Computational Geometry Aspects Of Monte Carlo Approaches .
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsComputational Geometry Aspects of MonteCarlo Approaches to PDE Problems inBiology, Chemistry, and MaterialsProf. Michael MascagniDepartment of Computer ScienceDepartment of MathematicsDepartment of Scientific ComputingGraduate Program in Molecular BiophysicsFlorida State University, Tallahassee, FL 32306 USAANDApplied and Computational Mathematics Division, Information Technology LaboratoryNational Institute of Standards and Technology, Gaithersburg, MD 20899-8910 USAE-mail: mascagni@fsu.edu or mascagni@nist.govURL: http://www.cs.fsu.edu/ mascagniResearch supported by ARO, DOE, NASA, NATO, NIST, National Science Fund (Bulgaria) &NSF, with equipment donated by Intel and Nvidia
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsOutline of the TalkMonte Carlo Methods for PDEsA Little History on Monte Carlo Methods for PDEsSome Examples Using This for Computing Elliptic ProblemsThe Walk on Spheres MethodBiochemical ProblemsGeneralization of the Monte Carlo ApproachComputational GeometryImportant Computational Paradigm: Computing CapacitanceCapacitance as a Computational ParadigmParallel ScalabilityScalability PlotsConclusions
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsDedicated to My Probability Professors at Courant:Raghu Varadhan and Monroe Donsker
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsIn Memoriam: Nikolai Simonov, 1956–2019Figure: Died in an avalanche in the Altai Mountains, May 6, 2019
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsMonte Carlo Methods for PDEsA Little History on Monte Carlo Methods for PDEsEarly History of MCMs for PDEs1. Courant, Friedrichs, and Lewy: Their pivotal 1928 paper hasprobabilistic interpretations and MC algorithms for linear ellipticand parabolic problems2. Fermi/Ulam/von Neumann: Atomic bomb calculations were doneusing Monte Carlo methods for neutron transport, their successinspired much post-War work especially in nuclear reactor design3. Kac and Donsker: Used large deviation calculations to estimateeigenvalues of a linear Schrödinger equation4. Forsythe and Leibler: Derived a MCM for solving special linearsystems related to discrete elliptic PDE problems
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsMonte Carlo Methods for PDEsA Little History on Monte Carlo Methods for PDEsIntegration: The Classic Monte Carlo ApplicationR11. Consider computing I 0 f (x) dx2. Conventionalquadrature methods:NXI wi f (xi )i 1I Standard quadrature is of this form with deterministic error boundsI If we hold work, f (xi ), constant as dimension increases we see theMC advantage vs. the curse of dimensionality3. Monte Carlo method has two parts to estimate a numericalquantity of interest, II The random process/variable: xi U[0, 1] i.i.d.I The estimator or score: f (xi )I One averages and uses a confidence interval for an"error bound#NNNX1 X11 X222Ī f (xi ), var (I) (f (xi ) Ī) f (xi ) N Ī ,NN 1N 1i 1i 1var (Ī) var (I),NqI Ī k var (Ī)i 1
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsMonte Carlo Methods for PDEsA Little History on Monte Carlo Methods for PDEsOther Early Monte Carlo ApplicationsI Numerical linear algebra based on sums: S PMi 1ai1. PDefine pi 0 as the probability of choosing index i, withMi 1 pi 1, and pi 0 whenever ai 6 02. Then ai /pi with index i chosenwith {pi } is an unbiased estimate of PMaiS, as E[ai /pi ] i 1 p pi SiI Can be used to solve linear systems of the form x Hx bI Consider the linear system: x Hx b, if H H 1, thenthe following iterative method converges:x n 1 : Hx n b, x 0 0,Pk 1and in particular we have x k i 0 H i b, and similarly theNeumann series converges:N Xi 0H i (I H) 1 , N Xi 0I Formally, the solution is x (I H) 1 b H i Xi 0Hi 11 H
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsMonte Carlo Methods for PDEsA Little History on Monte Carlo Methods for PDEsMore Modern Monte Carlo ApplicationsI Methods for partial differential and integral equations based onrandom walks/Markov chains (no need to find a discreteapproximation to the PDE/IE)1. Integral equation methods are similar in construction to the linearsystem methods2. PDEs can be solved by using the Feynman-Kac formula3. Some Monte Carlo methods can now beat deterministic solvers(electrostatics)I Efficient methods that exploit fast probabilistic application of alinear operatorI Modern sampling methods linear algebra (SVD) based looselyon the Johnson-Lindestrauss projection methodI Generation of random fieldsI Stochastic DEs and PDEsI Financial computingI Uncertainty quantification (UQ)
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsThe Walk on Spheres MethodThe First Passage (FP) Probability is the Green’sFunctionBack to our canonical elliptic boundary value problem:1 u(x) 0, x Ω2u(x) f (x), x ΩI Distribution of z is uniform on the sphereI Mean of the values of u(z) over the sphere is u(x)I u(x) has mean-value property and harmonicI Also, u(x) satisfies the boundary conditionu(x) Ex [f (X x (t Ω ))](1)
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsThe Walk on Spheres MethodThe First Passage (FP) Probability is the Green’sFunction ª:z µx; starting point ª µfirstŦpassage location
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsThe Walk on Spheres MethodThe First Passage (FP) Probability is the Green’sFunctionReinterpreting as an average of the boundary valuesZu(x) p(x, y ) f (y ) dy(2) ΩAnother representation in terms of an integral over the boundaryZ g(x, y )u(x) f (y ) dy n Ω(3)g(x, y ) – Green’s function of the Dirichlet problem in Ω p(x, y ) g(x, y ) n(4)
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsThe Walk on Spheres Method‘Walk on Spheres’ (WOS) and ‘Green’s Function FirstPassage’ (GFFP) AlgorithmsI Green’s function is known direct simulation of exit points and computation of thesolution through averaging boundary valuesI Green’s function is unknown simulation of exit points from standard subdomains of Ω,e.g. spheres Markov chain of ‘Walk on Spheres’ (or GFFP algorithm)x0 x, x1 , . . . , xNxi Ω and hits ε-shell is N O( ln(ε) ) stepsxN simulates exit point from Ω with O(ε) accuracy
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsThe Walk on Spheres Method‘Walk on Spheres’ (WOS) and ‘Green’s Function FirstPassage’ (GFFP) Algorithmsϵ Ωx; starting pointΩX x(τ Ω)first passage location
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsThe Walk on Spheres MethodArchitectural Considerations as We Move Towards theExascaleI Some trends in HPC architectures1. Memory per processor/core has inflected and is now decreasing2. Long-term trend is that memory bandwidth is the limiting factor forperformance and cost3. High clock rates and high bandwidth communication lead to highenergy consumption and hot boxes that need coolingI These Monte Carlo algorithms avoid all three of issues due totheir innate performance1. Minimal memory usage has always been a benefit of Monte Carlomethods2. Independent sampling means that the communication tocomputation ratio is extremely small and tunableI Monte Carlo is a very simple computational paradigm to explorefundamental aspects of parallelism, algorithmic resilience,fault-tolerance
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsThe Walk on Spheres MethodContinuum Biochemical ElectrostaticsMotivationI Experimental Data: Folding, stability & binding behavior ofbiomolecules can be modulated by changes in salt concentrationI Physical Model: Implicit solvent-based Poisson-Boltzmann modelcan provide accurate predictions of salt dependent behavior ofbiomoleculesI Mathematical Model: Elliptic boundary-value problemsSpecific ProblemsI Electrostatic free energy for linear case: only finite number ofelectrostatic potential point valuesI Dependence of energy on geometry: needs accurate treatmentI Singularities in solution: have to be taken into accountanalyticallyI Behavior at infinity: must be exactly enforcedI Functional dependence on salt concentration: needs accurateestimate
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsThe Walk on Spheres MethodMathematical Model: Molecular GeometryFigure: Biomolecule with dielectric i and region region Gi is in solution withdielectric e and region Ge . On the boundary of the biomolecule, electrostaticpotential and normal component of dielectric displacement continue
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsBiochemical ProblemsMathematical Model: Partial Differential EquationsI Poisson equation for the electrostatic potential, Φi , and pointcharges, Qm , inside a molecule (in CGS units): i Φi (x) 4πMXQm δ(x x (m) ) 0 , x Gim 1I For 1-1 salt (such as NaCl) Poisson-Boltzmann equation (PBE): Φe (x) κ2 sinh(Φe (x)) 0 , x Ge ,but we only consider the linearized PBE: Φe (x) κ2 Φe (x) 0 , x GeI For one-surface model: continuity condition on the dielectricboundary Φi ΦeΦi Φe , i e, y Γ n(y ) n(y )
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsBiochemical ProblemsElectrostatic Potential and EnergyI Point values of the potential: Φ(x) Φrf (x) Φc (x)Here, singular part of Φ:Φc (x) MXm 1Qm x x (m) I Reaction field electrostatic free energy of a molecule is linearcombination of point values of the regular part of the electrostaticpotential:M1XWrf Φrf (x (m) )Qm ,2m 1I Electrostatic solvation free energy difference between theenergy for a molecule in solvent with a given salt concentrationand the energy for the same molecule in vacuum:elec Gsolv Wrf ( i , e , κ) Wrf ( i , 1, 0)
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsBiochemical ProblemsThe Feynman-Kac FormulaI If we set f (x) 0 and have g(x) 6 0, the solution is Z τ Ωg(βy (s)) dsu(y ) E0I By linear superposition, the solution to Poisson equation is givenprobabilistically as Z τ Ω u(y ) Eg(βy (s)) ds f (βy (τ Ω ))0I The linearized Poisson-Boltzmann equation is given by u(x) κ2 u(x) 0, x Ω, u(x) f (x), x Ω, u 0 as x and has Wiener integral representation: R τ Ω 2u(y ) E f (βy (τ Ω ))e 0 κ ds
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsBiochemical Problems‘Walk-on-Spheres’ AlgorithmI Walk-on-spheres (WOS) algorithm for general domains with aregular boundaryI Define a Markov chain {xi , i 1, 2, . . .}I Set x0 x (m) for some m, xi xi 1 di ωi , i 1, 2, . . ., where1. di d(xi 1 ) is distance from xi 1 to Γ2. {ωi } is sequence of independent unit isotropic vectors3. xi is the exit point from the ball, B(xi 1 , d(xi 1 )), for a Brownianmotion starting at xi 1I Outside the molecule, on every step, walk-on-spheres terminatesκdiwith probability 1 q(κ, di ), where q(κ, di ) to dealsinh(κdi )with LPBE
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsBiochemical Problems‘Walk-on-Spheres’ and ‘Walk-in-Subdomains’I For general domains, an efficient way to simulate exit points is acombination of1. Inside the molecule: ‘walk-in-subdomains’2. Outside the molecule ‘walk-on-spheres’I The whole domain, Gi , is represented as a union of intersectingsubdomains:M[Gi Gmm 1I ‘Walk-in-Subdomains’: Simulate exit point separately in every Gm1. x0 x, x1 , . . . , xN – Markov chain, every xi 1 is an exit point fromthe corresponding subdomain for Brownian motion starting at xi2. For spherical subdomains, B(xim , Rim ), exit points are distributed inaccordance with the Poisson kernel:1 xi xim 2 (Rim )24πRim xi xi 1 3
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsBiochemical ProblemsMonte Carlo EstimatesI The estimate for the reaction-field potential point value:ξ[Φrf ](x (m) ) Φc (x1 )NinsX Fj (κ) (Φc (xjins ) Φc (xj,ins))(5)j 2 I Here {xj,ins} is a sequence of boundary points, after which therandom walker moves inside the domain, Gi , to xjinsI The estimate for the reaction-field energy:ξ[Wrf ] M1XQm ξ[Φrf ](x (m) )2m 1(6)
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsBiochemical ProblemsA Picture: The Algorithm for a Single Spherical Atom
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsBiochemical ProblemsThe Algorithm in Pictures: Walk Inside
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsBiochemical ProblemsThe Algorithm in Pictures: Walk 11110000000000011111111111
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsBiochemical ProblemsThe Algorithm in Pictures: Walk Outside
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsBiochemical ProblemsThe Algorithm in Pictures: Walk Outside
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsSome Examples Using This for Computing Elliptic ProblemsBiochemical ProblemsMonte Carlo Algorithm’s Computational ComplexityCost of a single trajectoryI Number of steps is random walk is not dependent on M, thenumber of atomsI The cost of finding the nearest sphere is M log2 (M) due tooptimizations2"Simulation""Theory-prediction"CPU Time(sec)1.510.50010002000300040005000Number of AtomsFigure: The CPU time per atom per trajectory is plotted as function of number of atoms. For smallnumber of atoms the CPU time scales linearly and for large number of atoms it asymptoticallyscales logarithmically
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsGeneralization of the Monte Carlo ApproachAccuracy: Monte Carlo vs. Deterministic
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsGeneralization of the Monte Carlo ApproachComputational GeometryGeometry: Problem DescriptionsThere are many geometric problems that arise in this algorithm:I Efficiently determining if a point is on the surface of the moleculeor inside of it (for interior walks)I Efficiently determining the closest sphere to a given exterior point(for walks outside molecule)I Efficiently determining if a query point is inside of the convex hullof the moleculeI Efficiently finding the largest possible sphere enclosing a querypoint for external walks
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsGeneralization of the Monte Carlo ApproachImportant Computational Paradigm: Computing CapacitancePorous Media: Complicated Interfaces
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsGeneralization of the Monte Carlo ApproachImportant Computational Paradigm: Computing CapacitanceComputing Capacitance ProbabilisticallyI Hubbard-Douglas: can compute permeability of nonskew objectvia capacitanceI Recall that C Qu , if we hold the conductor, Ω, at unit potentialu 1, then C total charge the conductor’s surface, ΩI The PDE system for the potential is u 0, x / Ω;u 1, x Ω;u 0 as x I Recall u(x) Ex [f (X x (t Ω ))] 1, if the walker hits Ω, or 0, if thewalker probability of walker starting at x hitting Ω beforeescaping to infinityI Charge density is first passage measure on the exterior of ΩI Construct a sphere, S(R), such that Ω S(R)RI Capacitance is C R P(t Ω ) R S(R) u(x)ds, wherethe starting point of the walk is chosen uniformly on the S(R)I Note, this definition is independent of R
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsGeneralization of the Monte Carlo ApproachCapacitance as a Computational ParadigmCapacitance as a Computational ParadigmI We have found that the running time of these codes hinged onthe efficiency of geometrical computationsI The probabilistic capacitance computation is key in1. Permeability computation via Hubbard-Douglas2. Many other electrostatics computations in areas of chemistry,computer science, physics, and materials science, including ourown PBE computations3. In the ZENO code written for materials property computations atNISTI We took the NIST ZENO code and analyzed it to determineareas for improvement
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsGeneralization of the Monte Carlo ApproachCapacitance as a Computational ParadigmThe NIST ZENO CodeI ZENO uses Monte Carlo sampling to compute a variety ofmaterials propertiesI ZENO is a FORTRAN 77 code that had a very suggestive CPUdistanceprofile98.55%(30.43%)309977 22.12%45.99%4097018175 4097842470 listersort22.12%(22.12%)4097018175 minsphere45.99%(7.02%)4097842470 38.98%4097842470 pythag38.98%(38.98%)4097842470
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsGeneralization of the Monte Carlo ApproachCapacitance as a Computational ParadigmWhat is the Computational Geometric Problem?I ZENO is named for the paradox, and getting close the boundaryfor producing approximate first-passage locations is done withthe ZENO algorithm, which is WOS by another nameI The usual geometry in these problems is additiveI As mentioned above the primitive computation geometrically isgiven a query point, find the closest point to the boundary, andthat is used as the radius to construct the WOS radiusI The usual ZENO problem is the so-called exterior problem,which is what is done in permeability: computing the probabilityof first-passage from using relative capacitance of a boundingsphere
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsGeneralization of the Monte Carlo ApproachCapacitance as a Computational ParadigmDifferent Approaches to the Computational GeometryI ZENO used a linear pass thought the additive list ofsubcomponents of the geometry, far from optimalI This is a problem that computational geometers have studied,mostly in very high dimensionsI Here is a comparison of some low-dimensional techniques thatuse a hierarchical k-D tree decomposition of the additivecomponentsTypeLinear ListANN (k-D Tree)NanoFLANN (k-D)CGAL (k-D Tree)Time ble: Zeno C , 1M paths, Intelr Xeonr CPU E5-2630 v2 @ 2.60 GHz
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsGeneralization of the Monte Carlo ApproachParallel ScalabilityParallel Scalability using MPINodes111212Processesper node112112Time 0xTable: Zeno C , 1M paths, Intelr Xeonr CPU E5-2630 v2 @ 2.60 GHz, 12physical & 24 logical coresNodes111212Processesper node112112Time xTable: Zeno C , 10M paths, Intelr Xeonr CPU E5-2630 v2 @ 2.60 GHz,12 physical & 24 logical cores
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsGeneralization of the Monte Carlo ApproachScalability PlotsSingle Node Scalability4518Single Node Scalability: Dual 6-core CPUsTime (s)4016Speedup143012251020815610452Time (s)Speedup3500081624324048Figure: Single Node Scalability5664
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsGeneralization of the Monte Carlo ApproachScalability PlotsAlmost Linear Scalability4550Strong Scalability: 1 Process per Node45404035353030Time e: Scalability Many Nodes5664SpeedupTime (s)25
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsConclusionsConclusionsI Monte Carlo can be an efficient method for the numericalsolution of PDEs as evidenced by1. Financial computing2. Numerous problems in electrostatics3. Problems in computational materialsI The computation of capacitance is a good model problem forcomputational investigation1. Computational geometry is the bottleneck that can be overcomewith proper choice of data-structure/algorithm2. Allows for almost perfect parallelization, across cores and multicorenodesI C version of ZENO uses SPRNG
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsConclusionsFuture WorkI We expect to be able to improve all the codes we havedeveloped1. New C version of ZENO2. Current C version of the WOS-PBE codeI We want to have a WOS-based implementation of thecapacitance code using CUDA on NVidia GPUs1. Must use the analogous implementation that was developed forvectorizing neutron transport2. Will take advantage of the GPU version of SPRNGI Plan to move to nonlinear PB solver using branching processesI Two orders of magnitude improvement with almost perfectparallel scaling will permit the rapid solution of very largeproblems
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsConclusionsBibliography[T. Mackoy, R. C. Harris, J. Johnson, M. Mascagni andM. O. Fenley (2011)] Numerical Optimization of aWalk-on-Spheres Solver for the Linear Poisson-BoltzmannEquation Communications in Computational Physics, in thepress.[A. Rasulov, G. Raimova and M. Mascagni (2010)] Monte CarloSolution of Some Nonlinear Parabolic Initial-Value ProblemsMathematics and Computers in Simulation, 80(6): 1118–1123.[M. Fenley, M. Mascagni, J. McClain, A. Silalahi and N. Simonov(2010)] Using Correlated Monte Carlo Sampling for EfficientlySolving the Linearized Poisson-Boltzmann Equation Over aBroad Range of Salt Concentrations Journal of Chemical Theoryand Computation, 6(1): 300–314.
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsConclusionsBibliography[N. Simonov and M. Mascagni and M. O. Fenley (2007)] MonteCarlo Based Linear Poisson-Boltzmann Approach MakesAccurate Salt-Dependent Solvation Energy Predictions PossibleJournal of Chemical Physics, 187(18), article #185105, 6 pages.[M. Mascagni and N. A. Simonov (2004)] Monte Carlo Methodsfor Calculating Some Physical Properties of Large MoleculesSIAM Journal on Scientific Computing, 26(1): 339–357.[N. A. Simonov and M. Mascagni (2004)] Random WalkAlgorithms for Estimating Effective Properties of Digitized PorousMedia Monte Carlo Methods and Applications, 10: 599–608.[M. Mascagni and N. A. Simonov (2004)] The Random Walk onthe Boundary Method for Calculating Capacitance Journal ofComputational Physics, 195: 465–473.
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsConclusionsBibliography[C.-O. Hwang and M. Mascagni (2003)] Analysis andComparison of Green’s Function First-Passage Algorithms with“Walk on Spheres” Algorithms Mathematics and Computers inSimulation, 63: 605–613.[C.-O. Hwang and M. Mascagni (2001)] Efficient modified “walkon spheres" algortihm for the linearized Poisson-Boltzmannequation Applied Physics Letters, 78: 787–789.[C.-O. Hwang, J. A. Given and M. Mascagni (2001)] TheSimulation-Tabulation Method for Classical Diffusion Monte CarloJournal of Computational Physics, 174: 925–946.[C.-O. Hwang, J. A. Given and M. Mascagni (2000)] On the RapidCalculation of Permeability for Porous Media Using BrownianMotion Paths Physics of Fluids, 12: 1699–1709.
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and MaterialsConclusionsCopyright Notice Michael Mascagni, 2021All Rights Reserved
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and Materials Monte Carlo Methods for PDEs A Little History on Monte Carlo Methods for PDEs Early History of MCMs for PDEs 1.Courant, Friedrichs, and Lewy: Their pivotal 1928 paper has probabilistic interpretations and MC algorithms for linear elliptic
The Markov Chain Monte Carlo Revolution Persi Diaconis Abstract The use of simulation for high dimensional intractable computations has revolutionized applied math-ematics. Designing, improving and understanding the new tools leads to (and leans on) fascinating mathematics, from representation theory through micro-local analysis. 1 IntroductionCited by: 343Page Count: 24File Size: 775KBAuthor: Persi DiaconisExplore furtherA simple introduction to Markov Chain Monte–Carlo .link.springer.comHidden Markov Models - Tutorial And Examplewww.tutorialandexample.comA Gentle Introduction to Markov Chain Monte Carlo for .machinelearningmastery.comMarkov Chain Monte Carlo Lecture Noteswww.stat.umn.eduA Zero-Math Introduction to Markov Chain Monte Carlo .towardsdatascience.comRecommended to you b
Quasi Monte Carlo has been developed. While the convergence rate of classical Monte Carlo (MC) is O(n¡1 2), the convergence rate of Quasi Monte Carlo (QMC) can be made almost as high as O(n¡1). Correspondingly, the use of Quasi Monte Carlo is increasing, especially in the areas where it most readily can be employed. 1.1 Classical Monte Carlo
Computational Geometry 4 Lectures Michaelmas Term 2003 1 Tutorial Sheet Dr ID Reid Overview Computational geometry is concerned with efcient algorithms and representa-tions for geometric computation. Techniques from computational geometry are used in: . Applications of projective transformations. Lecture 3: Convexity of point-sets, convex .
(Computational Information Geometry and Applications). . @misc{jga-compgeom-flatspaces-2009, title "Computational Geometry in Dually Flat Spaces", author "Frank Nielsen", year "2009"} c 2009, Frank Nielsen — p. 2/129. . present generalizations of common algorithms and data-structures in computational geometry: smallest enclosing .
and computational geometry. Finally, we give a few representative applications of computational semi-algebraic geometry in Section 37.11. 969 Preliminary version (July 19, 2017). To appear in the Handbook of Discrete and Computational Geometry, J.E. Goodman, J. O'Rourke, and C. D. Tóth (editors), 3rd edition, CRC Press, Boca Raton, FL, 2017.
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