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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2015, 2024 Peter Caspers
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<https://www.quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
/*! \file laplaceinterpolation.hpp
\brief Laplace interpolation of missing values
*/
#include <ql/experimental/math/laplaceinterpolation.hpp>
#include <ql/math/matrix.hpp>
#include <ql/math/matrixutilities/bicgstab.hpp>
#include <ql/math/matrixutilities/sparsematrix.hpp>
#include <ql/methods/finitedifferences/meshers/fdm1dmesher.hpp>
#include <ql/methods/finitedifferences/meshers/fdmmeshercomposite.hpp>
#include <ql/methods/finitedifferences/meshers/predefined1dmesher.hpp>
#include <ql/methods/finitedifferences/operators/fdmlinearopcomposite.hpp>
#include <ql/methods/finitedifferences/operators/fdmlinearoplayout.hpp>
#include <ql/methods/finitedifferences/operators/secondderivativeop.hpp>
#include <ql/methods/finitedifferences/operators/triplebandlinearop.hpp>
namespace QuantLib {
LaplaceInterpolation::LaplaceInterpolation(std::function<Real(const std::vector<Size>&)> y,
std::vector<std::vector<Real>> x,
Real relTol,
Size maxIterMultiplier)
: y_(std::move(y)), x_(std::move(x)), relTol_(relTol), maxIterMultiplier_(maxIterMultiplier) {
// set up the mesher
std::vector<Size> dim;
coordinateIncluded_.resize(x_.size());
for (Size i = 0; i < x_.size(); ++i) {
coordinateIncluded_[i] = x_[i].size() > 1;
if (coordinateIncluded_[i])
dim.push_back(x_[i].size());
}
numberOfCoordinatesIncluded_ = dim.size();
if (numberOfCoordinatesIncluded_ == 0) {
return;
}
QL_REQUIRE(!dim.empty(), "LaplaceInterpolation: singular point or no points given");
layout_ = ext::make_shared<FdmLinearOpLayout>(dim);
std::vector<ext::shared_ptr<Fdm1dMesher>> meshers;
for (auto & i : x_) {
if (i.size() > 1)
meshers.push_back(ext::make_shared<Predefined1dMesher>(i));
}
auto mesher = ext::make_shared<FdmMesherComposite>(layout_, meshers);
// set up the Laplace operator and convert it to matrix
struct LaplaceOp : public FdmLinearOpComposite {
explicit LaplaceOp(const ext::shared_ptr<FdmMesher>& mesher) {
for (Size direction = 0; direction < mesher->layout()->dim().size(); ++direction) {
if (mesher->layout()->dim()[direction] > 1)
map_.push_back(SecondDerivativeOp(direction, mesher));
}
}
std::vector<TripleBandLinearOp> map_;
Size size() const override { QL_FAIL("no impl"); }
void setTime(Time t1, Time t2) override { QL_FAIL("no impl"); }
Array apply(const array_type& r) const override { QL_FAIL("no impl"); }
Array apply_mixed(const Array& r) const override { QL_FAIL("no impl"); }
Array apply_direction(Size direction, const Array& r) const override {
QL_FAIL("no impl");
}
Array solve_splitting(Size direction, const Array& r, Real s) const override {
QL_FAIL("no impl");
}
Array preconditioner(const Array& r, Real s) const override { QL_FAIL("no impl"); }
std::vector<SparseMatrix> toMatrixDecomp() const override {
std::vector<SparseMatrix> decomp;
decomp.reserve(map_.size());
for (auto const& m : map_)
decomp.push_back(m.toMatrix());
return decomp;
}
};
SparseMatrix op = LaplaceOp(mesher).toMatrix();
// set up the linear system to solve
Size N = layout_->size();
SparseMatrix g(N, N, 5 * N);
Array rhs(N, 0.0), guess(N, 0.0);
Real guessTmp = 0.0;
struct f_A {
const SparseMatrix& g;
explicit f_A(const SparseMatrix& g) : g(g) {}
Array operator()(const Array& x) const { return prod(g, x); }
};
auto rowit = op.begin1();
Size count = 0;
std::vector<Real> corner_h(dim.size());
std::vector<Size> corner_neighbour_index(dim.size());
for (auto const& pos : *layout_) {
const auto& coord = pos.coordinates();
Real val =
y_(numberOfCoordinatesIncluded_ == x_.size() ? coord : fullCoordinates(coord));
QL_REQUIRE(rowit != op.end1() && rowit.index1() == count,
"LaplaceInterpolation: op matrix row iterator ("
<< (rowit != op.end1() ? std::to_string(rowit.index1()) : "na")
<< ") does not match expected row count (" << count << ")");
if (val == Null<Real>()) {
bool isCorner = true;
for (Size d = 0; d < dim.size() && isCorner; ++d) {
if (coord[d] == 0) {
corner_h[d] = meshers[d]->dplus(0);
corner_neighbour_index[d] = 1;
} else if (coord[d] == layout_->dim()[d] - 1) {
corner_h[d] = meshers[d]->dminus(dim[d] - 1);
corner_neighbour_index[d] = dim[d] - 2;
} else {
isCorner = false;
}
}
if (isCorner) {
// handling of the "corners", all second derivs are zero in the op
// this directly generalizes Numerical Recipes, 3rd ed, eq 3.8.6
Real sum_corner_h =
std::accumulate(corner_h.begin(), corner_h.end(), Real(0.0), std::plus<>());
for (Size j = 0; j < dim.size(); ++j) {
std::vector<Size> coord_j(coord);
coord_j[j] = corner_neighbour_index[j];
Real weight = 0.0;
for (Size i = 0; i < dim.size(); ++i) {
if (i != j)
weight += corner_h[i];
}
weight = dim.size() == 1 ? Real(1.0) : Real(weight / sum_corner_h);
g(count, layout_->index(coord_j)) = -weight;
}
g(count, count) = 1.0;
} else {
// point with at least one dimension with non-trivial second derivative
for (auto colit = rowit.begin(); colit != rowit.end(); ++colit)
g(count, colit.index2()) = *colit;
}
rhs[count] = 0.0;
guess[count] = guessTmp;
} else {
g(count, count) = 1;
rhs[count] = val;
guess[count] = guessTmp = val;
}
++count;
++rowit;
}
interpolatedValues_ = BiCGstab(f_A(g), maxIterMultiplier_ * N, relTol_).solve(rhs, guess).x;
}
std::vector<Size>
LaplaceInterpolation::projectedCoordinates(const std::vector<Size>& coordinates) const {
std::vector<Size> tmp;
for (Size i = 0; i < coordinates.size(); ++i) {
if (coordinateIncluded_[i])
tmp.push_back(coordinates[i]);
}
return tmp;
}
std::vector<Size>
LaplaceInterpolation::fullCoordinates(const std::vector<Size>& projectedCoordinates) const {
std::vector<Size> tmp(coordinateIncluded_.size(), 0);
for (Size i = 0, count = 0; i < coordinateIncluded_.size(); ++i) {
if (coordinateIncluded_[i])
tmp[i] = projectedCoordinates[count++];
}
return tmp;
}
Real LaplaceInterpolation::operator()(const std::vector<Size>& coordinates) const {
QL_REQUIRE(coordinates.size() == x_.size(), "LaplaceInterpolation::operator(): expected "
<< x_.size() << " coordinates, got "
<< coordinates.size());
if (numberOfCoordinatesIncluded_ == 0) {
Real val = y_(coordinates);
return val == Null<Real>() ? 0.0 : val;
} else {
return interpolatedValues_[layout_->index(numberOfCoordinatesIncluded_ == x_.size() ?
coordinates :
projectedCoordinates(coordinates))];
}
}
void laplaceInterpolation(Matrix& A,
const std::vector<Real>& x,
const std::vector<Real>& y,
Real relTol,
Size maxIterMultiplier) {
std::vector<std::vector<Real>> tmp;
tmp.push_back(y);
tmp.push_back(x);
if (y.empty()) {
tmp[0].resize(A.rows());
std::iota(tmp[0].begin(), tmp[0].end(), 0.0);
}
if (x.empty()) {
tmp[1].resize(A.columns());
std::iota(tmp[1].begin(), tmp[1].end(), 0.0);
}
LaplaceInterpolation interpolation(
[&A](const std::vector<Size>& coordinates) {
return A(coordinates[0], coordinates[1]);
},
tmp, relTol, maxIterMultiplier);
for (Size i = 0; i < A.rows(); ++i) {
for (Size j = 0; j < A.columns(); ++j) {
if (A(i, j) == Null<Real>())
A(i, j) = interpolation({i, j});
}
}
}
} // namespace QuantLib
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