Update README.md

.gitignore

@@ -1,3 +1,15 @@
+# VSCode
+.vscode/
+
+# HPC scripts and logs
+hpc/
+
+# Calculation outputs
+*.dat
+*.csv
+*.hdf5
+*.out
+
 # Temporary and scratch files
 temp/
 scratch/

CPU.jl

@@ -1,73 +0,0 @@
-include("common.jl")
-using TensorOperations, KrylovKit, LinearAlgebra
-
-"A Hamiltonian that can be applied to a vector"
-struct HOperator{T}
-    d::Int
-    n::Int
-    N::Int
-    L::T
-    μ::T
-    ∂1::Matrix{Complex{T}}
-    Vs::Array{Complex{T}}
-    hermitian::Bool
-    function HOperator{T}(V_twobody::Function, d::Int, n::Int, N::Int, L::T, ϕ::T, μ::T, n_image::Int) where {T<:Float}
-        k = -N÷2:N÷2-1
-        ∂1 = exp(-im * ϕ) .* ∂_1DOF.(L, N, k, k')
-        Vs = calculate_Vs(V_twobody, d, n, N, L, ϕ, n_image)
-        return new{T}(d, n, N, L, μ, ∂1, Vs, ϕ == 0.0)
-    end
-end
-
-Base.size(H::HOperator, i::Int)::Int = (i == 1 || i == 2) ? H.N^(H.d * (H.n - 1)) : throw(ArgumentError("HOperator only has 2 dimesions"))
-Base.size(H::HOperator)::Dims{2} = (size(H, 1), size(H, 2))
-
-"Dimensions of a vector to which H can be applied"
-vectorDims(H::HOperator)::Dims = tuple(fill(H.N, H.d * (H.n - 1))...)
-
-"Apply H on v and store the result in out"
-function LinearAlgebra.mul!(out::Array{Complex{T}}, H::HOperator{T}, v::Array{Complex{T}})::Array{Complex{T}} where {T<:Float}
-    #LinearMaps.check_dim_mul(out,H,v) --- dimensions don't match
-    # apply V operator
-    @. out = H.Vs * v
-    # apply K opereator
-    coeff = -1 / (2 * H.μ)
-    coords = H.n - 1
-    nconList_v_template = -collect(1:H.d*(coords))
-    for dim = 1:H.d
-        for coord1 = 1:coords
-            for coord2 = 1:coord1
-                i1 = which_index(H.n, dim, coord1)
-                i2 = which_index(H.n, dim, coord2)
-                nconList_1 = [-i1, 1]
-                nconList_2 = [-i2, 2]
-                nconList_v = copy(nconList_v_template)
-                if i1 == i2
-                    nconList_2[1] = 1
-                else
-                    nconList_v[i1] = 1
-                end
-                nconList_v[i2] = 2
-                v_new = @ncon((H.∂1, H.∂1, v), (nconList_1, nconList_2, nconList_v))
-                out = axpy!(coeff, v_new, out)
-            end
-        end
-    end
-    return out
-end
-
-"Apply H on v and return the result"
-function (H::HOperator{T})(v::Array{Complex{T}})::Array{Complex{T}} where {T<:Float}
-    out = similar(v)
-    return mul!(out, H, v)
-end
-
-tolerance = 1e-6
-
-"Wrapper for KrylovKit.eigsolve"
-function eig(H::HOperator{T}, levels::Int; resonances = !H.hermitian)::Tuple{Vector{Complex{T}},Any,Any} where {T<:Float}
-    x₀ = rand(Complex{T}, vectorDims(H))
-    evals, evecs, info = eigsolve(H, x₀, levels, resonances ? :LI : :SR; ishermitian = H.hermitian, tol = tolerance)
-    resonances || info.converged < levels && throw(error("Not enough convergence")) # don't check convergence for resonances
-    return evals, evecs, info
-end

GPU.jl

@@ -1,98 +0,0 @@
-include("common.jl")
-using KrylovKit, LinearAlgebra, CUDA, CUDA.CUTENSOR
-
-@assert CUDA.functional() && CUDA.has_cuda() && CUDA.has_cuda_gpu() "CUDA not available"
-
-"A Hamiltonian that can be applied to a vector"
-struct HOperator{T}
-    d::Int
-    n::Int
-    N::Int
-    K_diag::CuTensor{Complex{T}}
-    K_mixed::CuTensor{Complex{T}}
-    Vs::CuArray{Complex{T}}
-    hermitian::Bool
-    function HOperator{T}(V_twobody::Function, d::Int, n::Int, N::Int, L::T, ϕ::T, μ::T, n_image::Int) where {T<:Float}
-        k = -N÷2:N÷2-1
-        K_partial = (exp(-im * ϕ) * im / sqrt(2 * μ)) .* ∂_1DOF.(L, N, k, k')
-        K_diag = CuTensor(CuArray(K_partial * K_partial), ['a', 'A'])
-        K_mixed = CuTensor(CuArray(K_partial), ['a', 'A']) * CuTensor(CuArray(K_partial), ['b', 'B'])
-        Vs = calculate_Vs(V_twobody, d, n, N, L, ϕ, n_image)
-        return new{T}(d, n, N, K_diag, K_mixed, Vs, ϕ == 0.0)
-    end
-end
-
-Base.size(H::HOperator, i::Int)::Int = (i == 1 || i == 2) ? H.N^(H.d * (H.n - 1)) : throw(ArgumentError("HOperator only has 2 dimesions"))
-Base.size(H::HOperator)::Dims{2} = (size(H, 1), size(H, 2))
-
-"Dimensions of a vector to which H can be applied"
-vectorDims(H::HOperator)::Dims = tuple(fill(H.N, H.d * (H.n - 1))...)
-
-"cuTENSOR contraction and accumulation (C = A * B + C)"
-function contract_accumulate!(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,
-                          one(eltype(C)), C.data, C.inds, CUTENSOR.CUTENSOR_OP_IDENTITY, CUTENSOR.CUTENSOR_OP_IDENTITY)
-    return C
-end
-
-"Apply H on v and store the result in out"
-function LinearAlgebra.mul!(out::CuArray{Complex{T}}, H::HOperator{T}, v::CuArray{Complex{T}})::CuArray{Complex{T}} where {T<:Float}
-    #LinearMaps.check_dim_mul(out,H,v) --- dimensions don't match
-    ctx = context()
-    # apply V operator
-    NVTX.@range "V" @. out = H.Vs * v
-    synchronize(ctx)
-    # apply K opereator
-    coords = H.n - 1
-    inds_template = ('a' - 1) .+ collect(1:H.d*(coords))
-    v_t = CuTensor(v, copy(inds_template))
-    out_t = CuTensor(out, copy(inds_template))
-    for dim = 1:H.d
-        for coord1 = 1:coords
-            for coord2 = 1:coord1
-                i1 = which_index(H.n, dim, coord1)
-                i2 = which_index(H.n, dim, coord2)
-                @assert v_t.inds == inds_template "v indices permuted"
-                if i1 == i2
-                    @assert H.K_diag.inds[2] == 'A' "K_diag indices permuted"
-                    H.K_diag.inds[1] = 'a' - 1 + i1
-                    v_t.inds[i1] = 'A'
-                    #synchronize(ctx)
-                    NVTX.@range "K-diag" out_t = contract_accumulate!(out_t, H.K_diag, v_t)
-                    v_t.inds[i1] = 'a' - 1 + i1
-                else
-                    @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"
-    synchronize(ctx)
-    return out_t.data
-end
-
-"Apply H on v and return the result"
-function (H::HOperator{T})(v::CuArray{Complex{T}})::CuArray{Complex{T}} where {T<:Float}
-    out = similar(v)
-    return mul!(out, H, v)
-end
-
-tolerance = 1e-6
-
-"Wrapper for KrylovKit.eigsolve"
-function eig(H::HOperator{T}, levels::Int; resonances = !H.hermitian)::Tuple{Vector{Complex{T}},Any,Any} where {T<:Float}
-    x₀ = CUDA.rand(Complex{T}, vectorDims(H)...) # ... added
-    synchronize()
-    evals, evecs, info = eigsolve(H, x₀, levels, resonances ? :LI : :SR; ishermitian = H.hermitian, tol = tolerance)
-    resonances || info.converged < levels && throw(error("Not enough convergence")) # don't check convergence for resonances
-    return evals, evecs, info
-end

Hamiltonian.jl

@@ -0,0 +1,145 @@
+include("common.jl")
+using TensorOperations, KrylovKit, LinearAlgebra, CUDA, cuTENSOR, NVTX
+
+@enum Hamiltonian_backend cpu_tensor gpu_cutensor
+
+"A Hamiltonian that can be applied to a vector"
+struct Hamiltonian{T}
+    s::system{T}
+    K_partial::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')
+        K_diag = nothing
+        K_mixed = nothing
+        if mode == gpu_cutensor
+            K_diag = CuTensor(CuArray(K_partial * K_partial), ['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)
+    end
+end
+
+Base.size(H::Hamiltonian, i::Int)::Int = (i == 1 || i == 2) ? H.s.N^(H.s.d * (H.s.n - 1)) : throw(ArgumentError("Hamiltonian only has 2 dimesions"))
+Base.size(H::Hamiltonian)::Dims{2} = (size(H, 1), size(H, 2))
+
+"Dimensions of a vector to which 'H' can be applied"
+vectorDims(H::Hamiltonian)::Dims = tuple(fill(H.s.N, H.s.d * (H.s.n - 1))...)
+
+"Apply 'H' on 'v' and store the result in 'out' using the 'cpu_tensor' backend"
+function LinearAlgebra.mul!(out::Array{Complex{T}}, H::Hamiltonian{T}, v::Array{Complex{T}})::Array{Complex{T}} where {T<:Float}
+    #LinearMaps.check_dim_mul(out,H,v) --- dimensions don't match
+    # apply V operator
+    @. out = H.Vs * v
+    # apply K opereator
+    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 coord2 = 1:coord1
+                i1 = which_index(H.s, dim, coord1)
+                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_2[1] = 1
+                else
+                    nconList_v[i1] = 1
+                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
+    cuTENSOR.contraction!(one(eltype(C)), 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
+
+"Apply 'H' on 'v' and store the result in 'out' using the 'gpu_cutensor' backend"
+function LinearAlgebra.mul!(out::CuArray{Complex{T}}, H::Hamiltonian{T}, v::CuArray{Complex{T}})::CuArray{Complex{T}} where {T<:Float}
+    #LinearMaps.check_dim_mul(out,H,v) --- dimensions don't match
+    ctx = context()
+    # apply V operator
+    NVTX.@range "V" @. out = H.Vs * v
+    synchronize(ctx)
+    # apply K opereator
+    coords = H.s.n - 1
+    inds_template = ('a' - 1) .+ collect(1:H.s.d*(coords))
+    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 coord2 = 1:coord1
+                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_diag.inds[2] == 'A' "K_diag indices permuted"
+                    H.K_diag.inds[1] = 'a' - 1 + i1
+                    v_t.inds[i1] = 'A'
+                    #synchronize(ctx)
+                    NVTX.@range "K-diag" out_t = contract_accumulate!(out_t, H.K_diag, v_t)
+                    v_t.inds[i1] = 'a' - 1 + i1
+                else
+                    @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"
+    synchronize(ctx)
+    return out_t.data
+end
+
+"Apply 'H' on 'v' and return the result"
+function (H::Hamiltonian)(v)
+    out = similar(v)
+    return mul!(out, H, v)
+end
+
+tolerance = 1e-6
+
+"Wrapper for KrylovKit.eigsolve"
+function eig(H::Hamiltonian{T}, levels::Int; resonances = !H.hermitian)::Tuple{Vector,Vector,KrylovKit.ConvergenceInfo} where {T<:Float}
+    if H.mode == cpu_tensor
+        x₀ = rand(Complex{T}, vectorDims(H)...)
+    elseif H.mode == gpu_cutensor
+        x₀ = CUDA.rand(Complex{T}, vectorDims(H)...)
+        synchronize()
+    end
+    evals, evecs, info = eigsolve(H, x₀, levels, resonances ? :LI : :SR; ishermitian = H.hermitian, tol = tolerance, krylovdim = levels * 8)
+    info.converged < levels && throw(error("Not enough convergence"))
+    if H.hermitian evals = real.(evals) end
+    if H.mode == gpu_cutensor # to avoid possible GPU memory leak
+        CUDA.reclaim()
+        GC.gc(true)
+    end
+    return evals, evecs, info
+end

LocalPreferences.toml

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

Project.toml

@@ -0,0 +1,7 @@
+[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

@@ -0,0 +1,27 @@
+# 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.

benchmark.jl

@@ -1,21 +1,21 @@
-using CUDA
+include("Hamiltonian.jl")
 
 GPU_mode = !("CPU" in ARGS) && CUDA.functional() && CUDA.has_cuda() && CUDA.has_cuda_gpu()
 
 println("Running with ",Threads.nthreads()," thread(s)")
 
 if GPU_mode
-    include("GPU.jl")
+    mode=gpu_cutensor
     println("Available GPUs:")
     print("    ")
     println.(name.(devices()))
 else
-    include("CPU.jl")
+    mode=cpu_tensor
 end
 
 T=Float32
 
-function V_test(r2::T)::T
+function V_test(r2)
     return -4*exp(-r2/4)
 end
 
@@ -27,11 +27,11 @@ end
 
 N=10
 n_image=1
-μ=0.5
 
-for L::T in 5.0:14.0
+for L in 5.0:14.0
     println("Constructing H operator...")
-    @time H=HOperator{T}(V_test,3,3,N,L,convert(T,μ),n_image)
+    s=system{T}(3,3,N,L)
+    @time H=Hamiltonian{T}(s,V_test,0,n_image,mode)
     println("Applying H 1000 times...")
     if GPU_mode
         v=CUDA.rand(Complex{T},vectorDims(H)...)

calculations/3b_bound.jl

@@ -0,0 +1,19 @@
+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

@@ -0,0 +1,36 @@
+# 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

@@ -0,0 +1,24 @@
+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

@@ -0,0 +1,24 @@
+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

@@ -0,0 +1,24 @@
+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

@@ -0,0 +1,67 @@
+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

@@ -1,53 +1,64 @@
 Float = Union{Float32,Float64}
 
+"A few-body system defined by its physical parameters"
+struct system{T}
+    d::Int
+    n::Int
+    N::Int
+    L::T
+    μ::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))
+end
+
 norm_square(x::Array{Int})::Int = sum(x .* x)
 
 "Eq (46): Partial derivative matrix element for 1 degree of freedom"
-function ∂_1DOF(L::T, N::Int, k::Int, l::Int)::Complex{T} where {T<:Float}
+function ∂_1DOF(s::system{T}, k::Int, l::Int)::Complex{T} where {T<:Float}
     if k == l
-        return -im * (π / L)
+        return -im * (π / s.L)
     else
-        return (π / L) * (-1)^(k - l) * exp(-im * π * (k - l) / N) / sin(π * (k - l) / N)
+        return (π / s.L) * (-1)^(k - l) * exp(-im * π * (k - l) / s.N) / sin(π * (k - l) / s.N)
     end
 end
 
 "Which index (dimension of the multidimensional array) corresponds to spatial dimension 'dim' and particle 'p'?"
-which_index(n::Int, dim::Int, p::Int)::Int = (dim - 1) * (n - 1) + p
+which_index(s::system, dim::Int, p::Int)::Int = (dim - 1) * (s.n - 1) + p
 
 "Δk (distance in terms of lattice paramter) between two particles along the given dimension"
-function get_Δk(n::Int, N::Int, i::CartesianIndex, dim::Int, p1::Int, p2::Int)::Int
+function get_Δk(s::system, i::CartesianIndex, dim::Int, p1::Int, p2::Int)::Int
     if p1 == p2
         return 0
-    elseif p1 == n
+    elseif p1 == s.n
-        return -(i[which_index(n, dim, p2)] - N ÷ 2 - 1)
+        return -(i[which_index(s, dim, p2)] - s.N ÷ 2 - 1)
-    elseif p2 == n
+    elseif p2 == s.n
-        return i[which_index(n, dim, p1)] - N ÷ 2 - 1
+        return i[which_index(s, dim, p1)] - s.N ÷ 2 - 1
     else
-        return i[which_index(n, dim, p1)] - i[which_index(n, dim, p2)]
+        return i[which_index(s, dim, p1)] - i[which_index(s, dim, p2)]
     end
 end
 
 "Calculate diagonal elements of the V matrix"
-function calculate_Vs(V_twobody::Function, d::Int, n::Int, N::Int, L::T, ϕ::T, n_image::Int)::Array{Complex{T}} where {T<:Float}
+function calculate_Vs(s::system{T}, V_twobody::Function, ϕ::T, n_image::Int)::Array{Complex{T}} where {T<:Float}
-    coeff² = (exp(im * ϕ) * L / N)^2
+    coeff² = (exp(im * ϕ) * s.L / s.N)^2
-    images = collect.(Iterators.product(fill(-n_image:n_image, d)...)) # TODO: Learn how to use tuples instead of vectors
+    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(N, d * (n - 1))...)
+    Vs = zeros(Complex{T}, fill(s.N, s.d * (s.n - 1))...)
     Threads.@threads for i in CartesianIndices(Vs)
-        for p1 in 1:n
+        for p1 in 1:s.n
-            for p2 in (p1 + 1):n
+            for p2 in (p1 + 1):s.n
-                min_Δk = Array{Int}(undef, d)
+                min_Δk = Array{Int}(undef, s.d)
-                for dim in 1:d
+                for dim in 1:s.d
-                    Δk = get_Δk(n, N, i, dim, p1, p2)
+                    Δk = get_Δk(s, i, dim, p1, p2)
-                    if Δk > N ÷ 2
+                    if Δk > s.N ÷ 2
-                        min_Δk[dim] = Δk - N
+                        min_Δk[dim] = Δk - s.N
-                    elseif Δk < -N ÷ 2
+                    elseif Δk < -s.N ÷ 2
-                        min_Δk[dim] = Δk + N
+                        min_Δk[dim] = Δk + s.N
                     else
                         min_Δk[dim] = Δk
                     end
                 end
                 for image in images
-                    Δk² = norm_square(min_Δk .- (N .* image))
+                    Δk² = norm_square(min_Δk .- (s.N .* image))
                     Vs[i] += V_twobody(Δk² * coeff²)
                 end
             end

example.ipynb

@@ -6,8 +6,9 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "# prerequisite packages: KrylovKit, TensorOperations, LinearAlgebra, CUDA#tb/cutensor\n",
+    "# prerequisite packages: KrylovKit, TensorOperations, LinearAlgebra, CUDA#tb/cutensor, Plots\n",
-    "include(\"CPU.jl\") # using CPU mode\n",
+    "include(\"Hamiltonian.jl\")\n",
+    "mode = cpu_tensor # using CPU mode\n",
     "T = Float32 # single-precision mode"
    ]
   },
@@ -23,12 +24,12 @@
     "d = 3\n",
     "n = 3\n",
     "N = 6\n",
-    "L::T = 12\n",
+    "L = 12\n",
-    "ϕ::T = 0.0\n",
+    "ϕ = 0.0\n",
-    "mu::T = 0.5\n",
     "n_imag = 1\n",
     "\n",
-    "H = HOperator{T}(V_gauss, d, n, N, L, ϕ, mu, n_imag)\n",
+    "s = system{T}(d, n, N, L)\n",
+    "H = Hamiltonian{T}(s, V_gauss, ϕ, n_imag, mode)\n",
     "@time evals, evecs, info = eig(H, 5)\n",
     "print(info.numops, \" operations : \")\n",
     "println(evals)"
@@ -48,12 +49,12 @@
     "d = 3\n",
     "n = 2\n",
     "N = 32\n",
-    "L::T = 16\n",
+    "L = 16\n",
-    "ϕ::T = 0.5\n",
+    "ϕ = 0.5\n",
-    "mu::T = 0.5\n",
     "n_imag = 0\n",
     "\n",
-    "H = HOperator{T}(V_gauss, d, n, N, L, ϕ, mu, n_imag)\n",
+    "s = system{T}(d, n, N, L)\n",
+    "H = Hamiltonian{T}(s, V_gauss, ϕ, n_imag, mode)\n",
     "@time evals, evecs, info = eig(H, 20)\n",
     "print(info.numops, \" operations : \")\n",
     "print(evals)\n",

helper.jl

@@ -0,0 +1,5 @@
+"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)]

testing.ipynb

@@ -0,0 +1,71 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "include(\"Hamiltonian.jl\")\n",
+    "\n",
+    "println(\"Running with \",Threads.nthreads(),\" thread(s)\")\n",
+    "println(\"Available GPUs:\")\n",
+    "println.(name.(devices()))\n",
+    "\n",
+    "T=Float32\n",
+    "\n",
+    "function V_test(r2)\n",
+    "    return -4*exp(-r2/4)\n",
+    "end\n",
+    "\n",
+    "function test(mode)\n",
+    "    for (n,N) in [(2,16),(3,8)]\n",
+    "        println(\"\\n$n-body system with N=$N\")\n",
+    "        n_image=0\n",
+    "        for L::T in 5.0:9.0\n",
+    "            print(\"L=$L\")\n",
+    "            s=system{T}(3,n,N,L)\n",
+    "            H=Hamiltonian{T}(s,V_test,0.0,n_image,mode)\n",
+    "            evals,_,_ = eig(H,5)\n",
+    "            println(real.(evals))\n",
+    "        end\n",
+    "    end\n",
+    "end"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "test(cpu_tensor)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "test(gpu_cutensor)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Julia 1.8.5",
+   "language": "julia",
+   "name": "julia-1.8"
+  },
+  "language_info": {
+   "file_extension": ".jl",
+   "mimetype": "application/julia",
+   "name": "julia",
+   "version": "1.8.5"
+  },
+  "orig_nbformat": 4
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}