points");
            std::ofstream file(filename);
            for (const auto& r : grid) {
                // formatted output
                file << std::fixed << std::setprecision(6);
                for (int i=0; i<NDIM; ++i) file << r[i] << " ";
                file << std::endl;
            }
            file.close();
        }
 
    protected:
        double volume_element=0.1;
        double radius=3;
        bool do_print=false;
    };
 
    template<std::size_t NDIM>
    class dftgrid : public gridbase {
    public:
        GridBuilder builder;
        explicit dftgrid(const double volume_element, const double radius) {
            // increase number of radial grid points until the volume element is below the threshold
            double current_ve=1.0;
            std::size_t nradial=10;
            while (current_ve>volume_element) {
                nradial+=10;
                print("trying nradial",nradial);
                GridBuilder tmp;
                tmp.set_angular_order(7);
                tmp.set_nradial(nradial);
                tmp.make_grid();
                double volume=4./3. *M_PI * std::pow(radius,3.0);
                auto npoints=tmp.get_points().size();
                current_ve=volume/npoints;
                print("volume, npoints, volume element",volume,npoints,current_ve);
            }
            builder.set_nradial(nradial);
            builder.set_angular_order(7);
        }
 
 
        explicit dftgrid(const std::size_t nradial, const std::size_t angular_order, const coord_3d origin=coord_3d()) {
            static_assert(NDIM==3,"DFT Grids only defined for NDIM=3");
            builder.set_nradial(nradial);
            builder.set_angular_order(angular_order);
            builder.set_origin(origin);
            builder.make_grid();
        }
 
        std::vector<Vector<double,NDIM>> get_grid() const {
            return builder.get_points();
        }
 
    };
 
    /// grid with random points around the origin, with a Gaussian distribution
    template<std::size_t NDIM>
    class randomgrid : public gridbase {
    public:
        randomgrid(const double volume_element, const double radius, const Vector<double,NDIM> origin=Vector<double,NDIM>(0.0))
            : gridbase(), origin(origin) {
            this->volume_element=volume_element;
            this->radius=radius;
        }
 
        std::vector<Vector<double,NDIM>> get_grid() const {
            std::vector<Vector<double, NDIM>> grid;
            long npoint_within_volume=volume()/volume_element;
            if (do_print) print("npoint_within_volume",npoint_within_volume);
 
            auto cell = FunctionDefaults<NDIM>::get_cell();
            auto is_in_cell = [&cell](const Vector<double, NDIM>& r) {
                for (int d = 0; d < NDIM; ++d) if (r[d] < cell(d, 0) or r[d] > cell(d, 1)) return false;
                return true;
            };
            double rad=radius;
            auto o=origin;
            auto is_in_sphere = [&rad,&o](const Vector<double, NDIM>& r) {
                return ((r-o).normf()<rad);
            };
 
            // set variance such that about 70% of all grid points sits within the radius
            double variance=radius;
            if (NDIM==2) variance=0.6*radius;
            if (NDIM==3) variance=0.5*radius;
            long maxrank=10*npoint_within_volume;
            long rank=0;
            for (int r=0; r<maxrank; ++r) {
                auto tmp = gaussian_random_distribution(origin, variance);
                if (not is_in_cell(tmp)) continue;
                if (is_in_sphere(tmp)) ++rank;
                grid.push_back(tmp);
                if (rank==npoint_within_volume) break;
            }
            if (do_print) {
                print("origin                ",origin);
                print("radius                ",radius);
                print("grid points in volume ",rank);
                print("total grid points     ",grid.size());
                print("ratio                 ",rank/double(grid.size()));
                print("volume element        ",volume()/rank);
            }
            return grid;
        }
 
        Vector<double,NDIM> get_origin() const {
            return origin;
        }
 
    private:
 
        double volume() const {
            MADNESS_CHECK(NDIM>0 and NDIM<4);
            if (NDIM==1) return 2.0*radius;
            if (NDIM==2) return constants::pi*radius*radius;
            if (NDIM==3) return 4.0 / 3.0 * constants::pi * std::pow(radius, 3.0);
        }
 
        static Vector<double,NDIM> gaussian_random_distribution(const Vector<double,NDIM> origin, double variance) {
            std::random_device rd{};
            std::mt19937 gen{rd()};
            Vector<double,NDIM> result;
            for (int i = 0; i < NDIM; ++i) {
                std::normal_distribution<> d{origin[i], variance};
                result[i]=d(gen);
            }
 
            return result;
        }
 
        Vector<double,NDIM> origin;
 
    };
 
    template<std::size_t NDIM>
    struct cartesian_grid {
        Vector<double,NDIM> lovec,hivec;
        std::vector<long> stride;
        long index=0;
        long n_per_dim;
        long total_n;
        Vector<double,NDIM> increment;
 
        cartesian_grid(const double volume_per_gridpoint, const double radius) {
            double length_per_gridpoint=std::pow(volume_per_gridpoint,1.0/NDIM);
            n_per_dim=ceil(2.0*radius/length_per_gridpoint);
            print("length per gridpoint, n_per_dim",length_per_gridpoint,n_per_dim);
            print("volume_per_gridpoint",std::pow(length_per_gridpoint,NDIM));
            initialize(-radius,radius);
            print("increment",increment);
        }
 
 
        cartesian_grid(const long n_per_dim, const double lo, const double hi)
                : n_per_dim(n_per_dim) {
            initialize(lo,hi);
        }
 
        cartesian_grid(const cartesian_grid<NDIM>& other) : lovec(other.lovec),
                                                            hivec(other.hivec), stride(other.stride), index(0), n_per_dim(other.n_per_dim),
                                                            total_n(other.total_n), increment(other.increment) {
        }
 
        cartesian_grid& operator=(const cartesian_grid<NDIM>& other) {
            cartesian_grid<NDIM> tmp(other);
            std::swap(*this,other);
            return *this;
        }
 
        void initialize(const double lo, const double hi) {
            lovec.fill(lo);
            hivec.fill(hi);
            increment=(hivec-lovec)*(1.0/double(n_per_dim-1));
            stride=std::vector<long>(NDIM,1l);
            total_n=std::pow(n_per_dim,NDIM);
            for (long i=NDIM-2; i>=0; --i) stride[i]=n_per_dim*stride[i+1];
 
        }
 
        double volume_per_gridpoint() const{
            double volume=1.0;
            for (int i=0; i<NDIM; ++i) volume*=(hivec[i]-lovec[i]);
            return volume/total_n;
        }
 
        void operator++() {
            index++;
        }
 
        bool operator()() const {
            return index < total_n;
        }
 
        Vector<double,NDIM> get_coordinates() const {
            Vector<double,NDIM> tmp(NDIM);
            for (int idim=0; idim<NDIM; ++idim) {
                tmp[idim]=(index/stride[idim])%n_per_dim;
            }
            return lovec+tmp*increment;
        }
 
    };
 
    /// given a molecule, return a suitable grid
    template<std::size_t NDIM>
    class molecular_grid : public gridbase {
 
    public:
        /// ctor takes centers of the grid and the grid parameters
        molecular_grid(const std::vector<Vector<double,NDIM>> origins, const LowRankFunctionParameters& params)
            : centers(origins)
        {
            if (centers.size()==0) centers.push_back(Vector<double,NDIM>(0) );
            if (params.gridtype()=="random") grid_builder=std::make_shared<randomgrid<NDIM>>(params.volume_element(),params.radius());
            // else if (params.gridtype()=="cartesian") grid_builder=std::make_shared<cartesian_grid<NDIM>>(params.volume_element(),params.radius());
            else if (params.gridtype()=="dftgrid") {
                if constexpr (NDIM==3) {
                    grid_builder=std::make_shared<dftgrid<NDIM>>(params.volume_element(),params.radius());
                } else {
                    MADNESS_EXCEPTION("no dft grid with NDIM != 3",1);
                }
            } else {
                MADNESS_EXCEPTION("no such grid type",1);
            }
        }
 
 
        /// ctor takes centers of the grid and the grid builder
        molecular_grid(const std::vector<Vector<double,NDIM>> origins, std::shared_ptr<gridbase> grid)
            : centers(origins), grid_builder(grid) {
            if (centers.size()==0) centers.push_back({0,0,0});
        }
 
        /// ctor takes molecule and grid builder
        molecular_grid(const Molecule& molecule, std::shared_ptr<gridbase> grid) : molecular_grid(molecule.get_all_coords_vec(),grid) {}
 
        std::vector<Vector<double,NDIM>> get_grid() const {
            MADNESS_CHECK_THROW(grid_builder,"no grid builder given in molecular_grid");
            MADNESS_CHECK_THROW(centers.size()>0,"no centers given in molecular_grid");
            std::vector<Vector<double,NDIM>> grid;
            for (const auto& coords : centers) {
                print("atom sites",coords);
                if (auto builder=dynamic_cast<dftgrid<NDIM>*>(grid_builder.get())) {
                    if constexpr (NDIM==3) {
                        dftgrid<NDIM> b1(builder->builder.get_nradial(),builder->builder.get_angular_order(),coords);
                        auto atomgrid=b1.get_grid();
                        grid.insert(grid.end(),atomgrid.begin(),atomgrid.end());
                    } else {
                        MADNESS_EXCEPTION("no DFT grid for NDIM /= 3",1);
                    }
                } else if (auto builder=dynamic_cast<randomgrid<NDIM>*>(grid_builder.get())) {
                    randomgrid<NDIM> b1(builder->get_volume_element(),builder->get_radius(),coords);
                    auto atomgrid=b1.get_grid();
                    grid.insert(grid.end(),atomgrid.begin(),atomgrid.end());
                } else {
                    MADNESS_EXCEPTION("no such grid builder",1);
                }
            }
            return grid;
        }
 
    private:
        std::vector<Vector<double,NDIM>> centers;
        std::shared_ptr<gridbase> grid_builder;
 
    };
 
template<std::size_t PDIM>
struct particle {
    std::array<int,PDIM> dims;
 
    /// default constructor
    particle() = default;
 
    /// convenience for particle 1 (the left/first particle)
    static particle particle1() {
        particle p;
        for (int i=0; i<PDIM; ++i) p.dims[i]=i;
        return p;
    }
    /// convenience for particle 2 (the right/second particle)
    static particle particle2() {
        particle p;
        for (int i=0; i<PDIM; ++i) p.dims[i]=i+PDIM;
        return p;
    }
 
    /// return the other particle
    particle complement() const {
        MADNESS_CHECK(is_first() or is_last());
        if (is_first()) return particle2();
        return particle1();
    }
 
    particle(const int p) : particle(std::vector<int>(1,p)) {}
    particle(const int p1, const int p2) : particle(std::vector<int>({p1,p2})) {}
    particle(const int p1, const int p2,const int p3) : particle(std::vector<int>({p1,p2,p3})) {}
    particle(const std::vector<int> p) {
        for (int i=0; i<PDIM; ++i) dims[i]=p[i];
    }
 
    std::string str() const {
        std::stringstream ss;
        ss << *this;
        return ss.str();
    }
 
 
    /// type conversion to std::array
    std::array<int,PDIM> get_array() const {
        return dims;
    }
 
 
    /// assuming two particles only
    bool is_first() const {return dims[0]==0;}
    /// assuming two particles only
    bool is_last() const {return dims[0]==(PDIM);}
 
    template<std::size_t DUMMYDIM=PDIM>
    typename std::enable_if_t<DUMMYDIM==1, std::tuple<int>>
    get_tuple() const {return std::tuple<int>(dims[0]);}
 
    template<std::size_t DUMMYDIM=PDIM>
    typename std::enable_if_t<DUMMYDIM==2, std::tuple<int,int>>
    get_tuple() const {return std::tuple<int,int>(dims[0],dims[1]);}
 
    template<std::size_t DUMMYDIM=PDIM>
    typename std::enable_if_t<DUMMYDIM==3, std::tuple<int,int,int>>
    get_tuple() const {return std::tuple<int,int,int>(dims[0],dims[1],dims[2]);}
};
 
template<std::size_t PDIM>
std::ostream& operator<<(std::ostream& os, const particle<PDIM>& p) {
    os << "(";
    for (auto i=0; i<PDIM-1; ++i) os << p.dims[i] << ";";
    os << p.dims[PDIM-1] << ")";
    return os;
}
 
/// the low-rank functor is what the LowRankFunction will represent
 
/// derive from this class :
/// must implement in inner product
/// may implement an operator()(const coord_nd&)
template<typename T, std::size_t NDIM, std::size_t LDIM=NDIM/2>
struct LRFunctorBase {
 
    virtual ~LRFunctorBase() {};
    virtual std::vector<Function<T,LDIM>> inner(const std::vector<Function<T,LDIM>>& rhs,
                                        const particle<LDIM> p1, const particle<LDIM> p2) const =0;
 
    virtual Function<T,LDIM> inner(const Function<T,LDIM>& rhs, const particle<LDIM> p1, const particle<LDIM> p2) const {
        return inner(std::vector<Function<T,LDIM>>({rhs}),p1,p2)[0];
    }
 
    virtual T operator()(const Vector<T,NDIM>& r) const =0;
    virtual typename Tensor<T>::scalar_type norm2() const {
        MADNESS_EXCEPTION("L2 norm not implemented",1);
    }
 
    virtual World& world() const =0;
    friend std::vector<Function<T,LDIM>> inner(const LRFunctorBase& functor, const std::vector<Function<T,LDIM>>& rhs,
                                               const particle<LDIM> p1, const particle<LDIM> p2) {
        return functor.inner(rhs,p1,p2);
    }
    friend Function<T,LDIM> inner(const LRFunctorBase& functor, const Function<T,LDIM>& rhs,
                                               const particle<LDIM> p1, const particle<LDIM> p2) {
        return functor.inner(rhs,p1,p2);
    }
 
};
 
template<typename T, std::size_t NDIM, std::size_t LDIM=NDIM/2>
struct LRFunctorF12 : public LRFunctorBase<T,NDIM> {
//    LRFunctorF12() = default;
    LRFunctorF12(const std::shared_ptr<SeparatedConvolution<T,LDIM>> f12, const std::vector<Function<T,LDIM>>& a,
                 const std::vector<Function<T,LDIM>>& b) : f12(f12), a(a), b(b) {
 
        // if a or b are missing, they are assumed to be 1
        // you may not provide a or b, but if you do they have to have the same size because they are summed up
        if (a.size()>0 and b.size()>0)
            MADNESS_CHECK_THROW(a.size()==b.size(), "a and b must have the same size");
        if (a.size()==0) this->a.resize(b.size());
        if (b.size()==0) this->b.resize(a.size());
        MADNESS_CHECK_THROW(this->a.size()==this->b.size(), "a and b must have the same size");
    }
 
    /// delegate to the other ctor with vector arguments
    LRFunctorF12(const std::shared_ptr<SeparatedConvolution<T,LDIM>> f12,
                 const Function<T,LDIM>& a, const Function<T,LDIM>& b)
                 : LRFunctorF12(f12,std::vector<Function<T,LDIM>>({a}), std::vector<Function<T,LDIM>>({b})) {}
 
 
private:
    std::shared_ptr<SeparatedConvolution<T,LDIM>> f12;  ///< a two-particle function
    std::vector<Function<T,LDIM>> a,b;   ///< the lo-dim functions
public:
 
    World& world() const {return f12->get_world();}
    std::vector<Function<T,LDIM>> inner(const std::vector<Function<T,LDIM>>& rhs,
                                        const particle<LDIM> p1, const particle<LDIM> p2) const {
 
        std::vector<Function<T,LDIM>> result;
        // functor is now \sum_i a_i(1) b_i(2) f12
        // result(1) = \sum_i \int a_i(1) f(1,2) b_i(2) rhs(2) d2
        //            = \sum_i a_i(1) \int f(1,2) b_i(2) rhs(2) d2
        World& world=rhs.front().world();
 
        const int nbatch=30;
        for (int i=0; i<rhs.size(); i+=nbatch) {
            std::vector<Function<T,LDIM>> rhs_batch;
            auto begin= rhs.begin()+i;
            auto end= (i+nbatch)<rhs.size() ? rhs.begin()+i+nbatch : rhs.end();
            std::copy(begin,end, std::back_inserter(rhs_batch));
            auto tmp2= zero_functions_compressed<T,LDIM>(world,rhs_batch.size());
 
            if (a.size()==0) tmp2=apply(world,*(f12),rhs_batch);
 
            for (int ia=0; ia<a.size(); ia++) {
                auto premultiply= p1.is_first() ? a[ia] : b[ia];
                auto postmultiply= p1.is_first() ? b[ia] : a[ia];
 
                auto tmp=copy(world,rhs_batch);
                if (premultiply.is_initialized()) tmp=rhs_batch*premultiply;
                auto tmp1=apply(world,*(f12),tmp);
                if (postmultiply.is_initialized()) tmp1=tmp1*postmultiply;
                tmp2+=tmp1;
            }
 
            for (auto& t : tmp2) result.push_back(t);
        }
        return result;
    }
 
    typename Tensor<T>::scalar_type norm2() const {
        const Function<T, LDIM> one = FunctionFactory<T, LDIM>(world()).f(
                [](const Vector<double, LDIM>& r) { return 1.0; });
        std::vector<Function<T, LDIM>> pre, post;
        std::size_t sz = a.size();
        if (sz == 0) {
            pre = std::vector<Function<T, LDIM>>(1, one);
            post = std::vector<Function<T, LDIM>>(1, one);
        } else {
            pre = (a.front().is_initialized()) ? a : std::vector<Function<T, LDIM>>(sz, one);
            post = (b.front().is_initialized()) ? b : std::vector<Function<T, LDIM>>(sz, one);
        }
 
        const SeparatedConvolution<T,LDIM>& f12a=*(f12);
        const SeparatedConvolution<T,LDIM> f12sq= SeparatedConvolution<T,LDIM>::combine(f12a,f12a);
 
        // \int f(1,2)^2 d1d2 = \int f(1,2)^2 pre(1)^2 post(2)^2 d1 d2
        // || \sum_i f(1,2) a_i(1) b_i(2) || = \int ( \sum_{ij} a_i(1) a_j(1) f(1,2)^2 b_i(1) b_j(2) ) d1d2
        std::vector<Function<T,LDIM>> aa,bb;
        for (std::size_t i=0; i<pre.size(); ++i) {
            aa=append(aa,(pre[i]*pre));
            bb=append(bb,(post[i]*post));
        }
//        typename Tensor<T>::scalar_type term1 =madness::inner(mul(world(),post,post),f12sq(mul(world(),pre,pre)));
        typename Tensor<T>::scalar_type term1 =madness::inner(bb,f12sq(aa));
        return sqrt(term1);
 
    }
 
    T operator()(const Vector<double,NDIM>& r) const {
 
        if (a.size()==0) return 0.0;
        auto split = [](const Vector<double,NDIM>& r) {
            Vector<double,LDIM> first, second;
            for (int i=0; i<LDIM; ++i) {
                first[i]=r[i];
                second[i]=r[i+LDIM];
            }
            return std::make_pair(first,second);
        };
 
        double gamma=f12->info.mu;
        auto [first,second]=split(r);
 
 
        double result=0.0;
        for (std::size_t ia=0; ia<a.size(); ++ia) {
            double result1=1.0;
            if (a[ia].is_initialized()) result1*=a[ia](first);
            if (b[ia].is_initialized()) result1*=b[ia](first);
            if (f12->info.type==OT_SLATER) result1*=exp(-gamma*(first-second).normf());
            else if (f12->info.type==OT_GAUSS) result1*=exp(-gamma* madness::inner(first-second,first-second));
            else {
                MADNESS_EXCEPTION("no such operator_type",1);
            }
            result+=result1;
        }
        return result;
 
    }
};
 
template<typename T, std::size_t NDIM, std::size_t LDIM=NDIM/2>
struct LRFunctorPure : public LRFunctorBase<T,NDIM> {
    LRFunctorPure() = default;
    LRFunctorPure(const Function<T,NDIM>& f) : f(f) {}
    World& world() const {return f.world();}
 
    Function<T, NDIM> f;    ///< a hi-dim function
 
    std::vector<Function<T,LDIM>> inner(const std::vector<Function<T,LDIM>>& rhs,
                                        const particle<LDIM> p1, const particle<LDIM> p2) const {
        return madness::innerXX<LDIM>(f,rhs,p1.get_array(),p2.get_array());
//        std::vector<Function<T,LDIM>> result;
//        for (const auto& r : rhs) result.push_back(madness::inner(f,r,p1.get_tuple(),p2.get_tuple()));
//        return result;
    }
 
    T operator()(const Vector<double,NDIM>& r) const {
        return f(r);
    }
 
    typename Tensor<T>::scalar_type norm2() const {
        return f.norm2();
    }
};
 
 
    /// LowRankFunction represents a hi-dimensional (NDIM) function as a sum of products of low-dimensional (LDIM) functions
 
    /// f(1,2) = \sum_i g_i(1) h_i(2)
    /// a LowRankFunction can be created from a hi-dim function directly, or from a composite like f(1,2) phi(1) psi(2),
    /// where f(1,2) is a two-particle function (e.g. a Slater function)
    template<typename T, std::size_t NDIM, std::size_t LDIM=NDIM/2>
    class LowRankFunction {
    public:
 
        World& world;
        double rank_revealing_tol=1.e-8;     // rrcd tol
        std::string orthomethod="canonical";
        bool do_print=false;
        std::vector<Function<T,LDIM>> g,h;
        const particle<LDIM> p1=particle<LDIM>::particle1();
        const particle<LDIM> p2=particle<LDIM>::particle2();
 
        LowRankFunction(World& world) : world(world) {}
 
        LowRankFunction(std::vector<Function<T,LDIM>> g, std::vector<Function<T,LDIM>> h,
                        double tol, std::string orthomethod) : world(g.front().world()),
                        rank_revealing_tol(tol), orthomethod(orthomethod), g(g), h(h) {
 
        }
 
        /// shallow copy ctor
        LowRankFunction(const LowRankFunction& other) : world(other.world),
            rank_revealing_tol(other.rank_revealing_tol), orthomethod(other.orthomethod),
            g(other.g), h(other.h) {
        }
 
        /// deep copy
        friend LowRankFunction copy(const LowRankFunction& other) {
            return LowRankFunction<T,NDIM>(madness::copy(other.g),madness::copy(other.h),other.rank_revealing_tol,other.orthomethod);
        }
 
        LowRankFunction& operator=(const LowRankFunction& f) { // Assignment required for storage in vector
            LowRankFunction ff(f);
            std::swap(ff.g,g);
            std::swap(ff.h,h);
            return *this;
        }
 
        /// function evaluation
        T operator()(const Vector<double,NDIM>& r) const {
            Vector<double,LDIM> first, second;
            for (int i=0; i<LDIM; ++i) {
                first[i]=r[i];
                second[i]=r[i+LDIM];
            }
            double result=0.0;
            for (int i=0; i<rank(); ++i) result+=g[i](first)*h[i](second);
            return result;
        }
 
        /*
         * arithmetic section
         */
 
        /// addition
        LowRankFunction operator+(const LowRankFunction& b) const {
            LowRankFunction<T,NDIM> result=copy(*this);
            result+=b;
            return result;
        }
        /// subtraction
        LowRankFunction operator-(const LowRankFunction& b) const {
            LowRankFunction<T,NDIM> result=copy(*this);
            result-=b;
            return result;
        }
 
        /// in-place addition
        LowRankFunction& operator+=(const LowRankFunction& b) {
 
            g=append(g,copy(b.g));
            h=append(h,copy(b.h));
            return *this;
        }
 
        /// in-place subtraction
        LowRankFunction& operator-=(const LowRankFunction& b) {
            g=append(g,-1.0*b.g);   // operator* implies deep copy of b.g
            h=append(h,copy(b.h));
            return *this;
        }
 
        /// scale by a scalar
        template<typename Q>
        LowRankFunction operator*(const Q a) const {
            return LowRankFunction<TensorResultType<T,Q>,NDIM>(g * a, Q(h),rank_revealing_tol,orthomethod);
        }
 
        /// out-of-place scale by a scalar (no type conversion)
        LowRankFunction operator*(const T a) const {
            return LowRankFunction(g * a, h,rank_revealing_tol,orthomethod);
        }
 
        /// multiplication with a scalar
        friend LowRankFunction operator*(const T a, const LowRankFunction& other) {
            return other*a;
        }
 
        /// in-place scale by a scalar (no type conversion)
        LowRankFunction& operator*=(const T a) {
            g=g*a;
            return *this;
        }
 
        /// l2 norm
        typename TensorTypeData<T>::scalar_type norm2() const {
            auto tmp1=matrix_inner(world,h,h);
            auto tmp2=matrix_inner(world,g,g);
            return sqrt(tmp1.trace(tmp2));
        }
 
        std::vector<Function<T,LDIM>> get_functions(const particle<LDIM>& p) const {
            MADNESS_CHECK(p.is_first() or p.is_last());
            if (p.is_first()) return g;
            return h;
        }
 
        std::vector<Function<T,LDIM>> get_g() const {return g;}
        std::vector<Function<T,LDIM>> get_h() const {return h;}
 
        long rank() const {return g.size();}
 
        /// return the size in GByte
        double size() const {
            double sz=get_size(world,g);
            sz+=get_size(world,h);
            return sz;
        }
 
        Function<T,NDIM> reconstruct() const {
            auto fapprox=hartree_product(g[0],h[0]);
            for (int i=1; i<g.size(); ++i) fapprox+=hartree_product(g[i],h[i]);
            return fapprox;
        }
 
        /// orthonormalize the argument vector
        std::vector<Function<T,LDIM>> orthonormalize(const std::vector<Function<T,LDIM>>& g) const {
 
            double tol=rank_revealing_tol;
            std::vector<Function<T,LDIM>> g2;
            auto ovlp=matrix_inner(world,g,g);
            if (orthomethod=="canonical") {
                tol*=0.01;
                print("orthonormalizing with method/tol",orthomethod,tol);
                g2=orthonormalize_canonical(g,ovlp,tol);
            } else if (orthomethod=="cholesky") {
                print("orthonormalizing with method/tol",orthomethod,tol);
                g2=orthonormalize_rrcd(g,ovlp,tol);
            }
            else {
                MADNESS_EXCEPTION("no such orthomethod",1);
            }
            double tight_thresh=FunctionDefaults<3>::get_thresh()*0.1;
            return truncate(g2,tight_thresh);
        }
 
 
        /// optimize the lrf using the lrfunctor
 
        /// @param[in]  nopt       number of iterations (wrt to Alg. 4.3 in Halko)
        void optimize(const LRFunctorBase<T,NDIM>& lrfunctor1, const long nopt=1) {
            timer t(world);
            t.do_print=do_print;
            for (int i=0; i<nopt; ++i) {
                // orthonormalize h
                h=truncate(orthonormalize(h));
                t.tag("ortho1");
                g=truncate(inner(lrfunctor1,h,p2,p1));
                t.tag("inner1");
                g=truncate(orthonormalize(g));
                t.tag("ortho2");
                h=truncate(inner(lrfunctor1,g,p1,p1));
                t.tag("inner2");
            }
        }
 
        /// remove linear dependencies using rank-revealing cholesky decomposition
        ///
        /// @return this with g orthonormal and h orthogonal
        void reorthonormalize_rrcd(double thresh=-1.0) {
 
            // with g~ being the orthonormalized g and h~ the orthonormalized h (from rrcd)
            // with
            //    ovlp = <g_i | g_j> = L L^T
            // we get
            //    | g~_i(1) > = | g_j(1) > piv (L^T)^(-1)        // with piv permuting/truncating, and LT^(-1) already rank-reduced
            //    | g_i(1) >  = | g~_j(1) >  L^T piv^T
            //                = | g_j(1) > piv LT^(-1) LT piv^T
            // thus:
            // f(1,2) = sum_i | g_i(1)>< h_i(2) |
            //        = | g_i(1)> piv_g LTg^(-1) LTg piv_g^(-1) <h_i(2)|                     (1)
            //        = | g_i(1)> piv_g LTg^(-1) ( LTg  piv_g^(-1) <h_i(2)| )                (2)
            //        = | g_i(1)> piv_g LTg^(-1) <h'_i(2)|                                   (3)
            //        = | g_i(1)> piv_g LTg^(-1) piv_h Lh Lh^(-1) piv_h^T <h'_i(2)|          (4)
            //        = (| g_i(1)> piv_g LTg^(-1) piv_h Lh ) ( Lh^(-1) piv_h^T LTg piv_g^(-1) <h_i(2)| )
            // Notes:
            //  piv^(-1) = piv^T since piv is a unitary permutation matrix
            // Thus
            //   | h~_i(2) > = |h_i(2) > piv_g Lg piv_h LTh^(-1)
 
            double tol=rank_revealing_tol;
 
            // no pivot_inverse is necessary..
            auto pivot_vec = [&] (const std::vector<Function<T,LDIM>>& v, const Tensor<integer>& ipiv) {
                std::size_t rank=ipiv.size();
                std::vector<Function<T,LDIM> > pv(rank);
                for(int i=0;i<ipiv.size();++i) pv[i]=v[ipiv[i]];       // all elements in [rank,v.size()) = 0 due to lindep
                return pv;
            };
 
            auto pivot_mat = [&] (const Tensor<T>& t, const Tensor<integer>& ipiv) {
                std::size_t rank=ipiv.size();
                Tensor<T> pt(t.dim(0),t.dim(1));
                // for(int i=0;i<ipiv.size();++i) pt(i,_)=t(ipiv[i],_);       // all elements in [rank,v.size()) = 0 due to lindep
                for(int i=0;i<ipiv.size();++i) pt(ipiv[i],_)=t(i,_);       // all elements in [rank,v.size()) = 0 due to lindep
                return pt;
            };
 
            auto pivot = [](const Tensor<integer>& ipiv) {
                Tensor<T> piv(ipiv.size(),ipiv.size());
                for (int i=0; i<ipiv.size(); ++i) piv(ipiv[i],i)=1.0;
                return piv;
            };
 
 
            auto LT = [&](const Tensor<T>& ovlp1) {
                Tensor<integer> piv;
                auto ovlp=copy(ovlp1);  // copy since ovlp will be destroyed
                // auto ovlp=matrix_inner(world,v,v);
                int rank;
                rr_cholesky(ovlp,tol,piv,rank); // destroys ovlp and gives back Upper ∆ Matrix from CD
                print("initial, final rank",ovlp.dim(0),rank);
                // piv=piv(Slice(0,rank-1));
                Tensor<T> p=pivot(piv);
                Tensor<T> L=inner(p,transpose(ovlp));
 
                double err1=(ovlp1-inner(L,L,1,1)).normf();
                L=L(_,Slice(0,rank-1));       // (new, old)
                double err2=(ovlp1-inner(L,L,1,1)).normf();
                print("err1, err2", err1,err2);
                return std::make_pair(L,piv);
            };
 
            // orthonormalize g: g_ortho= transform(g_piv,LT_g^(-1))
            auto ovlp_g=matrix_inner(world,g,g);
            // ovlp_g=0.5*(ovlp_g+transpose(ovlp_g));
            // for (int i=0; i<ovlp_g.dim(0); ++i) ovlp_g(i,i)+=1.e-12;
            auto [L_g,piv_g]=LT(ovlp_g);              // (1)
            L_g=pivot_mat(L_g,piv_g);
            double zero=(ovlp_g-inner(L_g,L_g,1,1)).normf();
            print("zero",zero);
            // auto hpiv_g=pivot_vec(h,piv_g);         // (2) : | h_i > piv_g
            // auto ovlp_hsmall=matrix_inner(world,hpiv_g,hpiv_g);           // (3) : < h_i | h_j >
            // auto ovlp1=inner(LT_g,inner(ovlp_hsmall,LT_g,1,1));        // (3) : LTg piv_g <h_i | h_i> piv_g  Lg
            // auto [LTh,piv_h]=LT(ovlp1);                // (3)
 
            // auto gpiv_g=pivot_vec(g,piv_g);         // | g_i > piv_g
            // auto gtrans=inner(pivot_mat(inverse(LT_g),piv_h),LTh,1,1);        // LTg^(-1) piv_g Lh
            // auto htrans=inner(pivot_mat(transpose(LT_g),piv_h),inverse(LTh));        // LTg^(-1) piv_g Lh
            auto gpiv_g=pivot_vec(g,piv_g);         // | g_i > piv_g
            auto gtrans=inverse(L_g);
            auto htrans=L_g;
            auto hpiv_g=pivot_vec(h,piv_g);         // | h_i > piv_g
 
            g=transform(world,gpiv_g,(gtrans));
            h=transform(world,hpiv_g,transpose(htrans));
            auto Gortho=matrix_inner(world,g,g);
            for (int i=0; i<Gortho.dim(0); ++i) Gortho(i,i)-=1.0;
            print("G_ortho.normf()",Gortho.normf(),Gortho.dim(0));
            print("end of line");
            // g=gpiv_g;
            // h=hpiv_g;
        }
 
        void reorthonormalize(double thresh=-1.0) {
            if (orthomethod=="canonical") {
                reorthonormalize_canonical(thresh);
            } else if (orthomethod=="cholesky") {
                reorthonormalize_rrcd(thresh);
            } else {
                MADNESS_EXCEPTION("no such orthomethod",1);
            }
        }
        /// after external operations g might not be orthonormal and/or optimal -- reorthonormalize
 
        /// orthonormalization similar to Bischoff, Harrison, Valeev, JCP 137 104103 (2012), Sec II C 3
        /// f   =\sum_i g_i h_i
        ///     = g X- (X+)^T (Y+)^T Y- h
        ///     = g X-  U S V^T  Y- h
        ///     = g (X- U) (S V^T Y-) h
        /// requires 2 matrix_inner and 2 transforms. g and h are optimal
        /// @param[in]  thresh        SVD threshold
        void reorthonormalize_canonical(double thresh=-1.0) {
            if (thresh<0.0) thresh=rank_revealing_tol;
            Tensor<T> ovlp_g = matrix_inner(world, g, g);
            Tensor<T> ovlp_h = matrix_inner(world, h, h);
 
            ovlp_g=0.5*(ovlp_g+transpose(ovlp_g));
            ovlp_h=0.5*(ovlp_h+transpose(ovlp_h));
 
            auto [eval_g, evec_g] = syev(ovlp_g);
            auto [eval_h, evec_h] = syev(ovlp_h);
 
            // get relevant part of the eigenvalues and eigenvectors
            // eigenvalues are sorted in ascending order
            auto get_slice = [](auto eval, double thresh) {
                // remove small/negative eigenvalues
                eval.screen(thresh);
                Slice s;
                for (int i=0; i<eval.size(); ++i) {
                    MADNESS_CHECK_THROW(eval[i]>=0.0,"negative eigenvalues in reorthonormalize");
                    if (eval[i]>thresh) {
                        return s=Slice(i,-1);       // from i to the end
                        break;
                    }
                }
                return s;
            };
 
            Slice gslice=get_slice(eval_g,1.e-12);
            Slice hslice=get_slice(eval_h,1.e-12);
 
            Tensor<T> Xplus=copy(evec_g(_,gslice));
            Tensor<T> Xminus=copy(evec_g(_,gslice));
            Tensor<T> Yplus=copy(evec_h(_,hslice));
            Tensor<T> Yminus=copy(evec_h(_,hslice));
            eval_g=copy(eval_g(gslice));
            eval_h=copy(eval_h(hslice));
 
            for (int i=0; i<eval_g.size(); ++i) Xplus(_,i)*=std::pow(eval_g(i),0.5);
            for (int i=0; i<eval_g.size(); ++i) Xminus(_,i)*=std::pow(eval_g(i),-0.5);
            for (int i=0; i<eval_h.size(); ++i) Yplus(_,i)*=std::pow(eval_h(i),0.5);
            for (int i=0; i<eval_h.size(); ++i) Yminus(_,i)*=std::pow(eval_h(i),-0.5);
 
            Tensor<T> M=madness::inner(Xplus,Yplus,0,0);    // (X+)^T Y+
            auto [U,s,VT]=svd(M);
 
            // truncate
            typename Tensor<T>::scalar_type s_accumulated=0.0;
            int i=s.size()-1;
            for (;i>=0; i--) {
                s_accumulated+=s[i];
                if (s_accumulated>thresh) {
                    i++;
                    break;
                }
            }
            for (int j=0; j<s.size(); ++j) VT(j,_)*=s[j];
            Tensor<T> XX=madness::inner(Xminus,U,1,0);
            Tensor<T> YY=madness::inner(Yminus,VT,1,1);
 
            g=truncate(transform(world,g,XX));
            h=truncate(transform(world,h,YY));
        }
 
 
        double check_orthonormality(const std::vector<Function<T,LDIM>>& v) const {
            Tensor<T> ovlp=matrix_inner(world,v,v);
            return check_orthonormality(ovlp);
        }
 
        double check_orthonormality(const Tensor<T>& ovlp) const {
            timer t(world);
            t.do_print=do_print;
            Tensor<T> ovlp2=ovlp;
            for (int i=0; i<ovlp2.dim(0); ++i) ovlp2(i,i)-=1.0;
            if (world.rank()==0 and do_print) {
                print("absmax",ovlp2.absmax());
                print("l2",ovlp2.normf()/ovlp2.size());
            }
            t.tag("check_orthonoramality");
            return ovlp.absmax();
        }
 
        /// compute the l2 error |functor - \sum_i g_ih_i|_2
 
        /// \int (f(1,2) - gh(1,2))^2 = \int f(1,2)^2 - 2\int f(1,2) gh(1,2) + \int gh(1,2)^2
        /// since we are subtracting large numbers the numerics are sensitive, and NaN may be returned..
        double l2error(const LRFunctorBase<T,NDIM>& lrfunctor1) const {
 
            timer t(world);
            t.do_print=do_print;
 
            // \int f(1,2)^2 d1d2
            double term1 = lrfunctor1.norm2();
            term1=term1*term1;
            t.tag("computing term1");
 
            // \int f(1,2) pre(1) post(2) \sum_i g(1) h(2) d1d2
//            double term2=madness::inner(pre*g,f12(post*h));
            double term2=madness::inner(g,inner(lrfunctor1,h,p2,p1));
            t.tag("computing term2");
 
            // g functions are orthonormal
            // \int gh(1,2)^2 d1d2 = \int \sum_{ij} g_i(1) g_j(1) h_i(2) h_j(2) d1d2
            //   = \sum_{ij} \int g_i(1) g_j(1) d1 \int h_i(2) h_j(2) d2
            //   = \sum_{ij} delta_{ij} \int h_i(2) h_j(2) d2
            //   = \sum_{i} \int h_i(2) h_i(2) d2
            double zero=check_orthonormality(g);
            if (zero>1.e-10) print("g is not orthonormal",zero);
            // double term3a=madness::inner(h,h);
            auto tmp1=matrix_inner(world,h,h);
            auto tmp2=matrix_inner(world,g,g);
            double term3=tmp1.trace(tmp2);
//            print("term3/a/diff",term3a,term3,term3-term3a);
            t.tag("computing term3");
 
            double arg=term1-2.0*term2+term3;
            if (arg<0.0) {
                print("negative l2 error");
                arg*=-1.0;
//                throw std::runtime_error("negative argument in l2error");
            }
            double error=sqrt(arg)/sqrt(term1);
            if (world.rank()==0 and do_print) {
                print("term1,2,3, error",term1, term2, term3, "  --",error);
            }
 
            return error;
        }
 
    };
 
    // This interface is necessary to compute inner products
    template<typename T, std::size_t NDIM>
    double inner(const LowRankFunction<T,NDIM>& a, const LowRankFunction<T,NDIM>& b) {
        World& world=a.world;
        return (matrix_inner(world,a.g,b.g).emul(matrix_inner(world,a.h,b.h))).sum();
    }
 
 
 
//    template<typename T, std::size_t NDIM>
//    LowRankFunction<T,NDIM> inner(const Function<T,NDIM>& lhs, const LowRankFunction<T,NDIM>& rhs,
//                                  const std::tuple<int> v1, const std::tuple<int> v2) {
//        World& world=rhs.world;
//        // int lhs(1,2) rhs(2,3) d2 = \sum \int lhs(1,2) g_i(2) h_i(3) d2
//        //                      = \sum \int lhs(1,2) g_i(2) d2 h_i(3)
//        LowRankFunction<T, NDIM + NDIM - 2> result(world);
//        result.h=rhs.h;
//        decltype(rhs.g) g;
//        for (int i=0; i<rhs.rank(); ++i) {
//            g.push_back(inner(lhs,rhs.g[i],{v1},{0}));
//        }
//        result.g=g;
//        return result;
//    }
 
    /**
     * inner product: LowRankFunction lrf; Function f, g; double d
     *  lrf(1,3) = inner(lrf(1,2), lrf(2,3))
     *  lrf(1,3) = inner(lrf(1,2), f(2,3))
     *  g(1) = inner(lrf(1,2), f(2))
     *  d = inner(lrf(1,2), f(1,2))
     *  d = inner(lrf(1,2), lrf(1,2))
     */
 
    ///  lrf(1,3) = inner(full(1,2), lrf(2,3))
 
    /// @param[in] f1 the first function
    /// @param[in] f2 the second function
    /// @param[in] p1 the integration variable of the first function
    /// @param[in] p2 the integration variable of the second function
    template<typename T, std::size_t NDIM, std::size_t PDIM>
    LowRankFunction<T,NDIM> inner(const Function<T,NDIM>& f1, const LowRankFunction<T,NDIM>& f2,
                                  const particle<PDIM> p1, const particle<PDIM> p2) {
        auto result=inner(f2,f1,p2,p1);
        std::swap(result.g,result.h);
        return result;
    }
 
    ///  lrf(1,3) = inner(lrf(1,2), full(2,3))
 
    /// @param[in] f1 the first function
    /// @param[in] f2 the second function
    /// @param[in] p1 the integration variable of the first function
    /// @param[in] p2 the integration variable of the second function
    template<typename T, std::size_t NDIM, std::size_t PDIM>
    LowRankFunction<T,NDIM> inner(const LowRankFunction<T,NDIM>& f1, const Function<T,NDIM>& f2,
                                  const particle<PDIM> p1, const particle<PDIM> p2) {
        static_assert(TensorTypeData<T>::iscomplex==false, "complex inner in LowRankFunction not implemented");
        World& world=f1.world;
        static_assert(2*PDIM==NDIM);
        // int f(1,2) k(2,3) d2 = \sum \int g_i(1) h_i(2) k(2,3) d2
        //                      = \sum g_i(1) \int h_i(2) k(2,3) d2
        LowRankFunction<T, NDIM> result(world);
        if (p1.is_last()) { // integrate over 2: result(1,3) = lrf(1,2) f(2,3)
            result.g = f1.g;
            change_tree_state(f1.h,reconstructed);
            result.h=innerXX<PDIM>(f2,f1.h,p2.get_array(),particle<PDIM>::particle1().get_array());
        } else if (p1.is_first()) { // integrate over 1: result(2,3) = lrf(1,2) f(1,3)
            result.g = f1.h;        // correct! second variable of f1 becomes first variable of result
            change_tree_state(f1.g,reconstructed);
            result.h=innerXX<PDIM>(f2,f1.g,p2.get_array(),particle<PDIM>::particle1().get_array());
        }
        return result;
    }
 
    ///  lrf(1,3) = inner(lrf(1,2), lrf(2,3))
 
    /// @param[in] f1 the first function
    /// @param[in] f2 the second function
    /// @param[in] p1 the integration variable of the first function
    /// @param[in] p2 the integration variable of the second function
    template<typename T, std::size_t NDIM, std::size_t PDIM>
    LowRankFunction<T,NDIM> inner(const LowRankFunction<T,NDIM>& f1, const LowRankFunction<T,NDIM>& f2,
                                  const particle<PDIM> p1, const particle<PDIM> p2) {
        World& world=f1.world;
        static_assert(2*PDIM==NDIM);
 
        // inner(lrf(1,2) ,lrf(2,3) ) = \sum_ij g1_i(1) <h1_i(2) g2_j(2)> h2_j(3)
        auto matrix=matrix_inner(world,f2.get_functions(p2),f1.get_functions(p1));
        auto htilde=transform(world,f2.get_functions(p2.complement()),matrix);
        auto gg=copy(world,f1.get_functions(p1.complement()));
        return LowRankFunction<T,NDIM>(gg,htilde,f1.rank_revealing_tol,f1.orthomethod);
    }
 
    ///  f(1) = inner(lrf(1,2), f(2))
 
    /// @param[in] f1 the first function
    /// @param[in] vf vector of the second functions
    /// @param[in] p1 the integration variable of the first function
    /// @param[in] p2 the integration variable of the second function, dummy variable for consistent notation
    template<typename T, std::size_t NDIM, std::size_t PDIM>
    std::vector<Function<T,NDIM-PDIM>> inner(const LowRankFunction<T,NDIM>& f1, const std::vector<Function<T,PDIM>>& vf,
                                  const particle<PDIM> p1, const particle<PDIM> p2=particle<PDIM>::particle1()) {
        World& world=f1.world;
        static_assert(2*PDIM==NDIM);
        MADNESS_CHECK(p2.is_first());
 
        // inner(lrf(1,2), f_k(2) ) = \sum_i g1_i(1) <h1_i(2) f_k(2)>
        auto matrix=matrix_inner(world,f1.get_functions(p1),vf);
        return transform(world,f1.get_functions(p1.complement()),matrix);
    }
 
    ///  f(1) = inner(lrf(1,2), f(2))
 
    /// @param[in] f1 the first function
    /// @param[in] vf the second function
    /// @param[in] p1 the integration variable of the first function
    /// @param[in] p2 the integration variable of the second function, dummy variable for consistent notation
    template<typename T, std::size_t NDIM, std::size_t PDIM>
    Function<T,NDIM> inner(const LowRankFunction<T,NDIM>& f1, const Function<T,PDIM>& f2,
                                        const particle<PDIM> p1, const particle<PDIM> p2=particle<PDIM>::particle1()) {
        return inner(f1,std::vector<Function<T,PDIM>>({f2}),p1,p2)[0];
    }
 
    template<typename T, std::size_t NDIM, std::size_t LDIM=NDIM/2>
    class LowRankFunctionFactory {
    public:
 
        const particle<LDIM> p1=particle<LDIM>::particle1();
        const particle<LDIM> p2=particle<LDIM>::particle2();
 
        LowRankFunctionParameters parameters;
        std::vector<Vector<double,LDIM>> origins;  ///< origins of the molecular grid
 
        LowRankFunctionFactory() = default;
        LowRankFunctionFactory(const LowRankFunctionParameters param, const std::vector<Vector<double,LDIM>> origins={})
                : parameters(param), origins(origins) {}
 
        LowRankFunctionFactory(const LowRankFunctionParameters param, const Molecule& molecule)
                : LowRankFunctionFactory(param,molecule.get_all_coords_vec()){}
 
        LowRankFunctionFactory(const LowRankFunctionFactory& other) = default;
 
        LowRankFunctionFactory& set_radius(const double radius) {
            parameters.set_user_defined_value("radius",radius);
            return *this;
        }
        LowRankFunctionFactory& set_volume_element(const double volume_element) {
            parameters.set_user_defined_value("volume_element",volume_element);
            return *this;
        }
        LowRankFunctionFactory& set_rank_revealing_tol(const double rrtol) {
            parameters.set_user_defined_value("tol",rrtol);
            return *this;
        }
        LowRankFunctionFactory& set_orthomethod(const std::string orthomethod) {
            parameters.set_user_defined_value("orthomethod",orthomethod);
            return *this;
        }
 
        LowRankFunction<T,NDIM> project(const LRFunctorBase<T,NDIM>& lrfunctor) const {
            World& world=lrfunctor.world();
            bool do_print=true;
            timer t1(world);
            t1.do_print=do_print;
            auto orthomethod=parameters.orthomethod();
            auto rank_revealing_tol=parameters.tol();
 
            // get sampling grid
            molecular_grid<LDIM> mgrid(origins,parameters);
            auto grid=mgrid.get_grid();
            if (world.rank()==0) print("grid size",grid.size());
 
            auto Y=Yformer(lrfunctor,grid,parameters.rhsfunctiontype());
            t1.tag("Yforming");
 
            auto ovlp=matrix_inner(world,Y,Y);  // error in symmetric matrix_inner, use non-symmetric form here!
            t1.tag("compute ovlp");
            auto g=truncate(orthonormalize_rrcd(Y,ovlp,rank_revealing_tol));
            t1.tag("rrcd/truncate/thresh");
            auto sz=get_size(world,g);
            if (world.rank()==0 and do_print) print("gsize",sz);
//            check_orthonormality(g);
 
            if (world.rank()==0 and do_print) {
                print("Y.size()",Y.size());
                print("g.size()",g.size());
            }
 
            auto h=truncate(inner(lrfunctor,g,p1,p1));
            t1.tag("Y backprojection with truncation");
            return LowRankFunction<T,NDIM>(g,h,parameters.tol(),parameters.orthomethod());
 
        }
 
        /// apply a rhs (delta or exponential) on grid points to the hi-dim function and form Y = A_ij w_j (in Halko's language)
        std::vector<Function<T,LDIM>> Yformer(const LRFunctorBase<T,NDIM>& lrfunctor1, const std::vector<Vector<double,LDIM>>& grid,
                                              const std::string rhsfunctiontype, const double exponent=30.0) const {
 
            World& world=lrfunctor1.world();
            std::vector<Function<double,LDIM>> Y;
            if (rhsfunctiontype=="exponential") {
                std::vector<Function<double,LDIM>> omega;
                double coeff=std::pow(2.0*exponent/constants::pi,0.25*LDIM);
                for (const auto& point : grid) {
                    omega.push_back(FunctionFactory<double,LDIM>(world)
                                            .functor([&point,&exponent,&coeff](const Vector<double,LDIM>& r)
                                                     {
                                                         auto r_rel=r-point;
                                                         return coeff*exp(-exponent*madness::inner(r_rel,r_rel));
                                                     }));
                }
                Y=inner(lrfunctor1,omega,p2,p1);
            } else {
                MADNESS_EXCEPTION("confused rhsfunctiontype",1);
            }
            auto norms=norm2s(world,Y);
            std::vector<Function<double,LDIM>> Ynormalized;
 
            for (int i=0; i<Y.size(); ++i) if (norms[i]>parameters.tol()) Ynormalized.push_back(Y[i]);
            normalize(world,Ynormalized);
            return Ynormalized;
        }
 
    };
 
 
} // namespace madness
 
#endif //MADNESS_LOWRANKFUNCTION_H