adaptive.h

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#pragma once
 
#include "backend/memory.h"
#include "ode.h"
#include "runge_kutta.h"
 
namespace dg
{
 
//MW: The reason these are global lambdas instead of functions is so that make_unique
//can deduce the correct types when used on dg::AdaptiveTimeloop:
//std::make_unique<dg::AdaptiveTimeloop<Vec>>( adapt, ode, dg::pid_control, dg::fast_l2norm, 1e-6, 1e-6);
static auto l2norm = [] ( const auto& x){ return sqrt( dg::blas1::dot(x,x));};
 
static auto fast_l2norm  = []( const auto& x){ return sqrt( dg::blas1::reduce(
            x, (double)0, dg::Sum(), dg::Square()));};
 
static auto i_control = []( auto dt, auto eps, unsigned embedded_order, unsigned order)
{
    using value_type = std::decay_t<decltype(dt[0])>;
    return dt[0]*pow( eps[0], -1./(value_type)embedded_order);
};
static auto pi_control = []( auto dt, auto eps, unsigned embedded_order, unsigned order)
{
    using value_type = std::decay_t<decltype(dt[0])>;
    if( dt[1] == 0)
        return i_control( dt, eps, embedded_order, order);
    value_type m_k1 = -0.8, m_k2 = 0.31;
    value_type factor = pow( eps[0], m_k1/(value_type)embedded_order)
                     * pow( eps[1], m_k2/(value_type)embedded_order);
    return dt[0]*factor;
};
static auto pid_control = []( auto dt, auto eps, unsigned embedded_order, unsigned order)
{
    using value_type = std::decay_t<decltype(dt[0])>;
    if( dt[1] == 0)
        return i_control( dt, eps, embedded_order, order);
    if( dt[2] == 0)
        return pi_control( dt, eps, embedded_order, order);
    value_type m_k1 = -0.58, m_k2 = 0.21, m_k3 = -0.1;
    //value_type m_k1 = -0.37, m_k2 = 0.27, m_k3 = -0.1;
    value_type factor = pow( eps[0], m_k1/(value_type)embedded_order)
                     * pow( eps[1], m_k2/(value_type)embedded_order)
                     * pow( eps[2], m_k3/(value_type)embedded_order);
    return dt[0]*factor;
};
 
static auto ex_control = [](auto dt, auto eps, unsigned embedded_order, unsigned order)
{
    using value_type = std::decay_t<decltype(dt[0])>;
    if( dt[1] == 0)
        return i_control( dt, eps, embedded_order, order);
    value_type m_k1 = -0.367, m_k2 = 0.268;
    value_type factor = pow( eps[0], m_k1/(value_type)embedded_order)
                      * pow( eps[0]/eps[1], m_k2/(value_type)embedded_order);
    return dt[0]*factor;
};
static auto im_control = []( auto dt, auto eps, unsigned embedded_order, unsigned order)
{
    using value_type = std::decay_t<decltype(dt[0])>;
    if( dt[1] == 0)
        return i_control( dt, eps, embedded_order, order);
    value_type m_k1 = -0.98, m_k2 = -0.95;
    value_type factor = pow( eps[0], m_k1/(value_type)embedded_order)
                     *  pow( eps[0]/eps[1], m_k2/(value_type)embedded_order);
    return dt[0]*dt[0]/dt[1]*factor;
};
static auto imex_control = []( auto dt, auto eps, unsigned embedded_order, unsigned order)
{
    using value_type = std::decay_t<decltype(dt[0])>;
    value_type h1 = ex_control( dt, eps, embedded_order, order);
    value_type h2 = im_control( dt, eps, embedded_order, order);
    // find value closest to zero
    return fabs( h1) < fabs( h2) ? h1 : h2;
};
 
 
namespace detail{
template<class value_type>
struct Tolerance
{
    // sqrt(size) is norm( 1)
    Tolerance( value_type rtol, value_type atol, value_type size) :
        m_rtol(rtol*sqrt(size)), m_atol( atol*sqrt(size)){}
    DG_DEVICE
    void operator()( value_type u0, value_type& delta) const{
        delta = delta/ ( m_rtol*fabs(u0) + m_atol);
    }
    private:
    value_type m_rtol, m_atol;
};
} //namespace detail
 
//%%%%%%%%%%%%%%%%%%%Adaptive%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
template<class Stepper>
struct Adaptive
{
    using stepper_type = Stepper;
    using container_type = typename Stepper::container_type; 
    using value_type = typename Stepper::value_type; 
    Adaptive() = default;
    template<class ...StepperParams>
    Adaptive(StepperParams&& ...ps): m_stepper(std::forward<StepperParams>(ps)...),
        m_next(m_stepper.copyable()), m_delta(m_stepper.copyable())
    {
        dg::blas1::copy( 1., m_next);
        m_size = dg::blas1::dot( m_next, 1.);
    }
    template<class ...Params>
    void construct(Params&& ...ps)
    {
        //construct and swap
        *this = Adaptive(  std::forward<Params>(ps)...);
    }
 
    stepper_type& stepper() { return m_stepper;}
    const stepper_type& stepper() const { return m_stepper;}
 
    template< class ODE,
              class ControlFunction = value_type (std::array<value_type,3>,
                      std::array< value_type,3>, unsigned , unsigned),
              class ErrorNorm = value_type( const container_type&)>
    void step( ODE&& ode,
              value_type t0,
              const container_type& u0,
              value_type& t1,
              container_type& u1,
              value_type& dt,
              ControlFunction control,
              ErrorNorm norm,
              value_type rtol,
              value_type atol,
              value_type reject_limit = 2
              )
    {
        // prevent overwriting u0 in case stepper fails
        m_stepper.step( std::forward<ODE>(ode), t0, u0, m_t_next, m_next, dt,
                m_delta);
        m_nsteps++;
        dg::blas1::subroutine( detail::Tolerance<value_type>( rtol, atol,
                    m_size), u0, m_delta);
        m_eps0 = norm( m_delta);
        m_dt0 = dt;
        if( m_eps0 > reject_limit || std::isnan( m_eps0) )
        {
            // if stepper fails, restart controller
            dt = control( std::array<value_type,3>{m_dt0, 0, m_dt2},
                    std::array<value_type,3>{m_eps0, m_eps1, m_eps2},
                    m_stepper.embedded_order(),
                    m_stepper.order());
            if( fabs( dt) > 0.9*fabs(m_dt0))
                dt = 0.9*m_dt0;
            //0.9*m_dt0 is a safety limit
            //that prevents an increase of the timestep in case the stepper fails
            m_failed = true; m_nfailed++;
            dg::blas1::copy( u0, u1);
            t1 = t0;
            return;
        }
        if( m_eps0 < 1e-30) // small or zero
        {
            dt = 1e14*m_dt0; // a very large number
            m_eps0 = 1e-30; // prevent storing zero
        }
        else
        {
            dt = control( std::array<value_type,3>{m_dt0, m_dt1, m_dt2},
                    std::array<value_type,3>{m_eps0, m_eps1, m_eps2},
                    m_stepper.embedded_order(),
                    m_stepper.order());
            // a safety net
            if( fabs(dt) > 100*fabs(m_dt0))
                dt = 100*m_dt0;
        }
        m_eps2 = m_eps1;
        m_eps1 = m_eps0;
        m_dt2 = m_dt1;
        m_dt1 = m_dt0;
        dg::blas1::copy( m_next, u1);
        t1 = m_t_next;
        m_failed = false;
    }
 
    bool failed() const {
        return m_failed;
    }
    const unsigned& nfailed() const {
        return m_nfailed;
    }
    unsigned& nfailed() {
        return m_nfailed;
    }
    const unsigned& nsteps() const {
        return m_nsteps;
    }
    unsigned& nsteps() {
        return m_nsteps;
    }
 
    const value_type& get_error( ) const{
        return m_eps0;
    }
    private:
    void reset_history(){
        m_eps1 = m_eps2 = 1.;
        m_dt1 = m_dt2 = 0.;
    }
    bool m_failed = false;
    unsigned m_nfailed = 0;
    unsigned m_nsteps = 0;
    Stepper m_stepper;
    container_type m_next, m_delta;
    value_type m_size, m_eps0 = 1, m_eps1=1, m_eps2=1;
    value_type m_t_next = 0;
    value_type m_dt0 = 0., m_dt1 = 0., m_dt2 = 0.;
};
struct EntireDomain
{
    template<class T>
    bool contains( T& t) const { return true;}
};
 
//
template<class ContainerType>
struct AdaptiveTimeloop : public aTimeloop<ContainerType>
{
    using value_type = dg::get_value_type<ContainerType>;
    using container_type = ContainerType;
    AdaptiveTimeloop( ) = default;
 
 
    AdaptiveTimeloop( std::function<void (value_type, const ContainerType&,
                value_type&, ContainerType&, value_type&)> step)  :
        m_step(step){
            m_dt_current = dg::Buffer<value_type>( 0.);
    }
    template<class Adaptive, class ODE, class ErrorNorm =
        value_type( const container_type&), class
        ControlFunction = value_type(std::array<value_type,3>,
                std::array<value_type,3>, unsigned, unsigned)>
    AdaptiveTimeloop(
          Adaptive&& adapt,
          ODE&& ode,
          ControlFunction control,
          ErrorNorm norm,
          value_type rtol,
          value_type atol,
          value_type reject_limit = 2)
    {
        // On the problem of capturing perfect forwarding in a lambda
        //https://stackoverflow.com/questions/26831382/capturing-perfectly-forwarded-variable-in-lambda
        //https://vittorioromeo.info/index/blog/capturing_perfectly_forwarded_objects_in_lambdas.html
 
        m_step = [=, cap = std::tuple<Adaptive, ODE>(std::forward<Adaptive>(adapt),
                std::forward<ODE>(ode))  ]( auto t0, auto y0, auto& t,
                auto& y, auto& dt) mutable
        {
            std::get<0>(cap).step( std::get<1>(cap), t0, y0, t, y, dt, control, norm,
                    rtol, atol, reject_limit);
        };
        m_dt_current = dg::Buffer<value_type>( 0.);
    }
 
    template<class ...Params>
    void construct( Params&& ...ps)
    {
        //construct and swap
        *this = AdaptiveTimeloop( std::forward<Params>( ps)...);
    }
 
    void set_dt( value_type dt){
        m_dt_current = dg::Buffer<value_type>(dt);
    }
 
    template< class Domain>
    void integrate_in_domain(
                  value_type t0,
                  const ContainerType& u0,
                  value_type& t1,
                  ContainerType& u1,
                  value_type dt,
                  Domain&& domain,
                  value_type eps_root
                  );
 
    virtual AdaptiveTimeloop* clone() const{return new
        AdaptiveTimeloop(*this);}
    private:
    virtual void do_integrate(value_type& t0, const container_type& u0,
            value_type t1, container_type& u1, enum to mode) const;
    std::function<void( value_type, const ContainerType&, value_type&,
            ContainerType&, value_type&)> m_step;
    virtual value_type do_dt( ) const { return m_dt_current.data();}
    dg::Buffer<value_type> m_dt_current ;
};
 
template< class ContainerType>
void AdaptiveTimeloop<ContainerType>::do_integrate(
              value_type& t_current,
              const ContainerType& u0,
              value_type t1,
              ContainerType& u1,
              enum to mode
              )const
{
    value_type deltaT = t1-t_current;
    bool forward = (deltaT > 0);
 
    value_type& dt_current = m_dt_current.data();
    if( dt_current == 0)
        dt_current = forward ? 1e-6 : -1e-6; // a good a guess as any
    if( (dt_current < 0 && forward) || ( dt_current > 0 && !forward) )
        throw dg::Error( dg::Message(_ping_)<<"Error in AdaptiveTimeloop: Timestep has wrong sign! You cannot change direction mid-step: dt "<<dt_current);
 
    blas1::copy( u0, u1 );
    while( (forward && t_current < t1) || (!forward && t_current > t1))
    {
        if( dg::to::exact == mode
                &&( (forward && t_current + dt_current > t1)
                || (!forward && t_current + dt_current < t1) ) )
            dt_current = t1-t_current;
        if( dg::to::at_least == mode
                &&( (forward && dt_current > deltaT)
                || (!forward && dt_current < deltaT) ) )
            dt_current = deltaT;
        // Compute a step and error
        try{
            m_step( t_current, u1, t_current, u1, dt_current);
        }catch ( dg::Error& e)
        {
            e.append( dg::Message(_ping_) << "Error in AdaptiveTimeloop::integrate");
            throw;
        }
        if( !std::isfinite(dt_current) || fabs(dt_current) < 1e-9*fabs(deltaT))
        {
            value_type dt_current0 = dt_current;
            dt_current = 0.;
            throw dg::Error(dg::Message(_ping_)<<"Adaptive integrate failed to converge! dt = "<<std::scientific<<dt_current0);
        }
    }
}
 
template< class ContainerType>
template< class Domain>
void AdaptiveTimeloop<ContainerType>::integrate_in_domain(
              value_type t0,
              const ContainerType& u0,
              value_type& t1,
              ContainerType& u1,
              value_type dt,
              Domain&& domain,
              value_type eps_root
              )
{
    value_type t_current = t0, dt_current = dt;
    blas1::copy( u0, u1 );
    ContainerType& current(u1);
    if( t1 == t0)
        return;
    bool forward = (t1 - t0 > 0);
    if( dt == 0)
        dt_current = forward ? 1e-6 : -1e-6; // a good a guess as any
 
    ContainerType last( u0);
    while( (forward && t_current < t1) || (!forward && t_current > t1))
    {
        //remember last step
        t0 = t_current;
        dg::blas1::copy( current, last);
        if( (forward && t_current+dt_current > t1) || (!forward && t_current +
                    dt_current < t1) )
            dt_current = t1-t_current;
        // Compute a step and error
        try{
            m_step( t_current, current, t_current, current, dt_current);
        }catch ( dg::Error& e)
        {
            e.append( dg::Message(_ping_) << "Error in AdaptiveTimeloop::integrate");
            throw;
        }
        if( !std::isfinite(dt_current) || fabs(dt_current) < 1e-9*fabs(t1-t0))
            throw dg::Error(dg::Message(_ping_)<<"integrate_in_domain failed to converge! dt = "<<std::scientific<<dt_current);
        if( !domain.contains( current) )
        {
            //start bisection between t0 and t1
            t1 = t_current;
 
            // t0 and last are inside
            dg::blas1::copy( last, current);
 
            int j_max = 50;
            for(int j=0; j<j_max; j++)
            {
                if( fabs(t1-t0) < eps_root*fabs(t1) + eps_root)
                {
                    return;
                }
                dt_current = (t1-t0)/2.;
                t_current = t0; //always start integrate from inside
                value_type failed = t_current;
                // Naively we would assume that the timestepper always succeeds
                // because dt_current is always smaller than the previous timestep
                // However, there are cases when this is not true so we need to
                // explicitly check!!
                // t_current = t0, current = last (inside)
                m_step( t_current, current, t_current, current, dt_current);
                if( failed == t_current)
                {
                    dt_current = (t1-t0)/4.;
                    break; // we need to get out of here
                }
 
                //stepper( t0, last, t_middle, u1, (t1-t0)/2.);
                if( domain.contains( current) )
                {
                    t0 = t_current;
                    dg::blas1::copy( current, last);
                }
                else
                {
                    t1 = t_current;
                    dg::blas1::copy( last, current);
                }
                if( (j_max - 1) == j)
                    throw dg::Error( dg::Message(_ping_)<<"integrate_in_domain: too many steps in root finding!");
            }
        }
    }
}
 
}//namespace dg