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0.9964532883337928 33 + 36 98.220606664100151 0.9986072472643934 46 + 37 100 0.9999916341793182 68 diff --git a/src/functional/basis/fermi_bk/dlr100_1e-4.dlr b/src/functional/basis/fermi_bk/dlr100_1e-4.dlr new file mode 100644 index 0000000..c1667cd --- /dev/null +++ b/src/functional/basis/fermi_bk/dlr100_1e-4.dlr @@ -0,0 +1,19 @@ + 1 -100 8.365820681786179e-06 -64 + 2 -88.42109914844504 0.004220873864096322 -31 + 3 -72.4572010929753 0.011089301917899029 -16 + 4 -46.130379100884332 0.03777378598644059 -9 + 5 -33.169238168527734 0.05755612971910736 -6 + 6 -21.712016793882524 0.08085290531861267 -5 + 7 -13.78759357071734 0.154495759334352 -4 + 8 -6.6129197120883759 0.2331165045436147 -3 + 9 -3.1036341342994562 0.36274785745880495 -2 + 10 0 0.5283736933296579 -1 + 11 3.1036341342994562 0.620910114647278 0 + 12 6.6129197120883759 0.6978097121084752 1 + 13 13.78759357071734 0.8023190329128382 2 + 14 21.712016793882524 0.8733673112048332 3 + 15 33.169238168527734 0.9276122594382147 4 + 16 46.130379100884332 0.9622262140135596 5 + 17 72.4572010929753 0.988910698082101 8 + 18 88.42109914844504 0.9957791261359037 15 + 19 100 0.9999916341793182 34 diff --git a/src/functional/basis/fermi_bk/dlr100_1e-6.dlr b/src/functional/basis/fermi_bk/dlr100_1e-6.dlr new file mode 100644 index 0000000..e7aa395 --- /dev/null +++ b/src/functional/basis/fermi_bk/dlr100_1e-6.dlr @@ -0,0 +1,23 @@ + 1 -100 8.365820681786179e-06 -66 + 2 -88.42109914844504 0.004220873864096322 -31 + 3 -72.4572010929753 0.011089301917899029 -16 + 4 -58.546141170436265 0.023876060166412 -12 + 5 -46.130379100884332 0.03696260493494304 -9 + 6 -33.169238168527734 0.06959342333241447 -7 + 7 -21.712016793882524 0.12028756817270558 -6 + 8 -13.78759357071734 0.185455057323639 -5 + 9 -10.153475198905109 0.23022451887642945 -4 + 10 -6.6129197120883759 0.28955096224714105 -3 + 11 -3.1036341342994562 0.370910114647278 -2 + 12 0 0.45429916052045566 -1 + 13 3.1036341342994562 0.5457008394795443 0 + 14 6.6129197120883759 0.620910114647278 1 + 15 10.153475198905109 0.7526912290167369 2 + 16 13.78759357071734 0.8345156279950772 3 + 17 21.712016793882524 0.8848877405617853 4 + 18 33.169238168527734 0.9172578139975386 5 + 19 46.130379100884332 0.944873939833588 8 + 20 58.546141170436265 0.9622262140135596 11 + 21 72.4572010929753 0.988910698082101 15 + 22 88.42109914844504 0.9957791261359037 22 + 23 100 0.9999916341793182 34 diff --git a/src/functional/basis/fermi_bk/dlr100_1e-8.dlr b/src/functional/basis/fermi_bk/dlr100_1e-8.dlr new file mode 100644 index 0000000..75865a3 --- /dev/null +++ b/src/functional/basis/fermi_bk/dlr100_1e-8.dlr @@ -0,0 +1,27 @@ + 1 -100 8.365820681786179e-06 -76 + 2 -95.408303567634543 0.0035467116662072345 -40 + 3 -88.42109914844504 0.011089301917899029 -24 + 4 -72.4572010929753 0.022073547340693667 -16 + 5 -58.546141170436265 0.03417427055325602 -11 + 6 -46.130379100884332 0.06014378408635279 -9 + 7 -33.169238168527734 0.09884048354358091 -7 + 8 -27.154749462724212 0.12663268879516665 -6 + 9 -21.712016793882524 0.15803770394837513 -5 + 10 -13.78759357071734 0.1976809670871618 -4 + 11 -10.153475198905109 0.23581315333517103 -3 + 12 -6.6129197120883759 0.308991518668704 -2 + 13 -3.1036341342994562 0.38725214254119505 -1 + 14 0 0.47162630667034205 0 + 15 3.1036341342994562 0.558991518668704 1 + 16 6.6129197120883759 0.6533845328792967 2 + 17 10.153475198905109 0.7352401580435444 3 + 18 13.78759357071734 0.7943572076715961 4 + 19 21.712016793882524 0.845504240665648 5 + 20 27.154749462724212 0.8880475719728813 6 + 21 33.169238168527734 0.9191470946813873 7 + 22 46.130379100884332 0.944873939833588 9 + 23 58.546141170436265 0.9652032883337928 12 + 24 72.4572010929753 0.979839301917899 16 + 25 88.42109914844504 0.9898789069987692 23 + 26 95.408303567634543 0.997075729446744 35 + 27 100 0.9999916341793182 70 diff --git a/src/sample/sample.jl b/src/sample/sample.jl index d927811..75176e2 100644 --- a/src/sample/sample.jl +++ b/src/sample/sample.jl @@ -40,6 +40,14 @@ function SemiCircle(Euv, β, isFermi::Bool, Grid, type::Symbol, symmetry::Symbol end pbp[1:npo] = -pbp[2npo+1:-1:npo+2] + # if IsMatFreq + # if isFermi + # kernel = Spectral.kernelFermiΩ + # else + # end + # else + # end + g1 = _Green(Val(IsMatFreq), Euv, β, isFermi, Grid, symmetry, degree, pbp, npo, regularized) # g1 = _Green(IsMatFreq, Euv, β, isFermi, Grid, symmetry, degree, pbp, npo, regularized) @@ -54,6 +62,22 @@ function SemiCircle(dlr, type::Symbol, Grid = dlrGrid(dlr, type); degree = 24, r return SemiCircle(dlr.Euv, dlr.β, dlr.isFermi, Grid, type, dlr.symmetry; rtol = dlr.rtol, degree = degree, regularized = regularized) end +# @inline function kernelΩ(isFermi, symmetry, regularized::Bool = false) where {T<:AbstractFloat,isFermi,symmetry} +# if symmetry == :none +# if regularized +# return isFermi ? kernelFermiΩ : kernelBoseΩ_regular +# else +# return isFermi ? kernelFermiΩ : kernelBoseΩ +# end +# elseif symmetry == :ph +# return isFermi ? kernelFermiΩ_PH : kernelBoseΩ_PH +# elseif symmetry == :pha +# return isFermi ? kernelFermiΩ_PHA : kernelBoseΩ_PHA +# else +# error("Symmetry $symmetry is not implemented!") +# end +# end + function dlrGrid(dlr, type::Symbol) if type == :τ return dlr.τ @@ -77,6 +101,8 @@ function _Green(::Val{IsMatFreq}, Euv, β, isFermi, Grid, symmetry, n, pbp, npo, xl, wl = gausslegendre(n) xj, wj = gaussjacobi(n, 1 / 2, 0.0) # println(IsMatFreq) + type = Val(isFermi) + sym = Val(symmetry) G = IsMatFreq ? zeros(ComplexF64, length(Grid)) : zeros(Float64, length(Grid)) # G = getG(isMatFreq, Grid) @@ -90,8 +116,8 @@ function _Green(::Val{IsMatFreq}, Euv, β, isFermi, Grid, symmetry, n, pbp, npo, continue end ker = IsMatFreq ? - Spectral.kernelΩ(Val(isFermi), Val(symmetry), τ, Euv * x, β, regularized) : - Spectral.kernelT(Val(isFermi), Val(symmetry), τ, Euv * x, β, regularized) + Spectral.kernelΩ(type, sym, τ, Euv * x, β, regularized) : + Spectral.kernelT(type, sym, τ, Euv * x, β, regularized) G[τi] += (b - a) / 2 * wl[jj] * ker * sqrt(1 - x^2) end end @@ -100,8 +126,8 @@ function _Green(::Val{IsMatFreq}, Euv, β, isFermi, Grid, symmetry, n, pbp, npo, for jj = 1:n x = (a + b) / 2 + (b - a) / 2 * xj[jj] ker = IsMatFreq ? - Spectral.kernelΩ(Val(isFermi), Val(symmetry), τ, Euv * x, β, regularized) : - Spectral.kernelT(Val(isFermi), Val(symmetry), τ, Euv * x, β, regularized) + Spectral.kernelΩ(type, sym, τ, Euv * x, β, regularized) : + Spectral.kernelT(type, sym, τ, Euv * x, β, regularized) G[τi] += ((b - a) / 2)^1.5 * wj[jj] * ker * sqrt(1 + x) end @@ -111,8 +137,8 @@ function _Green(::Val{IsMatFreq}, Euv, β, isFermi, Grid, symmetry, n, pbp, npo, for jj = 1:n x = (a + b) / 2 + (b - a) / 2 * (-xj[n-jj+1]) ker = IsMatFreq ? - Spectral.kernelΩ(Val(isFermi), Val(symmetry), τ, Euv * x, β, regularized) : - Spectral.kernelT(Val(isFermi), Val(symmetry), τ, Euv * x, β, regularized) + Spectral.kernelΩ(type, sym, τ, Euv * x, β, regularized) : + Spectral.kernelT(type, sym, τ, Euv * x, β, regularized) G[τi] += ((b - a) / 2)^1.5 * wj[n-jj+1] * ker * sqrt(1 - x) end end diff --git a/src/vertex3/QR.jl b/src/vertex3/QR.jl new file mode 100644 index 0000000..8f13753 --- /dev/null +++ b/src/vertex3/QR.jl @@ -0,0 +1,207 @@ +module FQR + +using LinearAlgebra, Printf +using StaticArrays +# using GenericLinearAlgebra + +const Float = Float64 + +### faster, a couple of less digits +using DoubleFloats +# const Float = Double64 +const Double = Double64 + +# similar speed as DoubleFloats +# using MultiFloats +# const Float = Float64x2 +# const Double = Float64x2 + +### a couple of more digits, but slower +# using Quadmath +# const Float = Float128 + +### 64 digits by default, but a lot more slower +# const Float = BigFloat + +###################### traits to the functional QR ############################ +abstract type Grid end +abstract type FineMesh end + +dot(mesh, g1, g2) = error("QR.dot is not implemented!") +mirror(g) = error("QR.mirror for $(typeof(g)) is not implemented!") +irreducible(g) = error("QR.irreducible for $(typeof(g)) is not implemented!") +################################################################################# + +mutable struct Basis{D,Grid,Mesh} + ############ fundamental parameters ################## + Λ::Float # UV energy cutoff * inverse temperature + rtol::Float # error tolerance + + ############### DLR grids ############################### + N::Int # number of basis + grid::Vector{Grid} # grid for the basis + error::Vector{Float} # the relative error achieved by adding the current grid point + + ############### linear coefficients for orthognalization ####### + Q::Matrix{Double} # , Q = R^{-1}, Q*R'= I + R::Matrix{Double} + + ############ fine mesh ################# + mesh::Mesh + + function Basis{d,Grid}(Λ, rtol, mesh::Mesh) where {d,Grid,Mesh} + _Q = Matrix{Float}(undef, (0, 0)) + _R = similar(_Q) + return new{d,Grid,Mesh}(Λ, rtol, 0, [], [], _Q, _R, mesh) + end +end + +function addBasis!(basis::Basis{D,G,M}, grid, verbose) where {D,G,M} + basis.N += 1 + push!(basis.grid, grid) + + basis.Q, basis.R = GramSchmidt(basis) + + # println(maximum(basis.mesh.residual)) + # update the residual on the fine mesh + updateResidual!(basis) + + # println(maximum(basis.mesh.residual)) + # the new rtol achieved by adding the new grid point + push!(basis.error, sqrt(maximum(basis.mesh.residual))) + + (verbose > 0) && @printf("%3i %s -> error=%16.8g, Rmin=%16.8g\n", basis.N, "$(grid)", basis.error[end], basis.R[end, end]) +end + +function addBasisBlock!(basis::Basis{D,G,M}, idx, verbose) where {D,G,M} + _norm = sqrt(basis.mesh.residual[idx]) # the norm derived from the delta update in updateResidual + addBasis!(basis, basis.mesh.candidates[idx], verbose) + _R = basis.R[end, end] # the norm derived from the GramSchmidt + + @assert abs(_norm - _R) < basis.rtol * 100 "inconsistent norm on the grid $(basis.grid[end]) $_norm - $_R = $(_norm-_R)" + if abs(_norm - _R) > basis.rtol * 10 + @warn("inconsistent norm on the grid $(basis.grid[end]) $_norm - $_R = $(_norm-_R)") + end + + ## set the residual of the selected grid point to be zero + basis.mesh.selected[idx] = true + basis.mesh.residual[idx] = 0 # the selected mesh grid has zero residual + + # println(mirror(basis.mesh, idx)) + for grid in mirror(basis.mesh, idx) + addBasis!(basis, grid, verbose) + end +end + +function updateResidual!(basis::Basis{D}) where {D} + mesh = basis.mesh + + # q = Float.(basis.Q[end, :]) + q = Double.(basis.Q[:, end]) + + Threads.@threads for idx in 1:length(mesh.candidates) + if mesh.selected[idx] == false + candidate = mesh.candidates[idx] + pp = sum(q[j] * dot(mesh, basis.grid[j], candidate) for j in 1:basis.N) + _residual = mesh.residual[idx] - pp * pp + # @assert isnan(_residual) == false "$pp and $([q[j] for j in 1:basis.N]) => $([dot(mesh, basis.grid[j], candidate) for j in 1:basis.N])" + # println("working on $candidate : $_residual") + if _residual < 0 + if _residual < -basis.rtol + @warn("warning: residual smaller than 0 at $candidate got $(mesh.residual[idx]) - $(pp)^2 = $_residual") + end + mesh.residual[idx] = 0 + else + mesh.residual[idx] = _residual + end + end + end +end + +""" +Gram-Schmidt process to the last grid point in basis.grid +""" +function GramSchmidt(basis::Basis{D,G,M}) where {D,G,M} + _Q = zeros(Double, (basis.N, basis.N)) + _Q[1:end-1, 1:end-1] = basis.Q + + _R = zeros(Double, (basis.N, basis.N)) + _R[1:end-1, 1:end-1] = basis.R + _Q[end, end] = 1 + + newgrid = basis.grid[end] + + overlap = [dot(basis.mesh, basis.grid[j], newgrid) for j in 1:basis.N-1] + + for qi in 1:basis.N-1 + _R[qi, end] = basis.Q[:, qi]' * overlap + _Q[:, end] -= _R[qi, end] * _Q[:, qi] # q + end + + _norm = dot(basis.mesh, newgrid, newgrid) - _R[:, end]' * _R[:, end] + _norm = sqrt(abs(_norm)) + + @assert _norm > eps(Double(1)) * 100 "$_norm is too small as a denominator!\nnewgrid = $newgrid\nexisting grid = $(basis.grid)\noverlap=$overlap\nR=$_R\nQ=$_Q" + + _R[end, end] = _norm + _Q[:, end] /= _norm + + return _Q, _R +end + +function test(basis::Basis{D}) where {D} + println("testing orthognalization...") + KK = zeros(Double, (basis.N, basis.N)) + Threads.@threads for i in 1:basis.N + g1 = basis.grid[i] + for (j, g2) in enumerate(basis.grid) + KK[i, j] = dot(basis.mesh, g1, g2) + end + end + maxerr = maximum(abs.(KK - basis.R' * basis.R)) + println("Max overlap matrix R'*R Error: ", maxerr) + + maxerr = maximum(abs.(basis.R * basis.Q - I)) + println("Max R*R^{-1} Error: ", maxerr) + + II = basis.Q' * KK * basis.Q + maxerr = maximum(abs.(II - I)) + println("Max Orthognalization Error: ", maxerr) + + # KK = zeros(Float, (basis.N, basis.N)) + # Threads.@threads for i in 1:basis.N + # g1 = basis.grid[i] + # for (j, g2) in enumerate(basis.grid) + # KK[i, j] = dot(basis.mesh, g1, g2) + # end + # end + # println(maximum(abs.(KK' - KK))) + # A = cholesky(KK, Val{true}()) + # println(maximum(abs.(A.L * A.U - KK))) + # println(maximum(abs.(A.L' - A.U))) +end + +# function testResidual(basis, proj) +# # residual = [Residual(basis, proj, basis.grid[i, :]) for i in 1:basis.N] +# # println("Max deviation from zero residual: ", maximum(abs.(residual))) +# println("Max deviation from zero residual on the DLR grids: ", maximum(abs.(basis.residualFineGrid[basis.gridIdx]))) +# end + +function qr!(basis::Basis{dim,G,M}; initial = [], N = 10000, verbose = 0) where {dim,G,M} + #### add the grid in the idx vector first + + for i in initial + addBasisBlock!(basis, i, verbose) + end + + ####### add grids that has the maximum residual + maxResidual, idx = findmax(basis.mesh.residual) + while sqrt(maxResidual) > basis.rtol && basis.N < N + addBasisBlock!(basis, idx, verbose) + maxResidual, idx = findmax(basis.mesh.residual) + end + @printf("rtol = %.16e\n", sqrt(maxResidual)) + return basis +end + +end \ No newline at end of file diff --git a/src/vertex3/dlr1000_1e-9/plot.py b/src/vertex3/dlr1000_1e-9/plot.py new file mode 100644 index 0000000..8bc71e1 --- /dev/null +++ b/src/vertex3/dlr1000_1e-9/plot.py @@ -0,0 +1,24 @@ +import matplotlib.pyplot as plt +import numpy as np + +plt.gca().set_aspect('equal') +plt.style.use('science') + +grid = np.loadtxt("basis.dat") +finegrid = np.loadtxt("finegrid.dat") +Nfine = len(finegrid) +residual = np.loadtxt("residual.dat") +residual = np.sqrt(np.reshape(residual, (Nfine, Nfine))) + +xv, yv = np.meshgrid(finegrid, finegrid) +# plt.imshow(xv, yv, residual) +plt.contourf(xv, yv, residual, 16) +plt.colorbar() + +plt.scatter(grid[:, 0], grid[:, 1], c="yellow", alpha=0.5, s=6) +plt.xscale("log") +plt.yscale("log") +plt.xlabel("$\\omega_1$") +plt.ylabel("$\\omega_2$") +plt.savefig("residual.pdf") +plt.show() diff --git a/src/vertex3/dlr100_1e-9/plot.py b/src/vertex3/dlr100_1e-9/plot.py new file mode 100644 index 0000000..da53959 --- /dev/null +++ b/src/vertex3/dlr100_1e-9/plot.py @@ -0,0 +1,96 @@ +import matplotlib.pyplot as plt +import numpy as np +from scipy.spatial import Delaunay +from scipy.interpolate import NearestNDInterpolator +from scipy import optimize + +plt.gca().set_aspect('equal') +plt.style.use('science') + +grid = np.loadtxt("basis.dat") +finegrid = np.loadtxt("finegrid.dat") +Nfine = len(finegrid) +residual = np.loadtxt("residual.dat") + +print(len(residual)) +print(len([residual[i] + for i in range(len(residual)) if np.sqrt(residual[i]) > 0.2e-9])) + +# print("grid: ", list(zip(finegrid, finegrid))) +test = [np.array([x, y]) + for x in finegrid for y in finegrid] +interp = NearestNDInterpolator(test, -residual) + + +def interplate(x0): + # print("inter ", x0) + # print("got ", interp([np.array([x0[0], x0[1]])])) + return interp(x0[0], x0[1]) + + +residual = np.sqrt(np.reshape(residual, (Nfine, Nfine))) + +xv, yv = np.meshgrid(finegrid, finegrid) +# plt.imshow(xv, yv, residual) +plt.contourf(xv, yv, residual, 16) +plt.colorbar() + + +<<<<<<< HEAD +# tri = Delaunay(np.log(grid), qhull_options="QJ") +======= +tri = Delaunay(np.log(grid), qhull_options="QJ") +>>>>>>> b7b189f8832d73d71f65147bc5276a44071e4823 +# plt.triplot(grid[:, 0], grid[:, 1], tri.simplices) + +# print(len(grid)) +# print(len(tri.simplices)) +# print(tri.simplices[0]) + +# coordx = [] +# coordy = [] + +<<<<<<< HEAD +# for sim in tri.simplices: +# initial = [grid[s] for s in sim] +# x = (initial[0][0]+initial[1][0]+initial[2][0])/3 +# y = (initial[0][1]+initial[1][1]+initial[2][1])/3 +# # initial.append([x, y]) +# x0 = np.array([x, y]) +# initial = np.array(initial) +# # print(initial.shape) +# mimum = optimize.fmin(interplate, x0, +# xtol=0.05, ftol=1e-10, initial_simplex=initial) +# coordx.append(mimum[0]) +# coordy.append(mimum[1]) +# print(initial) + + +plt.scatter(grid[:, 0], grid[:, 1], c="yellow", alpha=0.5, s=6) +# plt.scatter(coordx, coordy, c="red", alpha=0.5, s=8) +======= +for sim in tri.simplices: + initial = [grid[s] for s in sim] + x = (initial[0][0]+initial[1][0]+initial[2][0])/3 + y = (initial[0][1]+initial[1][1]+initial[2][1])/3 + # initial.append([x, y]) + x0 = np.array([x, y]) + initial = np.array(initial) + # print(initial.shape) + # mimum = optimize.fmin(interplate, x0, + # xtol=0.05, ftol=1e-10, initial_simplex=initial) + # coordx.append(mimum[0]) + # coordy.append(mimum[1]) + # print(initial) + + +plt.scatter(grid[:, 0], grid[:, 1], c="red", alpha=0.5, s=6) +plt.scatter(coordx, coordy, c="red", alpha=0.5, s=15) +>>>>>>> b7b189f8832d73d71f65147bc5276a44071e4823 + +plt.xscale("log") +plt.yscale("log") +plt.xlabel("$\\omega_1$") +plt.ylabel("$\\omega_2$") +plt.savefig("residual.pdf") +plt.show() diff --git a/src/vertex3/dlr100_1e-9/residual.pdf b/src/vertex3/dlr100_1e-9/residual.pdf new file mode 100644 index 0000000..caf95b5 Binary files /dev/null and b/src/vertex3/dlr100_1e-9/residual.pdf differ diff --git a/src/vertex3/frequency.jl b/src/vertex3/frequency.jl new file mode 100644 index 0000000..c4c135e --- /dev/null +++ b/src/vertex3/frequency.jl @@ -0,0 +1,340 @@ +include("QR.jl") +# using QR +using Lehmann +using StaticArrays, Printf + +const Float = FQR.Float +const Double = FQR.Double +const DotF = FQR.Float +const Tiny = DotF(1e-5) + +struct FreqGrid{D} <: FQR.Grid + sector::Int # sector + omega::SVector{D,Float} # actual location of the grid point + coord::SVector{D,Int} # integer coordinate of the grid point on the fine meshes +end + +Base.show(io::IO, grid::FreqGrid{2}) = print(io, "ω$(grid.sector) = ($(@sprintf("%12.4f", grid.omega[1])), $(@sprintf("%12.4f", grid.omega[2])))") + +struct FreqFineMesh{D} <: FQR.FineMesh + color::Int # D+1 sectors + symmetry::Int # symmetrize colors and (omega1, omega2) <-> (omega2, omega1) + candidates::Vector{FreqGrid{D}} # vector of grid points + selected::Vector{Bool} + residual::Vector{Double} + + ## for frequency mesh only ### + fineGrid::Vector{Float} # fine grid for each dimension + cache1::Vector{DotF} # cache for exp(-x) + cache2::Matrix{DotF} # cache for exp(-x-y) + + + function FreqFineMesh{D}(Λ, rtol; sym = 0) where {D} + # initialize the residual on fineGrid with + _finegrid = Float.(fineGrid(Λ, rtol)) + separationTest(D, _finegrid) + Nfine = length(_finegrid) + + _cache1 = zeros(DotF, Nfine) + _cache2 = zeros(DotF, (Nfine, Nfine)) + for (xi, x) in enumerate(_finegrid) + _cache1[xi] = exp(-DotF(x)) + for (yi, y) in enumerate(_finegrid) + _cache2[xi, yi] = exp(-DotF(x) - DotF(y)) + end + end + + color = D + 1 + # color = 1 + mesh = new{D}(color, sym, [], [], [], _finegrid, _cache1, _cache2) + + if D == 2 + for (xi, x) in enumerate(_finegrid) + for (yi, y) in enumerate(_finegrid) + coord = (xi, yi) + for sector in 1:color + if irreducible(D, sector, coord, sym) # if grid point is in the reducible zone, then skip residual initalization + g = FreqGrid{D}(sector, (x, y), coord) + push!(mesh.candidates, g) + push!(mesh.residual, FQR.dot(mesh, g, g)) + push!(mesh.selected, false) + end + end + end + end + # elseif D == 3 + else + error("not implemented!") + end + println("fine mesh initialized.") + return mesh + end +end + +""" +composite expoential grid +""" +function fineGrid(Λ, rtol) + ############## use composite grid ############################################# + # degree = 8 + # ratio = Float(1.4) + # N = Int(floor(log(Λ) / log(ratio) + 1)) + # panel = [Λ / ratio^(N - i) for i in 1:N] + # grid = Vector{Float}(undef, 0) + # for i in 1:length(panel)-1 + # uniform = [panel[i] + (panel[i+1] - panel[i]) / degree * j for j in 0:degree-1] + # append!(grid, uniform) + # end + # append!(grid, Λ) + # println(grid) + # println("Composite expoential grid size: $(length(grid))") + # return grid + + ############# DLR based fine grid ########################################## + dlr = DLRGrid(Euv = Float64(Λ), beta = 1.0, rtol = Float64(rtol) / 100, isFermi = true, symmetry = :ph, rebuild = true) + # println("fine basis number: $(dlr.size)\n", dlr.ω) + degree = 2 + grid = Vector{Double}(undef, 0) + panel = Double.(dlr.ω) + for i in 1:length(panel)-1 + uniform = [panel[i] + (panel[i+1] - panel[i]) / degree * j for j in 0:degree-1] + append!(grid, uniform) + end + + println("fine grid size: $(length(grid)) within [$(grid[1]), $(grid[2])]") + return grid +end + +""" +Test the finegrids do not overlap +""" +function separationTest(D, finegrid) + if D == 2 + epsilon = eps(DotF(1)) * 10 + for (i, f) in enumerate(finegrid) + # either zero, or sufficiently large + @assert abs(f) < epsilon || abs(f) > Tiny "$i: $f should either smaller than $epsilon or larger than $Tiny" + for (j, g) in enumerate(finegrid) + # two frequencies are either the same, or well separated + @assert abs(f - g) < epsilon || abs(f - g) > Tiny "$i: $f and $j: $g should either closer than $epsilon or further than $Tiny" + fg = f + g + for (k, l) in enumerate(finegrid) + @assert abs(l - fg) < epsilon || abs(l - fg) > Tiny "$i: $f + $j: $g = $fg and $k: $l should either closer than $epsilon or further than $Tiny" + end + end + end + else + error("not implemented!") + end +end + +function coord2omega(mesh::FreqFineMesh{dim}, coord) where {dim} + fineGrid = mesh.fineGrid + if dim == 2 + return (fineGrid[coord[1]], fineGrid[coord[2]]) + elseif dim == 3 + return (fineGrid[coord[1]], fineGrid[coord[2]], fineGrid[coord[3]]) + else + error("not implemented!") + end +end + +function irreducible(D, sector, coord, symmetry) + if symmetry == 0 + return true + else + if D == 2 + # return (coord[1] <= coord[2]) && (sector == 1) + return (coord[1] <= coord[2]) + elseif D == 3 + # return (coord[1] <= coord[2] <= coord[3]) && (sector == 1) + return (coord[1] <= coord[2] <= coord[3]) + else + error("not implemented!") + end + end +end + +function FQR.irreducible(grid::FreqGrid{D}) where {D} + return irreducible(D, grid.sector, grid.coord, mesh.symmetry) +end + +function FQR.mirror(mesh::FreqFineMesh{D}, idx) where {D} + grid = mesh.candidates[idx] + coord, s = grid.coord, grid.sector + if mesh.symmetry == 0 + return [] + end + if D == 2 + x, y = coord + coords = unique([(x, y), (y, x),]) + # println(coords) + elseif D == 3 + x, y, z = coord + coords = unique([(x, y, z), (x, z, y), (y, x, z), (y, z, x), (z, x, y), (z, y, x)]) + else + error("not implemented!") + end + newgrids = FreqGrid{D}[] + # for s in 1:mesh.color + for c in coords + if s != grid.sector || c != Tuple(grid.coord) + push!(newgrids, FreqGrid{D}(s, coord2omega(mesh, c), c)) + end + end + # end + return newgrids +end + +""" +G(x, y) = (exp(-x)-exp(-y))/(x-y) +G(x, x) = -exp(-x) +""" +@inline function G2d(a::T, b::T, expa::T, expb::T) where {T} + if abs(a - b) > Tiny + return (expa - expb) / (a - b) + else + return -(expa + expb) / 2 + end +end + +""" +F(a, b, c) = (G(a, c)-G(a, c))/(a-b) where a != b, but a or b could be equal to c +""" +@inline function F2d(a::T, b::T, c::T, expa::T, expb::T, expc::T) where {T} + @assert abs(a - b) > Tiny "$a - $c > $Tiny" + return (G2d(a, c, expa, expc) - G2d(b, c, expb, expc)) / (a - b) +end + +""" +F(any, any, 0) +""" +@inline function Fii2d(ω1::T, ω2::T, expω1::T, expω2::T) where {T} + if ω1 < Tiny && ω2 < Tiny + return T(1) / 2 + elseif ω1 < Tiny && ω2 > Tiny + return (1 - ω2 - expω2) / ω2 / (ω1 - ω2) + elseif ω1 > Tiny && ω2 < Tiny + return (1 - ω1 - expω1) / ω1 / (ω2 - ω1) + elseif abs(ω1 - ω2) < Tiny + # @assert abs(ω1 - ω2) < eps(Float(1)) * 1000 "$ω1 - $ω2 = $(ω1-ω2)" + ω = (ω1 + ω2) / 2 + expω = (expω1 + expω2) / 2 + return T((1 - expω * (1 + ω)) / ω / ω) + else + return T((ω1 - ω2 + expω1 * ω2 - expω2 * ω1) / (ω1 * ω2 * (ω1 - ω2))) + end +end + +""" +F(a,b,c) +""" +@inline function Fij2d(a::T, b::T, c::T, expa::T, expb::T, expc::T) where {T} + if abs(a - b) > Tiny #a!=b + return F2d(a, b, c, expa, expb, expc) + else # a=b + if abs(a - c) > Tiny # a=b != c + return F2d(a, c, b, expa, expc, expb) + else # a==b==c: exp(-a)/2 + return (expa + expb + expc) / 6 + end + end +end + +""" +basis dot for 2D +""" +function FQR.dot(mesh::FreqFineMesh{2}, g1::FreqGrid{2}, g2::FreqGrid{2}) + # println("dot: ", g1, ", ", g2) + cache1 = mesh.cache1 + cache2 = mesh.cache2 + s1, s2 = g1.sector, g2.sector + c1, c2 = g1.coord, g2.coord + if s1 == s2 # F11, F22, F33 + ω1, ω2 = g1.omega[1] + g2.omega[1], g1.omega[2] + g2.omega[2] + expω1 = cache2[c1[1], c2[1]] + expω2 = cache2[c1[2], c2[2]] + return Fii2d(ω1, ω2, expω1, expω2) + elseif (s1 == 1 && s2 == 2) || (s1 == 2 && s2 == 3) || (s1 == 3 && s2 == 1) #F12, F23, F31 + a, b, c = g2.omega[2], g1.omega[1], g1.omega[2] + g2.omega[1] + ea, eb, ec = cache1[c2[2]], cache1[c1[1]], cache2[c1[2], c2[1]] + return Fij2d(a, b, c, ea, eb, ec) + else #F21, F32, F13 + a, b, c = g1.omega[2], g2.omega[1], g2.omega[2] + g1.omega[1] + ea, eb, ec = cache1[c1[2]], cache1[c2[1]], cache2[c2[2], c1[1]] + return Fij2d(a, b, c, ea, eb, ec) + end +end + +if abspath(PROGRAM_FILE) == @__FILE__ + + D = 2 + + lambda, rtol = 10, 1e-4 + mesh = FreqFineMesh{D}(lambda, rtol, sym = 0) + + # KK = zeros(3, 3) + # n = (2, 2) + # o = (mesh.fineGrid[n[1]], mesh.fineGrid[n[2]]) + # for i in 1:3 + # g1 = FreqGrid{2}(i, o, n) + # for j in 1:3 + # g2 = FreqGrid{2}(j, o, n) + # println(g1, ", ", g2) + # KK[i, j] = FQR.dot(mesh, g1, g2) + # end + # end + # display(KK) + # println() + + basis = FQR.Basis{D,FreqGrid{D}}(lambda, rtol, mesh) + FQR.qr!(basis, verbose = 1) + + lambda, rtol = 1000, 1e-8 + mesh = FreqFineMesh{D}(lambda, rtol, sym = 0) + basis = FQR.Basis{D,FreqGrid{D}}(lambda, rtol, mesh) + @time FQR.qr!(basis, verbose = 1) + + FQR.test(basis) + + mesh = basis.mesh + grids = basis.grid + open("basis.dat", "w") do io + for (i, grid) in enumerate(grids) + if grid.sector == 1 + println(io, grid.omega[1], " ", grid.omega[2]) + end + end + end + Nfine = length(mesh.fineGrid) + open("finegrid.dat", "w") do io + for i in 1:Nfine + println(io, basis.mesh.fineGrid[i]) + end + end + open("residual.dat", "w") do io + # println(mesh.symmetry) + residual = zeros(Double, Nfine, Nfine) + for i in 1:length(mesh.candidates) + if mesh.candidates[i].sector == 1 + x, y = mesh.candidates[i].coord + residual[x, y] = mesh.residual[i] + # println(x, ", ", y, " -> ", length(mirror(mesh, i))) + + for grid in FQR.mirror(mesh, i) + if grid.sector == 1 + xp, yp = grid.coord + residual[xp, yp] = residual[x, y] + # println(xp, ", ", yp) + end + end + end + end + + for i in 1:Nfine + for j in 1:Nfine + println(io, residual[i, j]) + end + end + end +end \ No newline at end of file diff --git a/src/vertex3/plot.py b/src/vertex3/plot.py new file mode 100644 index 0000000..d8cf127 --- /dev/null +++ b/src/vertex3/plot.py @@ -0,0 +1,40 @@ +import matplotlib.pyplot as plt +# from scipy import interpolate +import numpy as np +from scipy.integrate import simps + +plt.gca().set_aspect('equal') +plt.style.use('science') + + +grid = np.loadtxt("basis.dat") +finegrid = np.loadtxt("finegrid.dat") +Nfine = len(finegrid) +residual = np.loadtxt("residual.dat") +residual = np.reshape(residual, (Nfine, Nfine)) + +# f = interpolate.interp2d(finegrid, finegrid, residual, kind='linear') +total = simps(simps(residual, finegrid), finegrid) +print("total residual: ", np.sqrt(total)) + +error = np.sqrt(residual) + +dg = grid[1, :]-grid[0, :] +shift = np.sqrt(sum(dg*dg)) + +xv, yv = np.meshgrid(finegrid+shift, finegrid+shift) +# plt.imshow(xv, yv, residual) +plt.contourf(xv, yv, error.transpose(), 16) +plt.colorbar() + +plt.scatter(grid[:, 0]+shift, grid[:, 1]+shift, c="yellow", alpha=0.5, s=6) +plt.xlim([shift, finegrid[-1]+shift]) +plt.ylim([shift, finegrid[-1]+shift]) +# plt.xlim([1.0, finegrid[-1]+shift]) +# plt.ylim([1.0, finegrid[-1]+shift]) +plt.xscale("log") +plt.yscale("log") +plt.xlabel("$\\omega_1$") +plt.ylabel("$\\omega_2$") +# plt.savefig("residual.pdf") +plt.show() diff --git a/test/Manifest.toml b/test/Manifest.toml new file mode 100644 index 0000000..9533dab --- /dev/null +++ b/test/Manifest.toml @@ -0,0 +1,290 @@ +# This file is machine-generated - editing it directly is not advised + +[[ArgTools]] +uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" + +[[Artifacts]] +uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33" + +[[Base64]] +uuid = 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+[[TranscodingStreams]] +deps = ["Random", "Test"] +git-tree-sha1 = "216b95ea110b5972db65aa90f88d8d89dcb8851c" +uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa" +version = "0.9.6" + +[[UUIDs]] +deps = ["Random", "SHA"] +uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4" + +[[Unicode]] +uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5" + +[[Zlib_jll]] +deps = ["Libdl"] +uuid = "83775a58-1f1d-513f-b197-d71354ab007a" + +[[nghttp2_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" + +[[p7zip_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" diff --git a/test/dlr.jl b/test/dlr.jl index 4435cf8..db1d360 100644 --- a/test/dlr.jl +++ b/test/dlr.jl @@ -111,7 +111,7 @@ end # the accuracy greatly drops beyond Λ >= 1e8 and rtol<=1e-6 cases = [SemiCircle, MultiPole] Λ = [1e3, 1e5, 1e7] - rtol = [1e-8, 1e-10, 1e-12] + rtol = [1e-8, 1e-10] for case in cases for l in Λ for r in rtol diff --git a/test/spectral.jl b/test/spectral.jl index 22a207d..5a23e58 100644 --- a/test/spectral.jl +++ b/test/spectral.jl @@ -27,6 +27,7 @@ testKernelT(false, :none, 1.0, 10.0, 1.0, 1.0e-6) # small ϵ for bosonic case is particularly dangerous because the kernal diverges ~1/ϵ function testAccuracy(isFermi, symmetry, τGrid, ωGrid, β) + setprecision(128) maxErr = BigFloat(0.0) τ0, ω0, macheps = 0.0, 0.0, 0.0 for (τi, τ) in enumerate(τGrid) @@ -43,6 +44,7 @@ return maxErr, τ0, ω0, macheps end + println("testing kernel accuracy") β, Euv = 10000.0, 100.0 τGrid = [t for t in LinRange(-β + 1e-10, β, 100)] ωGrid = [w for w in LinRange(-Euv, Euv, 100)] @@ -63,4 +65,5 @@ ωGrid = [w for w in LinRange(-Euv, Euv, 100)] maxErr, τ0, ω0, macheps = testAccuracy(true, :none, τGrid, ωGrid, β) maxErr, τ0, ω0, macheps = testAccuracy(false, :none, τGrid, ωGrid, β) + println("kernel accuracy tested") end \ No newline at end of file