Improving Particle Methods Robert Krasny (Michigan) collaborators - - PowerPoint PPT Presentation
March 12-16, 2018 ICERM Workshop on Fast Algorithms for Generating Static and Dynamically Changing Point Configurations Improving Particle Methods Robert Krasny (Michigan) collaborators Weihua Geng (Southern Methodist University) Peter
March 12-16, 2018 ICERM Workshop on “Fast Algorithms for Generating Static and Dynamically Changing Point Configurations”
Weihua Geng (Southern Methodist University) Peter Bosler (Sandia National Laboratories) Ling Xu (Michigan)
NSF DMS-1418966 ONR N00014-14-1-0075 MICDE (Michigan Institute for Computational Discovery and Engineering)
Ω1 : protein domain : ✏1r2(x) =
Nc
X
k=1
qk(x yk) Ω2 : solvent domain : ✏2r2(x) ¯ 2(x) = 0 : PB equation Γ : molecular surface : 1 = 2 , ✏1 @1 @n = ✏2 @2 @n far-field boundary condition : lim
|x|→∞(x) = 0
G0(x, y) = 1 4⇡|x − y| : Coulomb potential G(x, y) = e−|x−y| 4⇡|x − y| : screened Coulomb potential , 2 = ¯ 2/✏2
1 2
⇣ 1 + ✏2
✏1
⌘ (x) = Z
Γ
h K1(x, y)@n(y) + K2(x, y)(y) i dSy + S1(x)
1 2
⇣ 1 + ✏1
✏2
⌘ @n(x) = Z
Γ
h K3(x, y)@n(y) + K4(x, y)(y) i dSy + S2(x) K1 = G0 − G , K2 = ✏2 ✏1 @nyG − @nyG0 K4 = @2
nxny(G − G0) , K3 = @nxG0 − ✏1 ✏2@nxG
S1(x) = 1 ✏1
Nc
X
k=1
qkG0(x, yk) , S2(x) = 1 ✏1
Nc
X
k=1
qk@nxG0(x, yk)
triangulation + centroid collocation xi : centroid , Ai : area , i = 1 : N
1 2
⇣ 1 + ✏2
✏1
⌘ φi =
N
X
j=1 j6=i
h K1(xi, xj)∂nφj + K2(xi, xj)φj i Aj + S1(xi)
1 2
⇣ 1 + ✏1
✏2
⌘ ∂nφi =
N
X
j=1 j6=i
h K3(xi, xj)∂nφj + K4(xi, xj)φj i Aj + S2(xi) ⇒ linear system : Ax = b
φi =
N
X
j=1 j6=i
qjG(xi, xj) , i = 1, . . . , N : particle-particle , O(N 2) ≈ X
c p
X
k=0
ak(xi, xc)M k(c) : particle-cluster , O(N log N)
k!Dk yG(x, xc)
Li-Johnston-K (2009) JCP 228
TABI : N = 20K → 500K APBS : grid = 65 × 332 → 513 × 3212 , hmax = 1.63 ˚ A → 0.13 ˚ A
10 10
1
10
1
10
2
10
3
error in Esol (%) CPU time (s) 3% 6% TABI−PB TABI−P APBS−PB APBS−P 10 10
1
10
1
10
2
10
3
10
4
error in Esol (%) memory (MB) TABI−PB TABI−P APBS−PB APBS−P
CPU time / error memory / error
Geng-K (2013) JCP 247
TABI parallel performance , N = 132196 , error in Esol ≈ 1.3% Poisson-Boltzmann # processors run time (s) speedup parallel efficiency (%) 1 799.3 1.00 100.0 2 410.0 1.95 97.5 4 223.8 3.57 89.3 8 123.7 6.46 80.9 Poisson # processors run time (s) speedup parallel efficiency (%) 1 324.4 1.00 100.0 2 166.7 1.95 97.3 4 92.6 3.50 87.6 8 51.0 6.36 79.5
TOOLS FOR PROTEIN SCIENCE Improvements to the APBS biomolecular solvation software suite
Elizabeth Jurrus,1 Dave Engel,1 Keith Star,1 Kyle Monson,1 Juan Brandi,1 Lisa E. Felberg,2 David H. Brookes,2 Leighton Wilson,3 Jiahui Chen,4 Karina Liles,1 Minju Chun,1 Peter Li,1 David W. Gohara,5 Todd Dolinsky,6 Robert Konecny,7 David R. Koes ,8 Jens Erik Nielsen,9 Teresa Head-Gordon,2 Weihua Geng,4 Robert Krasny,3 Guo-Wei Wei,10 Michael J. Holst,7
1,11*
1Pacific Northwest National Laboratory, Richland, Washington 2University of California, Berkeley, California 3University of Michigan, Ann Arbor, Michigan 4Southern Methodist University, Dallas, Texas
6FoodLogiQ, Durham, North Carolina 7University of California San Diego, San Diego, California 8University of Pittsburgh, Pittsburgh, Pennsylvania 9Protein Engineering, Novozymes A/S, Copenhagen, Denmark 10Michigan State University, East Lansing, Michigan 11Brown University, Providence, Rhode Island
93.9
20 40 60 80 Surface Potential (kJ/mol/e)
PDBID: 3dfg
given : q0(x) , u(x, t) , x 2 S problem : determine q(x, t) for t > 0
∂q ∂t(x, t) + u · rq(x, t) = 0
flow map : a ! x(a, t) , a 2 S ∂ ∂tx(a, t) = u(x(a, t), t) , x(a, 0) = a q(x(a, t), t) = q0(a)
particles : xj(t) ≈ x(aj, t) , xj(0) = aj , j = 1 : M panels : Pk(t) = x(Pk(0), t) , S = ∪N
k=1Pk(t) , k = 1 : N
icosahedral triangulation
N 20 80 320 1280 5120 20480 81920 327680 M 32 122 482 1922 7682 30722 122882 491522 ∆λ 63.43 33.87 17.22 8.64 4.33 2.16 1.08 0.54 Bosler-Kent-K-Jablonowski (2017) JCP 340
d dtxj(t) = u(xj, t) , xj(0) = aj qj = q0(aj)
j } → {xnew j
j
movie
d dtxj(t) = u(xj, t) , xj(0) = aj qj = q0(aj)
j } → {xnew j
j
{qnew
j
} = I({xnew
j
}; {xold
j }, {qold j })
{anew
j
} = I({xnew
j
}; {xold
j }, {aold j })
{qnew
j
} = q0({anew
j
}) I : STRIPACK/SSRFPACK (Renka 1997)
movie
test case 1 test case 2
0.5 1 2 4 8 16 32 10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10 ∆λ (degrees)
∞ 2
O(∆λ4) LR, l∞ LR, l2 LPM−d, l∞ LPM−d, l2 LPM−i, l∞ LPM−i, l2 0.5 1 2 4 8 16 32 10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10 10
1
∆λ (degrees)
∞ 2
O(∆λ4) LR, l∞ LR, l2 LPM−d, l∞ LPM−d, l2 LPM−i, l∞ LPM−i, l2
movie
movie
Gaussian-hills tracer , grid spacing ∆λ = 8.64 direct remeshing indirect remeshing
test case 1 test case 2
0.5 1 2 4 8 16 32 10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10 ∆λ (degrees)
∞ 2
O(∆λ4) LR, l∞ LR, l2 LPM−d, l∞ LPM−d, l2 LPM−i, l∞ LPM−i, l2 0.5 1 2 4 8 16 32 10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10 10
1
∆λ (degrees)
∞ 2
O(∆λ4) LR, l∞ LR, l2 LPM−d, l∞ LPM−d, l2 LPM−i, l∞ LPM−i, l2
Bosler-Wang-Jablonowski-K (2014) FDR 46
S
S
k=1Ak
k=1 k6=j
movie
movie
movie
∂ω ∂t + u · rω = 0 , r · u = 0 u = r⊥ψ , r2ψ = ω
x(a, t) , dx dt = u(x, t) , x(a, 0) = a u(x, t) = Z Kδ(x, y)ω(y, t)dy , Kδ(x, y) = r⊥ ln(|x y|2 + δ2) 4π ω(x, t) = ω0(a)
5 10 15
t
50 100 150 200 250 300
max(t)
(t) (t)
5 10 15
t
0.95 0.98 1 1.02 1.05
max(t) / max(0)
(t) / (0) (t) / (0))
ωmax(t) = maxxω(x, t) Γ(t) = Z ω(x, t)dA γ(t) = Z ω2(x, t)dA
maximum vorZcity : circulaZon : enstrophy :