NONLOCALITY AND STOCHASTICITY TWO EMERGENT DIRECTIONS FOR APPLIED MATHEMATICS Max Gunzburger Department of Scientific Computing Florida State University North Carolina State University, March 10, 2011 s C cientific omputing Florida State University 1/2748
STOCHASTICITY
RECENT DEVELOPMENTS IN NUMERICAL METHODS FOR SPDES Lots about uncertainty in this workshop: Brian Adams Managing Complexity in Simulation-Based Uncertainty Quantification Dan Cacuci Experimentally Validated Best Estimate + Uncertainty Modeling of Complex Systems: The Cornerstone of the Emerging Field of Predictive Science Chris Jones Mathematical Challenges of Climate Research: Data Assimilation and Uncertainty
The ubiquity of uncertainty quantification (UQ) there is uncertainty everywhere uncertainty should be quantified, or else the world will come to an end but deterministic models have served us quite well in lots of situations but knowing about uncertainty might be able to avert disastrous situations, might save money, and might end up with better policies at least that is the hope (or dream) one has about UQ
The ubiquity of probability common definition of UQ uncertainty quantification = given statistics about the inputs of a system, determine statistics about the output of the system but what is usually meant uncertainty quantification = given information about the pdf of the inputs of a system, determine information about the pdf of the output of the system
but, because there are other ways to quantify uncertainty, what it should really be uncertainty quantification = given information about the uncertainty in the inputs of a system, determine information about the uncertainty in the output of the system to a statistician UQ statistics
a word about randomness computations are always deterministic pseudo random number generators are deterministic algorithms that s why computations are repeatable lots of probability and statistics theorems to not apply a word about rare events they are not always rare - just because an event happening has probability zero, does not mean it doesn t happen
The ubiquity of SPDEs everything in the universe can be described by stochastic PDEs what we mean is, perhaps, everything in the world can be described by PDEs with random inputs if this were only true!!!!! to a probabilist SPDE PDE with random inputs
But in this talk, we define stochasticity = probabilistic approach to UQ for systems governed by PDEs SPDE = PDE with random inputs within this talk, we give a short course on numerical methods for SPDEs in particular, we will lecture on recent developments but first, we need to talk about random inputs in particular, we will go over what is assumed (justly or unjustly) about random inputs before we get into the short course about recent developments assumptions marked with a * are assumed for old developments as well
The only thing certain about uncertainty is that it is itself uncertain This is certainly true about uncertain inputs especially those that are defined probabilistically
The ubiquity of independence* random parameters are assumed to be independent - makes life simple - may or may not be true in practice
The ubiquity of hypercubes* random parameters are assumed to be defined on a hypercube (of possible infinite extent) can t easily do constrained parameters, e.g., γ 2 + β 2 14.75 * For all practical purposes this is also true for old developments if one has more than a few parameters
The ubiquity of Gauss or uniform* when in doubt, assume the inputs have a uniform or Gaussian PDF - if you know the mean or if you know a range (or impose one) for a random parameter, assume it is uniformly distributed - if you know the mean and variance, assume it s Guassian need for lots more model calibration determining statistical information about inputs
The ubiquity of making things positive* how can a coefficient in an elliptic equation be Gaussian? no problema assume it is e γ where γ is Gaussian (log-normal coefficient) why?
The ubiquity of correlation it is assumed that random fields are correlated colored noise this means one has to know something about how the random field is correlated one seldom knows how the random field is correlated, so, most often, it is assumed that correlation function = e (x x ) 2 where L = correlation length L 2 or e x x L A random field can be viewed as a function whose value at a point or at an instant of time is random
Now, we are ready for the short course on recent developments in numerical methods for SPDEs
A one-slide, very short course in recent developments in numerical methods for SPDEs SPDE = a partial differential equation with random inputs stochastic finite element method = discretize dependence of solution on spatial variables using a finite element method stochastic Galerkin method = discretize dependence of solution on random parameters using a Galerkin approach Karhunen-Loevy expansion = do an SVD of the correlation matrix to discretize a correlated random field into a finite set of uncorrelated parameters polynomial chaos method = approximate dependence of inputs and/or solutions on parameters using global orthogonal polynomials stochastic sampling method = sample random parameters at a selected set of sampling points, e.g., Monte Carlo sampling stochastic collocation method = sample random parameters at quadrature points for some quadrature rule sparse grid method = use a sparse grid quadrature rule, e.g., a Smolyak rule, to define the sampling points
What has to be assumed about the outputs for these recently developed methods to work? Remember, we already assumed a lot about the inputs for these recently developed methods to be applicable method working = is better than, e.g., Monte Carlo sampling
The ubiquity of the curse obtaining accurate statistics about the output of a system requires multiple simulations of the system or the simulation of a humongous coupled stochastic-spatial-temporal system recent developments in numerical methods for SPDEs are meant to mitigate the curse of dimensionality as the parameter dimension increases, the cost of obtaining accurate statistics grows quickly the idea is that by increasing accuracy, one can reduce costs e.g., the number of sample points can be reduced do polynomial chaos and sparse grid collocation methods really do this? in the best-case scenario, they do somewhat for moderate parameter dimension Very important fact: in principle, the accuracy of Monte Carlo sampling is independent of the parameter dimension
The ubiquity of smoothness the solution must be assumed to possess lots of derivatives with respect to the parameters quantities of interest are also assumed to possess lots of derivatives with respect to the parameters without the necessary smoothness, recently developed methods break down Often, the pointwise solution of the SPDE is not what one is interested in, but rather, a functional of that solution is of interest
THE UBIQUITY OF WHITE NOISE White noise random fields, are by far the most popular means for modeling uncertainty in inputs Gaussian white noise is by far the most popular white noise white noise is often invoked when one knows nothing about the nature of the uncertainty lots of beautiful math about white noise very popular in the modeling community The value of a white noise random field at a point or at an instant of time is identically distributed but is independent of and uncorrelated from its values at other points and other instants of time Recall that polynomial chaos and stochastic collocation only apply to correlated noise
Problemas numero uno: white noise has infinite energy the power spectrum of white noise is flat approximations of white noise such as 1 N t i=1 χ (ti 1,t i ]ω i where ω i, i = 1,..., N, are independently sampled from a given (usually Guassian) PDF, have unboundedly increasing energy as t 0 numero dos: white noise is not observed
Why do we get away with using white noise? one truncates white noise (band limit it to a finite spectrum) of course, then it is not longer white noise is is another model for noise at least for the additive noise case and for problems with smoothing solution operators, the solution is not white in fact, it can be much smoorter Is there an alternative to white noise? one that has finite energy one that is observed pink noise or more, generally, 1/f α noise
Noises having a 1/f α power spectrum α = 0 white noise α = 2 Brownian motion (or red noise) α = 1 pink noise finite energy for α > 1 ubiquitously observed in natural, social, man-made,... systems Is 1/f α noise the noise for you? it may not be what you want, but if you don t know anything about your noise, why not?
Realizations of approximate 1/f α noise for α = 0, 0.5, 1, 1.5, and 2
Corresponding approximate realizations of solution of 1D elliptic problem with 1/f α additive noise
NONLOCALITY
In many situations, atomistic and finer scale models invariably involve nonlocal interactions some interactions are of short range, e.g., effectively extending over a few neighboring atoms Leonard-Jones potential in molecular dynamics some interactions are of long range, e.g., effectively extending over many neighboring atoms Coulomb potential in molecular dynamics Classical continuum models, on the other hand, interactions are local in nature particles interact only through contact one of the truly seminal ideas in modeling the physical world (Cauchy)
Classical models have been extremely useful (this is an understatement!) however Classical models cannot accurately model or completely break down in many important situations plasticity, radiation, fracture,... Classical models, e.g., of elasticity, do not possess a length scale other than those characterizing the domain or that appear in spatially dependent coefficients, e.g., material properties think of the wave equation u tt u xx = f as a result, they cannot be used on their own for multiscale modeling gives rise to, e.g., AtC (atomistic-to-continuum) coupling models for which classical continuum models are coupled to atomistic models for example, classical elasticity coupled to molecular dynamics Thus, there is a huge industry devoted to fixing classical models
Higher-order gradient methods, e.g., au xx replaced by au xx bu xxxx now we have the length scale b/a now we have a multiscale model if b/a is small, then we reduce to the case u xx if b/a is large, then we reduce to the case u xxxx if b/a is neither small nor large, the we have u xx + a b u xxxx Nonlinear rheologies, e.g., nonlinear constitutive laws a length scale can be embedded into the constitutive law very popular in solid mechanics (e.g., plasticity), electromagnetism (e.g., ferromagnetics), fluids (polymers, mantle convection, or ice-sheet flows)
Nonlocal (integral) terms added to differential terms u xx replaced by u xx + k(x, y)u(y)dy a length scale can be easily be made to appear e.g., through an interaction radius for the kernel k(x, y) nonlinear integral terms also popular e.g., to model radiation effects well-used idea in time dependent problems e.g., to model problems with memory or delay also used for spatial modeling in all three types of fixes for classical models, still have spatial derivatives
Why we do not want spatial derivatives derivatives make it difficult to model singular behavior, e.g., crack propagation derivatives make it even more difficult to model nucleation of singularities, e.g., crack nucleation Why we do want nonlocality coupling (local) continuum models to (nonlocal) atomistic models is messy, e.g., ghost force effects especially at small scales, nonlocality is observed, e.g., in diffusion and solid mechanics experiments Why we do want multiscale having a model that accurately describes behaviors over a wide range of scales is the holy grail, or at least, the pot of gold at the end of the rainbow
Ideal model (perhaps) contains a length scale δ is a multiscale model when viewed on a scale of δ, it behaves like a nonlocal, discrete (e.g., atomistic) model when viewed on a scale much larger than δ, it behaves like a local, classical continuum model is free of spatial derivatives is itself a continuum model Such models exist for diffusion and mechanics lots of great math for nonlocal diffusion along with some simulations impressive simulations and some recent math for nonlocal mechanics
NONLOCAL DIFFUSION The parabolic heat equation modeling diffusion: u t (D(x) w(x) ) = b(x, t) in Ω R d The nonlocal equation modeling nonlocal diffusion: with u t + 2 ( u(x ) u(x) ) µ(x, x ) dx = b(x) in Ω R d B δ (x) µ(x, x ) = µ(x, x) 0 B δ (x) = {x Ω : x x δ}
suppose that and Ω Ω µ(x, x ) dx dx = 1 µ(x, x ) = µ(x, x) 0 so that µ(x, x ) can be interpreted as the joint probability density of moving between x and x then ( u(x ) u(x) ) µ(x, x ) dx B δ (x) = u(x )µ(x, x ) dx u(x) B δ (x) B δ (x) µ(x, x ) dx is the rate at which u enters x less the rate at which u departs x
the nonlocal diffusion equation can be derived from a nonlocal random walk you get drift if µ(x, x ) = µ(x, x) the nonlocal diffusion equation is an example of a differential Chapman-Kolmogorov equation
Constrained-value problems for the operator ( Lu = 2 u(x ) u(x) ) µ(x, x ) dx B δ (x) to obtain a solution of the nonlocal problem Lu(x) = b(x) data has to be specified on measurable sets thus, for example, we have the constrained-value problem { Lu(x) = b(x) for x Ω u(x) = g(x) where meas(ω o ) 0) and Ω Ω o = for x Ω o this is the nonlocal analog of the Dirichlet boundary-value problem for a second-order elliptic operator nonlocal analogs of Neumann problems can also be defined
What about spaces for well posedness of constrained-value problem? for appropriate (singular) kernels µ(x, x ) L : H s (Ω) H s (Ω) for 0 s 1 this is in contrast to second-order elliptic operators that map H 1 0(Ω) H 1 (Ω) this is also in contrast to fractional Laplacians that can map H s (Ω) H s (Ω) for s 1 2 note that for kernels such that s < 1 2, L is bounded when acting on functions with jump discontinuities note that the case s = 0 is allowable solving the constrained value problem results in no smoothing
Connections to differential operators if u(x) is smooth, then, L(u) second-order elliptic operator as δ 0 Nonlinear versions of nonlocal constrained-value problems are easily defined
NONLOCAL MECHANICS Generalizations of the nonlocal equation ρu tt = 2 µ(x, x ) ( u(x ) u(x) ) dx B δ (x) define nonlocal models for mechanics the peridynamics model for mechanics Everything said about the nonlocal diffusion operator holds for the operator 2 µ(x, x ) ( u(x ) u(x) ) dx B δ (x) and its peridynamic generalizations constrained-value problems analogous to boundary-value problems in classical elasticity can be defined reduced smoothness gains with respect to the data for solutions of constrainedvalue problems can again be obtained, including the L 2 L 2 case
for appropriately chosen kernels, the operator is bounded when acting on functions with jump discontinuities generalizations of the operators reduce to, as δ 0, to classical linear elasticity operators nonlinear versions can be defined Furthermore, peridynamics has been shown to reduce to molecular dynamics when viewed at the atomistic scale Furthermore, for appropriate kernels, displacements u having jump discontinuities are allowable cracks are allowable Generalizations are easily defined that also allow for the nucleation of cracks
Constrained-value problem in peridynamics may be discretized by a collocation method to obtain a discrete system that looks like the equations for a system of interacting particles in fact, peridynamics has been incorporated at Sandia into their molecular dynamics software package LAMMPS
Implications for finite element methods (based on weak formulations of constrainedvalue problems) discontinuous finite element spaces are conforming optimal accuracy can be obtained by using abrupt grid refinement, this remains true even if solutions contain jump discontinuities
a multiscale computational model is now realizable by merely varying the grid size h and choosing appropriate finite element spaces where nothing bad is happening, choose h δ standard continuous finite element spaces discrete equations are automatically the same as that for the finite element discretization of the corresponding PDE model where bad things are happening, choose h < δ discontinuous finite element spaces discrete equations automatically reduce to particle dynamics-type equations
Dispersion behavior