WORD SERIES FOR THE ANALYSIS OF SPLITTING SDE INTEGRATORS Alfonso Álamo/J. M. Sanz-Serna Universidad de Valladolid/Universidad Carlos III de Madrid 1
I. OVERVIEW 2
The importance of splitting integrators for (d/dt)x = f a (x) + f b (x) + keeps increasing: evolutionary PDEs, geometric integration,... Format: maps that advances one step given by composition ψ h = φ a λh φb µh, where h is the stepsize and φ a h, φb h,... are exact solution flows of (d/dt)x = f a (x), (d/dt)x = f b (x),... 3
Standard analysis of ψ h starts by seeing φ a h, φb h,... as (noncommuting) exponentials. BCH formula then used to write ψ h as a single exponential. The exponent provides the modified differential equation of the integrator, ie the (h-dependent) equation whose h flow formally coincides with ψ h. Difference between modified and true (d/dt)x = f a (x) + f b (x) + equations yields information on the properties of the integrator. 4
This is an indirect approach based on comparing differential equations rather than maps. It stands apart from usual techniques in numerical ODEs based on expansion of integrator map ψ h. For Runge-Kutta and related integrators the expansion of ψ h is best carried out by means of B-series (Hairer and Wanner 1974). B-series are parameterized by rooted trees. 5
Murua and SS (1999) developed a B-series technique to analyze splitting integrators (Hairer, Lubich, Wanner, GI Book, Chapter III.3). Recently Murua and SS suggested word series as an alternative to the B-series. Word series are parameterized by words from an alphabet. Use of word series based on simple algebraic systematic manipulations. 6
In numerical analysis, word series may be used to study the order of consistency of splitting integrators, find modified equations, etc. Outside numerical mathematics, word series may be used (Chartier, Murua, SS 2010 2015) to perform averaging, to find normal forms, to compute integrals of motion, etc. In this talk we present an example of the use of word series to analyze splitting methods for SDEs. (Also: Ito SDEs, modified equations,... ) 7
II. DEFINING WORD SERIES 8
A MOTIVATING (DETERMINISTIC) EXAMPLE: dx = a A λ a (t) dtf a (x). A finite or infinite index set (alphabet). f a (x) vector field in D dimensions. λ a (t) scalar-valued function. 9
Solution with x(0) = x 0 given by (cf Chen series) x(t) = x 0 + J a1 a n (t)f a1 a n (x 0 ), n=1 a 1,...,a n A with J a1 a n (t) given by t 0 λ an (t n ) dt n tn 0 λ an 1 (t n 1 ) dt n 1 t2 0 λ a1 (t 1 ) dt 1, and, recursively, f a1 a n (x) = x f a2 a n (x) f a1 (x). Note separation: λ a, J a1 a n / f a, f a1 a n. 10
WORDS AND THEIR BASIS FUNCTIONS W denotes set of all words from the alphabet A (including the empty word ); W n is the set of words with n letters. Associate with w = a 1 a n W its word basis function f w (x) = f a1 a n (x) (with f (x) x). (Similar to elementary differentials.) 11
DEFINITION If δ C W (ie δ maps words into scalars), the word series with coefficients δ is the formal series: W δ (x) = δ w f w (x). w W Note that the notion of word series is relative to the f a. For each t, the solution value x(t) corresponds to the coefficients J w (t) built above from the λ a (t). 12
SOME PARTICULAR CASES: EXAMPLE I: A consists of a single letter a. For each n there is a single word w = a a with n letters. If furthermore λ a (t) 1, J w (t), w W n is found to be t n /n!. The word series W α(t) (x 0 ) is the Taylor expansion in powers of t of the solution x(t) of (d/dt)x = f a (x), x(0) = x 0. 13
EXAMPLE II: A consists of a two letters a, b. For each n there are 2 n words with n letters. If furthermore λ a (t) 1, λ b (t) 1, α w (t) = t n /n! for w W n. The word series W J(t) (x 0 ) is the Taylor expansion of x(t), (d/dt)x = f a (x) + f b (x), written in terms of f a, f b. 14
STOCHASTIC SDEs: dx = f a (x)dt + f A db(t) (Stratonovich), fits in this framework with λ a (t) 1 and λ A (t)dt = db(t). Solution formally given by W J(t) (x 0 ) where now J w (t) is a stochastic process. (Stochastic Taylor expansion see Kloeden and Platen, Chapter 5.) 15
III. OPERATING WITH WORD SERIES 16
THE CONVOLUTION PRODUCT If δ, δ C W, their convolution product δ δ C W is (δ δ ) a1 a n = δ δ a 1 a n + Not commutative. n 1 j=1 δ a1 a j δ a j+1 a n + δ a1 a n δ Associative. Unit: 1 C W with 1 = 1 and 1 w = 0 for w. 17
THE GROUP G (Group of characters of shuffle Hopf algebra) Let denote shuffle product of words. (Eg ab c = abc + acb + cab.) The set G of those γ C W that satisfy the so-called shuffle relations: γ = 1 and, for each w, w W, γ w γ w = N γ wj if w w = N w j. j=1 j=1 is a noncommutative group for. 18
For fixed t (and ω Ω), the coefficients J w (t), w W satisfy the shuffle relations, ie they give an element of G. For γ G, the series W γ (x) has special properties: (1) W γ (x) acts on the vector space of all word series by composition: W δ ( Wγ (x) ) = W γ δ (x), δ C W. 19
(2) If χ : R D R is any smooth observable: χ ( W γ (x) ) = γ w D w (χ)(x), w W where, for a A, D a = f a is the Lie operator associated with f a and for w = a 1... a n, D w is the differential operator D a1... D an. (3) Equivariance with respect to arbitrary changes of variables x = x(x). 20
IIII. SPLITTING INTEGRATORS FOR SDES 21
Consider integrators for Stratonovich equation dx = f a1 (x)dt + + f an (x)dt +f A1 (x)db A1 + + f AN (x)db AN such that the one-step map ψ h is composition of flows of individual pieces dx = f ai (x)dt or dx = f Aj (x)db Aj (or blocks, eg dx = f a1 (x)dt + f A2 (x)db Aj ). Each flow entering the composition is (for fixed t and ω Ω) a word series with known coefficients in G. 22
By property (1), ψ h = W γ(t) (x), where the coefficients γ w (t) are readily found by using the convolution formula. γ w (t) compared with coefficients J w (t) of true solution. For strong local error to be O(h p ), p = 3/2, 2, 5/2,... γ w (t) = J w (t) as for all words of weight p 1/2 (weight = number of lower case letters + number of upper case letters/2). (This assumes all relevant basis functions are 0.) 23
By property (3) E (χ ( W γ (t)(x) )) = w W E ( γ w (t) ) D w (χ)(x), For weak local error to be O(h p ), p = 2, 3,..., E ( γ w (t) ) = E ( J w (t) ), for all words of weight p 1. (This assumes all relevant basis functions are 0.) 24
V. LANGEVIN DYNAMICS 25
Consider dq = M 1 p dt dp = F (q) dt γp dt + σm 1/2 db, M diagonal with diagonal elements m i > 0, i = 1,..., d, γ > 0, B d-dimensional Wiener process. 26
Leimkuhler and Matthews (2013) consider several integrators based on the split systems: a: dq = M 1 p dt, dp = 0. b: dq = 0, dp = F (q) dt. o: dq = 0, dp = γp dt + σm 1/2 db. 27
They use acronym aboba for the Strang-like splitting ψ h = φ a h/2 φb h/2 φo h φb h/2 φa h/2 ; baoab defined similarly. Both aboba and baoab possess strong local error O(h 3/2 ); weak O(h 3 ). In spite of similarity, baoab turns out to be clearly superior to aboba. 28
We use word series based on the vector fields: a: (M 1 p, 0). b: (0, F (q)). c: (0, γp). A j : (0, σ m j e j ), j = 1,..., d. 29
Sparsity pattern results in many zero word basis functions. For weight < 3, f w 0 only for A j, A j a, A j c, A j ab, A j ca, A j cc and some purely deterministic words (that cause no trouble). The coefficients of the methods in the word series expansion are easily found to be: 30
w weight aboba baoab A j 1/2 A j a 3/2 (h/2)j Aj (h/2)j Aj A j c 3/2 A j ab 5/2 0 (h 2 /4)J Aj A j ca 5/2 (h/2)j Aj c (h/2)j Aj c A j cc 5/2 Only difference in A j ab, where baoab provides a better approximation: correlation truth-approximation is 5/3 74% for baoab, 0 for aboba. 31
Symmetrically, for word A j ba, baoab has γ Aj ba = 0, while aboba has γ Aj ba = (h 2 /4)J Aj. But then the basis function is zero and aboba does not benefit from it. The same phenomena appear for longer words. Word analysis reveals that the source of discrepancy is the following algorithmic flaw in aboba: in any given time step, it uses the same value of the force in both kicks of a given step. 32