Determining superconducting vortices configurations with stochastic processes in a GPU-based code N. Molero Puerto1, R. Mayo García2 1 URJC 2 CIEMAT March, 27th – 29th 2018

Index The Physical Problem Experiment Simulations Results Conclusions

Many effects can be observed The Physical Problem Superconducting vortex lattice dynamics and vortex lattice pinning are strongly modified by arrays of nanodefects embedded in superconducting films By using arrays of holes (antidotes), which thread the films or dots embedded in the sample, this effect can be studied Many effects can be observed reconfiguration of the vortex lattice Effects induced by arrays made with different materials and different diameters of the pinning centers Channeling effects Etc. 3

Different interactions take part The Physical Problem Magnetoresistance measurements are a perfect tool to study these effects Resistance vs. applied magnetic fields shows deep minima when the vortex lattice matches the unit cell of the array due to geometric matching occurs when the vortex density is an integer multiple of the pinning center density Different interactions take part Vortex–vortex Vortex–artificially induced pinning center (array of nanodefects) Vortex–intrinsic and random pinning centers The magnetoresistance minima show up only when the temperature is close to Tc 4

The diameter of the Ni dots is 200 nm Experiment Samples are Nb film on top of array of submicrometric Ni dots which have been fabricated by electron beam lithography on Si substrates Size 40 mm2 400×600 nm2 rectangular arrays of Ni dots have been selected as the artificially fabricated pinning arrays for the present work The thickness of the Ni dots is 40 nm, while the thickness of the Nb film is 100 nm The diameter of the Ni dots is 200 nm 5

Minima appear at applied magnetic fields Experiment Minima appear at applied magnetic fields Hn = n· φ0 / (a· b) a and b are the lattice parameters of the rectangular array φ0 = 2.07· 10−15 Wb Other parameters l = 2.6 mm f0 = 3.08· 10-6 T2nm rp = 100 nm The number of vortices n per array unit cell can be known by simple inspection of the magnetoresistance curves, in which the first minimum corresponds to one vortex per unit cell, the second minimum to two vortices per unit cell, and so on 6

The Physical Problem 7

DiVoS finds the minimum of the following equation Simulations The DiVoS code can simulate squares, rectangles, and triangles of any size DiVoS finds the minimum of the following equation DiVoS does not take advantages neither of matching conditions with respect to the vortices lattices nor computational cutoff approximations to place the vortices 8

Simulations: GA Genetic Algorithm A population of candidate solutions (genes) to an optimization problem is evolved toward better solutions Previous objective function to be minimized by the movement of the vortices The evolution starts from a set of populations of randomly generated individuals Each candidate solution has a set of properties which is a repetitive application of the mutation, crossover, inversion, and selection operators The algorithm ends when either a maximum number of generations has been produced, or a satisfactory fitness level has been reached for the population 9

Simulations: SA Simulated Annealing Minimize a function by identifying with the energy of an imaginary physical system undergoing an annealing process Move from Fi to Fi+1 via a proposal. If the new state has: Lower energy, accept Fi+1 Higher energy, accept with probability A=exp(-DF/KT) Stochastic acceptance of higher energy states, allows the process to escape local minima When T is high, the acceptance of these uphill moves is higher, and local minima are discouraged As T is lowered, more concentrated search near current local minimum, since only few uphill moves will be allowed 10

Simulations: SA Simulated Annealing Reannealing interval, or epoch length, is the number of points to accept before reannealing (change the temperature) L = iterations at a particular temperature Larger decreases in T require correspondingly longer L to re-equilibrate Running long L at larger temperatures is not very useful  Decrease T rapidly at first Vortices Select 1 movement out of 8 randomly 1 by 1 (~Molecular Dynamics) All at the same time (Multidimensional Gaussian) 11

Simulations: SA Simulated Annealing After the epoch length (L steps), the comparison Fi+1 vs Fi is made After the comparison, L and T are updated Reannealing interval evolves with Lk+1=bLk with b>1 Thermostat Linear: Tk+1=aTk, (with 1<a<0) or Tk–a (with a>0) Exponential: Tk+1=Tk· 0.95a (with a≥1) Logarithmic: Tk+1=Tk/log(a) (with a≥10) 12

Simulations: BH Basin Hopping Similar to SA, though it represents an improvement Random perturbation of coords (location on error surface) Local minimization – find best (lowest) error locally Accept/reject new position based on function value at that point Acceptance test is usually Metropolis criterion from Monte Carlo (Metropolis-Hastings) methods MH is more generally used to generate random samples from a prob. distribution from which direct sampling is difficult (maybe we don’t know what it looks like) 13

Computational outcomes Results Computational outcomes Over 99% of the computing time gets into the evaluation of interactions Original scalability is of O(N²), being N the number of vortices Parallel scalability is of O(N²/2G), being G the number of GPU cores By using a cache mechanism, a GPU core is faster than a CPU for this problem 14

Experiments have been executed 5 times each Results GA 200 genes randomly created 1 gene per Nvdia core 500 generations Mutation and crossover rate were constant SA and BH Initial value of T was 25,000 K Temperature decreased as logarithmic function Population in the simulated annealing is 1 Experiments have been executed 5 times each Tesla K20 15

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BH presents better results than SA and GA Conclusions From the computational point of view, the code executes and scales pretty well BH presents better results than SA and GA Further tests are needed to be carried out in order to properly match experiment and simulation Current mathematical model does not correspond with experiments though A plethora of results can be found in “Determinación de configuraciones de vórtices superconductores con procesos estocásticos con aceleradores gráficos en entornos de supercomputación”. BSc Thesis. N. Molero Puerto 22

THANK YOU. rafael. mayo@ciemat. es CIEMAT Avda THANK YOU!!! rafael.mayo@ciemat.es CIEMAT Avda. Complutense, 40 – 28040 Madrid http://rdgroups.ciemat.es/web/sci-track/ 23