// AddtoRhs adds the essential bcs / constraints terms to the augmented fb vector
func (o EssentialBcs) AddToRhs(fb []float64, sol *Solution) {
// skip if there are no constraints
if len(o.Bcs) == 0 {
return
}
// add -At*λ to fb
la.SpMatTrVecMulAdd(fb, -1, o.Am, sol.L) // fb += -1 * At * λ
// assemble -rc = c - A*y into fb
ny := len(sol.Y)
for i, c := range o.Bcs {
fb[ny+i] = c.Fcn.F(sol.T, nil)
}
la.SpMatVecMulAdd(fb[ny:], -1, o.Am, sol.Y) // fb += -1 * A * y
}
func TestJacobian03(tst *testing.T) {
//verbose()
chk.PrintTitle("TestJacobian 03")
// grid
var g fdm.Grid2D
//g.Init(1.0, 1.0, 4, 4)
g.Init(1.0, 1.0, 6, 6)
//g.Init(1.0, 1.0, 11, 11)
// equations numbering
var e fdm.Equations
peq := utl.IntUnique(g.L, g.R, g.B, g.T)
e.Init(g.N, peq)
// K11 and K12
var K11, K12 la.Triplet
fdm.InitK11andK12(&K11, &K12, &e)
// assembly
F1 := make([]float64, e.N1)
fdm.Assemble(&K11, &K12, F1, nil, &g, &e)
// prescribed values
U2 := make([]float64, e.N2)
for _, eq := range g.L {
U2[e.FR2[eq]] = 50.0
}
for _, eq := range g.R {
U2[e.FR2[eq]] = 0.0
}
for _, eq := range g.B {
U2[e.FR2[eq]] = 0.0
}
for _, eq := range g.T {
U2[e.FR2[eq]] = 50.0
}
// functions
k11 := K11.ToMatrix(nil)
k12 := K12.ToMatrix(nil)
ffcn := func(fU1, U1 []float64) error { // K11*U1 + K12*U2 - F1
la.VecCopy(fU1, -1, F1) // fU1 := (-F1)
la.SpMatVecMulAdd(fU1, 1, k11, U1) // fU1 += K11*U1
la.SpMatVecMulAdd(fU1, 1, k12, U2) // fU1 += K12*U2
return nil
}
Jfcn := func(dfU1dU1 *la.Triplet, U1 []float64) error {
fdm.Assemble(dfU1dU1, &K12, F1, nil, &g, &e)
return nil
}
U1 := make([]float64, e.N1)
CompareJac(tst, ffcn, Jfcn, U1, 0.0075)
print_jac := false
if print_jac {
W1 := make([]float64, e.N1)
fU1 := make([]float64, e.N1)
ffcn(fU1, U1)
var Jnum la.Triplet
Jnum.Init(e.N1, e.N1, e.N1*e.N1)
Jacobian(&Jnum, ffcn, U1, fU1, W1)
la.PrintMat("K11 ", K11.ToMatrix(nil).ToDense(), "%g ", false)
la.PrintMat("Jnum", Jnum.ToMatrix(nil).ToDense(), "%g ", false)
}
test_ffcn := false
if test_ffcn {
Uc := []float64{0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 50.0, 25.0, 325.0 / 22.0, 100.0 / 11.0, 50.0 / 11.0,
0.0, 50.0, 775.0 / 22.0, 25.0, 375.0 / 22.0, 100.0 / 11.0, 0.0, 50.0, 450.0 / 11.0, 725.0 / 22.0,
25.0, 325.0 / 22.0, 0.0, 50.0, 500.0 / 11.0, 450.0 / 11.0, 775.0 / 22.0, 25.0, 0.0, 50.0, 50.0,
50.0, 50.0, 50.0, 50.0,
}
for i := 0; i < e.N1; i++ {
U1[i] = Uc[e.RF1[i]]
}
fU1 := make([]float64, e.N1)
min, max := la.VecMinMax(fU1)
io.Pf("min/max fU1 = %v\n", min, max)
}
}
func Test_nls04(tst *testing.T) {
//verbose()
chk.PrintTitle("nls04. finite differences problem")
// grid
var g fdm.Grid2D
g.Init(1.0, 1.0, 6, 6)
// equations numbering
var e fdm.Equations
peq := utl.IntUnique(g.L, g.R, g.B, g.T)
e.Init(g.N, peq)
// K11 and K12
var K11, K12 la.Triplet
fdm.InitK11andK12(&K11, &K12, &e)
// assembly
F1 := make([]float64, e.N1)
fdm.Assemble(&K11, &K12, F1, nil, &g, &e)
// prescribed values
U2 := make([]float64, e.N2)
for _, eq := range g.L {
U2[e.FR2[eq]] = 50.0
}
for _, eq := range g.R {
U2[e.FR2[eq]] = 0.0
}
for _, eq := range g.B {
U2[e.FR2[eq]] = 0.0
}
for _, eq := range g.T {
U2[e.FR2[eq]] = 50.0
}
// functions
k11 := K11.ToMatrix(nil)
k12 := K12.ToMatrix(nil)
ffcn := func(fU1, U1 []float64) error { // K11*U1 + K12*U2 - F1
la.VecCopy(fU1, -1, F1) // fU1 := (-F1)
la.SpMatVecMulAdd(fU1, 1, k11, U1) // fU1 += K11*U1
la.SpMatVecMulAdd(fU1, 1, k12, U2) // fU1 += K12*U2
return nil
}
Jfcn := func(dfU1dU1 *la.Triplet, U1 []float64) error {
fdm.Assemble(dfU1dU1, &K12, F1, nil, &g, &e)
return nil
}
JfcnD := func(dfU1dU1 [][]float64, U1 []float64) error {
la.MatCopy(dfU1dU1, 1, K11.ToMatrix(nil).ToDense())
return nil
}
prms := map[string]float64{
"atol": 1e-8,
"rtol": 1e-8,
"ftol": 1e-12,
"lSearch": 0.0,
}
// init
var nls_sps NlSolver // sparse analytical
var nls_num NlSolver // sparse numerical
var nls_den NlSolver // dense analytical
nls_sps.Init(e.N1, ffcn, Jfcn, nil, false, false, prms)
nls_num.Init(e.N1, ffcn, nil, nil, false, true, prms)
nls_den.Init(e.N1, ffcn, nil, JfcnD, true, false, prms)
defer nls_sps.Clean()
defer nls_num.Clean()
defer nls_den.Clean()
// results
U1sps := make([]float64, e.N1)
U1num := make([]float64, e.N1)
U1den := make([]float64, e.N1)
Usps := make([]float64, e.N)
Unum := make([]float64, e.N)
Uden := make([]float64, e.N)
// solution
Uc := []float64{0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 50.0, 25.0, 325.0 / 22.0, 100.0 / 11.0, 50.0 / 11.0,
0.0, 50.0, 775.0 / 22.0, 25.0, 375.0 / 22.0, 100.0 / 11.0, 0.0, 50.0, 450.0 / 11.0, 725.0 / 22.0,
25.0, 325.0 / 22.0, 0.0, 50.0, 500.0 / 11.0, 450.0 / 11.0, 775.0 / 22.0, 25.0, 0.0, 50.0, 50.0,
50.0, 50.0, 50.0, 50.0,
}
io.PfYel("\n---- sparse -------- Analytical Jacobian -------------------\n")
// solve
err := nls_sps.Solve(U1sps, false)
if err != nil {
chk.Panic(err.Error())
}
// check
fdm.JoinVecs(Usps, U1sps, U2, &e)
chk.Vector(tst, "Usps", 1e-14, Usps, Uc)
// plot
if false {
g.Contour("results", "fig_t_heat_square", nil, Usps, 11, false)
}
io.PfYel("\n---- dense -------- Analytical Jacobian -------------------\n")
// solve
err = nls_den.Solve(U1den, false)
if err != nil {
chk.Panic(err.Error())
}
// check
fdm.JoinVecs(Uden, U1den, U2, &e)
chk.Vector(tst, "Uden", 1e-14, Uden, Uc)
io.PfYel("\n---- sparse -------- Numerical Jacobian -------------------\n")
// solve
err = nls_num.Solve(U1num, false)
if err != nil {
chk.Panic(err.Error())
}
// check
fdm.JoinVecs(Unum, U1num, U2, &e)
chk.Vector(tst, "Unum", 1e-14, Unum, Uc)
}
// Radau5 step function
func radau5_step_mpi(o *Solver, y0 []float64, x0 float64, args ...interface{}) (rerr float64, err error) {
// factors
α := r5.α_ / o.h
β := r5.β_ / o.h
γ := r5.γ_ / o.h
// Jacobian and decomposition
if o.reuseJdec {
o.reuseJdec = false
} else {
// calculate only first Jacobian for all iterations (simple/modified Newton's method)
if o.reuseJ {
o.reuseJ = false
} else if !o.jacIsOK {
// Jacobian triplet
if o.jac == nil { // numerical
//if x0 == 0.0 { io.Pfgrey(" > > > > > > > > . . . numerical Jacobian . . . < < < < < < < < <\n") }
err = num.JacobianMpi(&o.dfdyT, func(fy, y []float64) (e error) {
e = o.fcn(fy, o.h, x0, y, args...)
return
}, y0, o.f0, o.w[0], o.Distr) // w works here as workspace variable
} else { // analytical
//if x0 == 0.0 { io.Pfgrey(" > > > > > > > > . . . analytical Jacobian . . . < < < < < < < < <\n") }
err = o.jac(&o.dfdyT, o.h, x0, y0, args...)
}
if err != nil {
return
}
// create M matrix
if o.doinit && !o.hasM {
o.mTri = new(la.Triplet)
if o.Distr {
id, sz := mpi.Rank(), mpi.Size()
start, endp1 := (id*o.ndim)/sz, ((id+1)*o.ndim)/sz
o.mTri.Init(o.ndim, o.ndim, endp1-start)
for i := start; i < endp1; i++ {
o.mTri.Put(i, i, 1.0)
}
} else {
o.mTri.Init(o.ndim, o.ndim, o.ndim)
for i := 0; i < o.ndim; i++ {
o.mTri.Put(i, i, 1.0)
}
}
}
o.njeval += 1
o.jacIsOK = true
}
// initialise triplets
if o.doinit {
o.rctriR = new(la.Triplet)
o.rctriC = new(la.TripletC)
o.rctriR.Init(o.ndim, o.ndim, o.mTri.Len()+o.dfdyT.Len())
xzmono := o.Distr
o.rctriC.Init(o.ndim, o.ndim, o.mTri.Len()+o.dfdyT.Len(), xzmono)
}
// update triplets
la.SpTriAdd(o.rctriR, γ, o.mTri, -1, &o.dfdyT) // rctriR := γ*M - dfdy
la.SpTriAddR2C(o.rctriC, α, β, o.mTri, -1, &o.dfdyT) // rctriC := (α+βi)*M - dfdy
// initialise solver
if o.doinit {
err = o.lsolR.InitR(o.rctriR, false, false, false)
if err != nil {
return
}
err = o.lsolC.InitC(o.rctriC, false, false, false)
if err != nil {
return
}
}
// perform factorisation
o.lsolR.Fact()
o.lsolC.Fact()
o.ndecomp += 1
}
// updated u[i]
o.u[0] = x0 + r5.c[0]*o.h
o.u[1] = x0 + r5.c[1]*o.h
o.u[2] = x0 + r5.c[2]*o.h
// (trial/initial) updated z[i] and w[i]
if o.first || o.ZeroTrial {
for m := 0; m < o.ndim; m++ {
o.z[0][m], o.w[0][m] = 0.0, 0.0
o.z[1][m], o.w[1][m] = 0.0, 0.0
o.z[2][m], o.w[2][m] = 0.0, 0.0
}
} else {
c3q := o.h / o.hprev
c1q := r5.μ1 * c3q
c2q := r5.μ2 * c3q
for m := 0; m < o.ndim; m++ {
o.z[0][m] = c1q * (o.ycol[0][m] + (c1q-r5.μ4)*(o.ycol[1][m]+(c1q-r5.μ3)*o.ycol[2][m]))
o.z[1][m] = c2q * (o.ycol[0][m] + (c2q-r5.μ4)*(o.ycol[1][m]+(c2q-r5.μ3)*o.ycol[2][m]))
o.z[2][m] = c3q * (o.ycol[0][m] + (c3q-r5.μ4)*(o.ycol[1][m]+(c3q-r5.μ3)*o.ycol[2][m]))
o.w[0][m] = r5.Ti[0][0]*o.z[0][m] + r5.Ti[0][1]*o.z[1][m] + r5.Ti[0][2]*o.z[2][m]
o.w[1][m] = r5.Ti[1][0]*o.z[0][m] + r5.Ti[1][1]*o.z[1][m] + r5.Ti[1][2]*o.z[2][m]
o.w[2][m] = r5.Ti[2][0]*o.z[0][m] + r5.Ti[2][1]*o.z[1][m] + r5.Ti[2][2]*o.z[2][m]
}
}
// iterations
o.nit = 0
o.η = math.Pow(max(o.η, o.ϵ), 0.8)
o.θ = o.θmax
o.diverg = false
var Lδw, oLδw, thq, othq, iterr, itRerr, qnewt float64
var it int
for it = 0; it < o.NmaxIt; it++ {
// max iterations ?
o.nit = it + 1
if o.nit > o.nitmax {
o.nitmax = o.nit
}
// evaluate f(x,y) at (u[i],v[i]=y0+z[i])
for i := 0; i < 3; i++ {
for m := 0; m < o.ndim; m++ {
o.v[i][m] = y0[m] + o.z[i][m]
}
o.nfeval += 1
err = o.fcn(o.f[i], o.h, o.u[i], o.v[i], args...)
if err != nil {
return
}
}
// calc rhs
if o.hasM {
// using δw as workspace here
la.SpMatVecMul(o.δw[0], 1, o.mMat, o.w[0]) // δw0 := M * w0
la.SpMatVecMul(o.δw[1], 1, o.mMat, o.w[1]) // δw1 := M * w1
la.SpMatVecMul(o.δw[2], 1, o.mMat, o.w[2]) // δw2 := M * w2
if o.Distr {
mpi.AllReduceSum(o.δw[0], o.v[0]) // v is used as workspace here
mpi.AllReduceSum(o.δw[1], o.v[1]) // v is used as workspace here
mpi.AllReduceSum(o.δw[2], o.v[2]) // v is used as workspace here
}
for m := 0; m < o.ndim; m++ {
o.v[0][m] = r5.Ti[0][0]*o.f[0][m] + r5.Ti[0][1]*o.f[1][m] + r5.Ti[0][2]*o.f[2][m] - γ*o.δw[0][m]
o.v[1][m] = r5.Ti[1][0]*o.f[0][m] + r5.Ti[1][1]*o.f[1][m] + r5.Ti[1][2]*o.f[2][m] - α*o.δw[1][m] + β*o.δw[2][m]
o.v[2][m] = r5.Ti[2][0]*o.f[0][m] + r5.Ti[2][1]*o.f[1][m] + r5.Ti[2][2]*o.f[2][m] - β*o.δw[1][m] - α*o.δw[2][m]
}
} else {
for m := 0; m < o.ndim; m++ {
o.v[0][m] = r5.Ti[0][0]*o.f[0][m] + r5.Ti[0][1]*o.f[1][m] + r5.Ti[0][2]*o.f[2][m] - γ*o.w[0][m]
o.v[1][m] = r5.Ti[1][0]*o.f[0][m] + r5.Ti[1][1]*o.f[1][m] + r5.Ti[1][2]*o.f[2][m] - α*o.w[1][m] + β*o.w[2][m]
o.v[2][m] = r5.Ti[2][0]*o.f[0][m] + r5.Ti[2][1]*o.f[1][m] + r5.Ti[2][2]*o.f[2][m] - β*o.w[1][m] - α*o.w[2][m]
}
}
// solve linear system
o.nlinsol += 1
var errR, errC error
if !o.Distr && o.Pll {
wg := new(sync.WaitGroup)
wg.Add(2)
go func() {
errR = o.lsolR.SolveR(o.δw[0], o.v[0], false)
wg.Done()
}()
go func() {
errC = o.lsolC.SolveC(o.δw[1], o.δw[2], o.v[1], o.v[2], false)
wg.Done()
}()
wg.Wait()
} else {
errR = o.lsolR.SolveR(o.δw[0], o.v[0], false)
errC = o.lsolC.SolveC(o.δw[1], o.δw[2], o.v[1], o.v[2], false)
}
// check for errors from linear solution
if errR != nil || errC != nil {
var errmsg string
if errR != nil {
errmsg += errR.Error()
}
if errC != nil {
if errR != nil {
errmsg += "\n"
}
errmsg += errC.Error()
}
err = errors.New(errmsg)
return
}
// update w and z
for m := 0; m < o.ndim; m++ {
o.w[0][m] += o.δw[0][m]
o.w[1][m] += o.δw[1][m]
o.w[2][m] += o.δw[2][m]
o.z[0][m] = r5.T[0][0]*o.w[0][m] + r5.T[0][1]*o.w[1][m] + r5.T[0][2]*o.w[2][m]
o.z[1][m] = r5.T[1][0]*o.w[0][m] + r5.T[1][1]*o.w[1][m] + r5.T[1][2]*o.w[2][m]
o.z[2][m] = r5.T[2][0]*o.w[0][m] + r5.T[2][1]*o.w[1][m] + r5.T[2][2]*o.w[2][m]
}
// rms norm of δw
Lδw = 0.0
for m := 0; m < o.ndim; m++ {
Lδw += math.Pow(o.δw[0][m]/o.scal[m], 2.0) + math.Pow(o.δw[1][m]/o.scal[m], 2.0) + math.Pow(o.δw[2][m]/o.scal[m], 2.0)
}
Lδw = math.Sqrt(Lδw / float64(3*o.ndim))
// check convergence
if it > 0 {
thq = Lδw / oLδw
if it == 1 {
o.θ = thq
} else {
o.θ = math.Sqrt(thq * othq)
}
othq = thq
if o.θ < 0.99 {
o.η = o.θ / (1.0 - o.θ)
iterr = Lδw * math.Pow(o.θ, float64(o.NmaxIt-o.nit)) / (1.0 - o.θ)
itRerr = iterr / o.fnewt
if itRerr >= 1.0 { // diverging
qnewt = max(1.0e-4, min(20.0, itRerr))
o.dvfac = 0.8 * math.Pow(qnewt, -1.0/(4.0+float64(o.NmaxIt)-1.0-float64(o.nit)))
o.diverg = true
break
}
} else { // diverging badly (unexpected step-rejection)
o.dvfac = 0.5
o.diverg = true
break
}
}
// save old norm
oLδw = Lδw
// converged
if o.η*Lδw < o.fnewt {
break
}
}
// did not converge
if it == o.NmaxIt-1 {
chk.Panic("radau5_step failed with it=%d", it)
}
// diverging => stop
if o.diverg {
rerr = 2.0 // must leave state intact, any rerr is OK
return
}
// error estimate
if o.LerrStrat == 1 {
// simple strategy => HW-VII p123 Eq.(8.17) (not good for stiff problems)
for m := 0; m < o.ndim; m++ {
o.ez[m] = r5.e0*o.z[0][m] + r5.e1*o.z[1][m] + r5.e2*o.z[2][m]
o.lerr[m] = r5.γ0*o.h*o.f0[m] + o.ez[m]
rerr += math.Pow(o.lerr[m]/o.scal[m], 2.0)
}
rerr = max(math.Sqrt(rerr/float64(o.ndim)), 1.0e-10)
} else {
// common
if o.hasM {
for m := 0; m < o.ndim; m++ {
o.ez[m] = r5.e0*o.z[0][m] + r5.e1*o.z[1][m] + r5.e2*o.z[2][m]
o.rhs[m] = o.f0[m]
}
if o.Distr {
la.SpMatVecMul(o.δw[0], γ, o.mMat, o.ez) // δw[0] = γ * M * ez (δw[0] is workspace)
mpi.AllReduceSumAdd(o.rhs, o.δw[0], o.δw[1]) // rhs += join_with_sum(δw[0]) (δw[1] is workspace)
} else {
la.SpMatVecMulAdd(o.rhs, γ, o.mMat, o.ez) // rhs += γ * M * ez
}
} else {
for m := 0; m < o.ndim; m++ {
o.ez[m] = r5.e0*o.z[0][m] + r5.e1*o.z[1][m] + r5.e2*o.z[2][m]
o.rhs[m] = o.f0[m] + γ*o.ez[m]
}
}
// HW-VII p123 Eq.(8.19)
if o.LerrStrat == 2 {
o.lsolR.SolveR(o.lerr, o.rhs, false)
rerr = o.rms_norm(o.lerr)
// HW-VII p123 Eq.(8.20)
} else {
o.lsolR.SolveR(o.lerr, o.rhs, false)
rerr = o.rms_norm(o.lerr)
if !(rerr < 1.0) {
if o.first || o.reject {
for m := 0; m < o.ndim; m++ {
o.v[0][m] = y0[m] + o.lerr[m] // y0perr
}
o.nfeval += 1
err = o.fcn(o.f[0], o.h, x0, o.v[0], args...) // f0perr
if err != nil {
return
}
if o.hasM {
la.VecCopy(o.rhs, 1, o.f[0]) // rhs := f0perr
if o.Distr {
la.SpMatVecMul(o.δw[0], γ, o.mMat, o.ez) // δw[0] = γ * M * ez (δw[0] is workspace)
mpi.AllReduceSumAdd(o.rhs, o.δw[0], o.δw[1]) // rhs += join_with_sum(δw[0]) (δw[1] is workspace)
} else {
la.SpMatVecMulAdd(o.rhs, γ, o.mMat, o.ez) // rhs += γ * M * ez
}
} else {
la.VecAdd2(o.rhs, 1, o.f[0], γ, o.ez) // rhs = f0perr + γ * ez
}
o.lsolR.SolveR(o.lerr, o.rhs, false)
rerr = o.rms_norm(o.lerr)
}
}
}
}
return
}