Testing script

.gitignore

@@ -1,9 +1,3 @@
-# VSCode
-.vscode/
-
-# HPC scripts and logs
-hpc/
-
 # Calculation outputs
 *.dat
 *.csv

Hamiltonian.jl

@@ -6,27 +6,23 @@ using TensorOperations, KrylovKit, LinearAlgebra, CUDA, cuTENSOR, NVTX
 "A Hamiltonian that can be applied to a vector"
 struct Hamiltonian{T}
     s::system{T}
-    K_partial::Matrix{Complex{T}}
+    K::Union{CuTensor{Complex{T}}, Matrix{Complex{T}}}
-    K_diag::Union{CuTensor{Complex{T}},Nothing}
-    K_mixed::Union{CuTensor{Complex{T}},Nothing}
     Vs::Union{Array{Complex{T}}, CuArray{Complex{T}}}
     hermitian::Bool
     mode::Hamiltonian_backend
 
     function Hamiltonian{T}(s::system{T}, V_twobody::Function, ϕ::Real, n_image::Int, mode::Hamiltonian_backend) where {T<:Float}
         @assert mode != gpu_cutensor || CUDA.functional() && CUDA.has_cuda() && CUDA.has_cuda_gpu() "CUDA not available"
-        k = -s.N÷2:s.N÷2-1
-        Vs = calculate_Vs(s, V_twobody, convert(T, ϕ), n_image)
         hermitian = ϕ == 0.0
-        K_partial = (exp(-im * convert(T, ϕ)) * im / sqrt(2 * s.μ)) .* ∂_1DOF.(Ref(s), k, k')
+        Vs = calculate_Vs(s, V_twobody, convert(T, ϕ), n_image)
-        K_diag = nothing
+        k = -s.N÷2:s.N÷2-1
-        K_mixed = nothing
+        ∂ = ∂_1DOF.(Ref(s), k, k')
+        K = exp(-2im * convert(T, ϕ)) .* (∂ * ∂)    # TODO: Calculate K matrix elements directly
         if mode == gpu_cutensor
-            K_diag = CuTensor(CuArray(K_partial * K_partial), ['a', 'A'])
+            K = CuTensor(CuArray(K), ['a', 'A'])
-            K_mixed = CuTensor(CuArray(K_partial), ['a', 'A']) * CuTensor(CuArray(K_partial), ['b', 'B'])
             Vs = CuArray(Vs)
         end
-        return new{T}(s, K_partial, K_diag, K_mixed, Vs, hermitian, mode)
+        return new{T}(s, K, Vs, hermitian, mode)
     end
 end
 
@@ -45,30 +41,22 @@ function LinearAlgebra.mul!(out::Array{Complex{T}}, H::Hamiltonian{T}, v::Array{
     coords = H.s.n - 1
     nconList_v_template = -collect(1:H.s.d*(coords))
     for dim = 1:H.s.d
-        for coord1 = 1:coords
+        for coord = 1:coords
-            for coord2 = 1:coord1
+            i = which_index(H.s, dim, coord)
-                i1 = which_index(H.s, dim, coord1)
+            nconList_K = [-i, 1]
-                i2 = which_index(H.s, dim, coord2)
-                nconList_1 = [-i1, 1]
-                nconList_2 = [-i2, 2]
             nconList_v = copy(nconList_v_template)
-                if i1 == i2
+            nconList_v[i] = 1
-                    nconList_2[1] = 1
+            v_new = @ncon((H.K, v), (nconList_K, nconList_v))
-                else
+            coeff = -1 / (2 * H.s.μs[coord])
-                    nconList_v[i1] = 1
+            out = axpy!(coeff, v_new, out)
-                end
-                nconList_v[i2] = 2
-                v_new = @ncon((H.K_partial, H.K_partial, v), (nconList_1, nconList_2, nconList_v))
-                out = axpy!(1, v_new, out)
-            end
         end
     end
     return out
 end
 
 "cuTENSOR contraction and accumulation (C = A * B + C)"
-function contract_accumulate!(C::CuTensor, A::CuTensor, B::CuTensor)::CuTensor
+function contract_accumulate!(alpha::Number, C::CuTensor, A::CuTensor, B::CuTensor)::CuTensor
-    cuTENSOR.contraction!(one(eltype(C)), A.data, A.inds, cuTENSOR.CUTENSOR_OP_IDENTITY, B.data, B.inds, cuTENSOR.CUTENSOR_OP_IDENTITY,
+    cuTENSOR.contraction!(alpha, A.data, A.inds, cuTENSOR.CUTENSOR_OP_IDENTITY, B.data, B.inds, cuTENSOR.CUTENSOR_OP_IDENTITY,
                           one(eltype(C)), C.data, C.inds, cuTENSOR.CUTENSOR_OP_IDENTITY, cuTENSOR.CUTENSOR_OP_IDENTITY)
     return C
 end
@@ -86,31 +74,16 @@ function LinearAlgebra.mul!(out::CuArray{Complex{T}}, H::Hamiltonian{T}, v::CuAr
     v_t = CuTensor(v, copy(inds_template))
     out_t = CuTensor(out, copy(inds_template))
     for dim = 1:H.s.d
-        for coord1 = 1:coords
+        for coord = 1:coords
-            for coord2 = 1:coord1
+            i = which_index(H.s, dim, coord)
-                i1 = which_index(H.s, dim, coord1)
-                i2 = which_index(H.s, dim, coord2)
             @assert v_t.inds == inds_template "v indices permuted"
-                if i1 == i2
+            @assert H.K.inds[2] == 'A' "K_diag indices permuted"
-                    @assert H.K_diag.inds[2] == 'A' "K_diag indices permuted"
+            H.K.inds[1] = 'a' - 1 + i
-                    H.K_diag.inds[1] = 'a' - 1 + i1
+            v_t.inds[i] = 'A'
-                    v_t.inds[i1] = 'A'
             #synchronize(ctx)
-                    NVTX.@range "K-diag" out_t = contract_accumulate!(out_t, H.K_diag, v_t)
+            coeff = -1 / (2 * H.s.μs[coord])
-                    v_t.inds[i1] = 'a' - 1 + i1
+            NVTX.@range "K" out_t = contract_accumulate!(coeff, out_t, H.K, v_t)
-                else
+            v_t.inds[i] = 'a' - 1 + i
-                    @assert H.K_mixed.inds[2] == 'A' && H.K_mixed.inds[4] == 'B' "K_mixed indices permuted"
-                    H.K_mixed.inds[1] = 'a' - 1 + i1
-                    H.K_mixed.inds[3] = 'a' - 1 + i2
-                    # OPTIMIZE: A and B can be swapped
-                    v_t.inds[i1] = 'A'
-                    v_t.inds[i2] = 'B'
-                    #synchronize(ctx)
-                    NVTX.@range "K-mixed" out_t = contract_accumulate!(out_t, H.K_mixed, v_t)
-                    v_t.inds[i1] = 'a' - 1 + i1
-                    v_t.inds[i2] = 'a' - 1 + i2
-                end
-            end
         end
     end
     @assert out_t.inds == inds_template "out indices permuted"

LocalPreferences.toml

@@ -1,2 +0,0 @@
-[TensorOperations]
-precompile_workload = true

Project.toml

@@ -1,7 +0,0 @@
-[deps]
-CUDA = "052768ef-5323-5732-b1bb-66c8b64840ba"
-KrylovKit = "0b1a1467-8014-51b9-945f-bf0ae24f4b77"
-NVTX = "5da4648a-3479-48b8-97b9-01cb529c0a1f"
-Preferences = "21216c6a-2e73-6563-6e65-726566657250"
-TensorOperations = "6aa20fa7-93e2-5fca-9bc0-fbd0db3c71a2"
-cuTENSOR = "011b41b2-24ef-40a8-b3eb-fa098493e9e1"

README.md

@@ -1,27 +0,0 @@
-# DVR-jl
-
-Solves the quantum $n$-body problem in finite volume (lattice) with periodic boundary conditions. Uses discrete variable representation (DVR) with optional support for complex scaling to study resonances. All details can be found in [H. Yu, N. Yapa, and S. König, Complex scaling in finite volume, Phys. Rev. C 109, 014316 (2024)](https://doi.org/10.1103/PhysRevC.109.014316).
-
-Written in Julia with optional CUDA GPU acceleration (experimental).
-
-## Installation
-
-Make sure you have Julia installed. Required packages can be installed with a single command:
-```bash
-julia --project=. -e 'import Pkg; Pkg.instantiate()'
-```
-
-## Usage
-
-See `calculations/3b_bound.jl` for an example on a 3-body bound state.
-See `calculations/3b_res_from_paper.jl` for an example of a 3-body resonance via complex scaling.
-
-## Planned features
-
-- [ ] Spin and isospin degrees of freedom for nuclear calculations
-- [ ] Multi-node HPC support
-- [ ] Parity and cubic symmetries ($S_4$)
-
-## Acknowledgments
-
-The author gratefully acknowledges the guidance from Sebastian König.

calculations/3b_bound.jl

@@ -1,19 +0,0 @@
-include("../Hamiltonian.jl")
-mode = cpu_tensor
-T = Float32
-
-V_gauss(r2) = -2 * exp(-r2 / 4)
-
-d = 3
-n = 3
-N = 20
-L = 15
-n_imag = 1
-ϕ = 0
-
-s = system{T}(d, n, N, L)
-H = Hamiltonian{T}(s, V_gauss, ϕ, n_imag, mode)
-@time evals, _, info = eig(H, 5)
-
-print(info.numops, " operations")
-display(evals)

calculations/3b_res_from_paper.jl

@@ -1,36 +0,0 @@
-# 10.1007/s00601-020-01550-8
-# Fig. 7
-# E_R = 4.18(8)
-
-#./En.run -d 3 -n 3 -N 16 -c pot=v_shifted_gauss,v0=2.0,r=1.5,a=3.0 -c n_eig=20 -c which=li -c tol=1e-6 -L 16 -c phi=0.3 -v
-
-include("../Hamiltonian.jl")
-mode = cpu_tensor
-T = Float32 # single-precision mode
-
-using Plots
-
-V_gauss(r2) =
-    2 * exp(-((sqrt(r2) - 3) / 1.5) ^ 2)
-
-d = 3
-n = 3
-N = 16
-L = 16
-n_imag = 0
-
-for ϕ::T in 0.2:0.05:0.4
-    s = system{T}(d, n, N, L)
-    H = Hamiltonian{T}(s, V_gauss, ϕ, n_imag, mode)
-    @time evals, _, info = eig(H, 20)
-    
-    print(info.numops, " operations")
-    display(evals)
-
-    scatter(real.(evals), imag.(evals); legend=false)
-    xlabel!("Re E")
-    ylabel!("Im E")
-    xlims!(0, 6)
-    ylims!(-0.6, 0)
-    savefig("temp/phi$(Int(round(ϕ * 100))).png")
-end

calculations/ComplexScaling-FV-P-res.jl

@@ -1,24 +0,0 @@
-include("../Hamiltonian.jl")
-mode = cpu_tensor
-T = Float32 # single-precision mode
-
-V_gauss(r2) =
-    -10 * exp(-(sqrt(r2)) ^ 2)
-
-d = 3
-n = 2
-N = 96
-ϕ = pi/6
-n_imag = 1
-
-open("ComplexScaling-FV-P-res.dat", "w") do f
-    for L = range(20, 35, length=16)
-        println("Calculating L=", L)
-        s = system{T}(d, n, N, L)
-        H = Hamiltonian{T}(s, V_gauss, ϕ, n_imag, mode)
-        @time evals, _, info = eig(H, 40)
-
-        dataline = vcat([L], hcat(real.(evals), imag.(evals))'[:])
-        println(f, join(dataline, '\t'))
-    end
-end

calculations/ComplexScaling-FV-S-bound-phi.jl

@@ -1,24 +0,0 @@
-include("../Hamiltonian.jl")
-mode = cpu_tensor
-T = Float32 # single-precision mode
-
-V_gauss(r2) =
-    -10 * exp(-(sqrt(r2)) ^ 2)
-
-d = 3
-n = 2
-N = 30
-L = 6
-n_imag = 1
-
-open("ComplexScaling-FV-S-bound-phi.dat", "w") do f
-    for ϕ = range(0.0, 0.5, length=11)
-        println("Calculating ϕ=", ϕ)
-        s = system{T}(d, n, N, L)
-        H = Hamiltonian{T}(s, V_gauss, ϕ, n_imag, mode)
-        @time evals, _, info = eig(H, 10, resonances = false)
-
-        dataline = vcat([ϕ], hcat(real.(evals), imag.(evals))'[:])
-        println(f, join(dataline, '\t'))
-    end
-end

calculations/ComplexScaling-FV-S-res-phi.jl

@@ -1,24 +0,0 @@
-include("../Hamiltonian.jl")
-mode = cpu_tensor
-T = Float32 # single-precision mode
-
-V_gauss(r2) =
-    2 * exp(- ((sqrt(r2)-3)/1.5) ^ 2)
-
-d = 3
-n = 2
-N = 96
-L = 30
-n_imag = 1
-
-open("ComplexScaling-FV-S-res-phi.dat", "w") do f
-    for ϕ = range(0.1, 0.6, length=26)
-        println("Calculating ϕ=", ϕ)
-        s = system{T}(d, n, N, L)
-        H = Hamiltonian{T}(s, V_gauss, ϕ, n_imag, mode)
-        @time evals, _, info = eig(H, 40, resonances = true)
-
-        dataline = vcat([ϕ], hcat(real.(evals), imag.(evals))'[:])
-        println(f, join(dataline, '\t'))
-    end
-end

calculations/EC-R2R-S.jl

@@ -1,67 +0,0 @@
-using Plots, Arpack
-
-include("../helper.jl")
-include("../Hamiltonian.jl")
-
-mode = cpu_tensor
-T = Float32 # single-precision mode
-
-V_r2(c) = r2 -> c * (-5 * exp(-r2/3) + 2 * exp(-r2/10))
-
-d = 3
-n = 2
-N = 48
-L = 30
-ϕ = pi/6
-n_imag = 1
-s = system{T}(d, n, N, L)
-
-train_cs = range(0.78, 0.45, length=5)
-train_ref = reverse([0.05387926313545913-0.008900278182520881im,
-    0.11254295298924327-0.020515067379548786im,
-    0.16060154707503538-0.03716539208626717im,
-    0.19741353362674618-0.05994519982799412im,
-    0.2219100763497223-0.08959449893439568im])
-
-extrapolate_cs = range(0.38, 0.22, length=5)
-extrapolate_ref = reverse([0.23165109150003316-0.12052751440975719im,
-    0.23190549514995962-0.1406687118589838im,
-    0.22763660218046278-0.1626190970863793im,
-    0.21807104244164865-0.18635600686249373im,
-    0.2020979906072586-0.21180157628258728im])
-
-training_E = ComplexF64[]
-training_vec = Array[]
-exact_E = ComplexF64[]
-extrapolated_E = ComplexF64[]
-
-for c in train_cs
-    println("Training c=", c)
-    H = Hamiltonian{T}(s, V_r2(c), ϕ, n_imag, mode)
-    @time evals, evecs, info = eig(H, 20, resonances = true)
-    i = nearestIndex(evals, pop!(train_ref))
-    push!(training_E, evals[i])
-    push!(training_vec, evecs[i])
-end
-
-N_EC = [sum(x .* y) for (x, y) in Iterators.product(training_vec, training_vec)]
-
-for c in extrapolate_cs
-    println("Extrapolating c=", c)
-    H = Hamiltonian{T}(s, V_r2(c), ϕ, n_imag, mode)
-    @time evals, _, info = eig(H, 40, resonances = true)
-    nearestE = nearest(evals, pop!(extrapolate_ref))
-    push!(exact_E, nearestE)
-    
-    # EC extrapolation
-    H_training_vec = H.(training_vec)
-    H_EC = [sum(x .* y) for (x, y) in Iterators.product(training_vec, H_training_vec)]
-
-    evals = eigvals(H_EC, N_EC)
-    push!(extrapolated_E, nearestE)
-end
-
-scatter(real.(training_E), imag.(training_E), label="training")
-scatter!(real.(exact_E), imag.(exact_E), label="exact")
-scatter!(real.(extrapolated_E), imag.(extrapolated_E), label="extrapolated")
-savefig("temp/EC-R2R-S.pdf")

common.jl

@@ -1,17 +1,37 @@
 Float = Union{Float32,Float64}
 
+norm_square(x) = sum(x .* x)
+reducedMass(m1, m2) = 1 / (1/m1 + 1/m2)
+
 "A few-body system defined by its physical parameters"
 struct system{T}
     d::Int
     n::Int
     N::Int
     L::T
-    μ::T
+    μs::Vector{T}
+    invU::Matrix{T}
 
-    system{T}(d::Int, n::Int, N::Int, L::Real, μ::Real=0.5) where {T<:Float} = new{T}(d, n, N, convert(T, L), convert(T, μ))
+    function system{T}(d::Int, n::Int, N::Int, L::Real) where {T<:Float}
+        μs = [1/((coord + 1)^2 * reducedMass(coord, 1)) for coord in 1:(n - 1)]
+
+        # TODO: Optimize
+        U = Matrix{T}(undef, n, n)
+        for i in CartesianIndices(U)
+            if i[1] + 1 == i[2]
+                U[i] = -i[1]
+            elseif i[1] >= i[2]
+                U[i] = 1
+            else
+                U[i] = 0
             end
+        end
+        U[n, :] .= 1/n
+        invU = inv(U)[:, 1:(n - 1)]
         
-norm_square(x::Array{Int})::Int = sum(x .* x)
+        return new{T}(d, n, N, convert(T, L), μs, invU)
+    end
+end
 
 "Eq (46): Partial derivative matrix element for 1 degree of freedom"
 function ∂_1DOF(s::system{T}, k::Int, l::Int)::Complex{T} where {T<:Float}
@@ -22,19 +42,20 @@ function ∂_1DOF(s::system{T}, k::Int, l::Int)::Complex{T} where {T<:Float}
     end
 end
 
-"Which index (dimension of the multidimensional array) corresponds to spatial dimension 'dim' and particle 'p'?"
+"Which index (dimension of the multidimensional array) corresponds to spatial dimension 'dim' of coordinate 'coord'?"
-which_index(s::system, dim::Int, p::Int)::Int = (dim - 1) * (s.n - 1) + p
+which_index(s::system, dim::Int, coord::Int)::Int = (dim - 1) * (s.n - 1) + coord
 
-"Δk (distance in terms of lattice paramter) between two particles along the given dimension"
+"Get the distance to the nearest image of the particle"
-function get_Δk(s::system, i::CartesianIndex, dim::Int, p1::Int, p2::Int)::Int
+function nearest(s::system, Δk)
-    if p1 == p2
+    # TODO: Optimize
-        return 0
+    while true
-    elseif p1 == s.n
+        if Δk >= s.N ÷ 2
-        return -(i[which_index(s, dim, p2)] - s.N ÷ 2 - 1)
+            Δk -= s.N
-    elseif p2 == s.n
+        elseif Δk < -s.N ÷ 2
-        return i[which_index(s, dim, p1)] - s.N ÷ 2 - 1
+            Δk += s.N
         else
-        return i[which_index(s, dim, p1)] - i[which_index(s, dim, p2)]
+            return Δk
+        end            
     end
 end
 
@@ -44,21 +65,17 @@ function calculate_Vs(s::system{T}, V_twobody::Function, ϕ::T, n_image::Int)::A
     images = collect.(Iterators.product(fill(-n_image:n_image, s.d)...)) # TODO: Learn how to use tuples instead of vectors
     Vs = zeros(Complex{T}, fill(s.N, s.d * (s.n - 1))...)
     Threads.@threads for i in CartesianIndices(Vs)
+        xs = reshape(collect(Tuple(i)), s.n - 1, s.d) .- (s.N ÷ 2 + 1)
+        rs = s.invU * xs
         for p1 in 1:s.n
-            for p2 in (p1 + 1):s.n
+            for p2 in 1:(p1 - 1)
-                min_Δk = Array{Int}(undef, s.d)
+                Δk = Array{T}(undef, s.d)
                 for dim in 1:s.d
-                    Δk = get_Δk(s, i, dim, p1, p2)
+                    Δk_temp = rs[p1, dim] - rs[p2, dim]
-                    if Δk > s.N ÷ 2
+                    Δk[dim] = nearest(s, Δk_temp)
-                        min_Δk[dim] = Δk - s.N
-                    elseif Δk < -s.N ÷ 2
-                        min_Δk[dim] = Δk + s.N
-                    else
-                        min_Δk[dim] = Δk
-                    end
                 end
                 for image in images
-                    Δk² = norm_square(min_Δk .- (s.N .* image))
+                    Δk² = norm_square(Δk .- (s.N .* image))
                     Vs[i] += V_twobody(Δk² * coeff²)
                 end
             end

helper.jl

@@ -1,5 +0,0 @@
-"Index of the nearest value in a list to a given reference point"
-nearestIndex(list::Array, ref) = argmin(norm.(list .- ref))
-
-"Nearest value in a list to a given reference point"
-nearest(list::Array, ref) = list[nearestIndex(list, ref)]

test_jacobi.jl

@@ -0,0 +1,21 @@
+include("Hamiltonian.jl")
+
+T=Float32
+
+function V_test(r2)
+    return -4*exp(-r2/4)
+end
+
+for (n,N) in [(2,16), (3,8)]
+    println("\n$n-body system with N=$N")
+    n_image=0
+    for L::T in 5.0:9.0
+        print("L=$L: ")
+        s=system{T}(3,n,N,L)
+        print("Constructing H...")
+        H=Hamiltonian{T}(s,V_test,0.0,n_image,cpu_tensor)
+        print("Diagonalizing...")
+        evals,_,_ = eig(H,5)
+        println(real.(evals))
+    end
+end