Matrix-valued stochastic processes

Transcription

1 Matrix-valued stochastic processes and applications Małgorzata Snarska (Cracow University of Economics) Grodek, February 2017 M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

2 Outline 1 Introduction 2 Challenges in dynamic systems analysis 3 Random matrix approach 4 A short introduction to free probability calculus 5 Additive matrix diffusion M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

3 Research objectives M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

4 Research objectives Aim: Introduce methods of free probability theory in the context of stochastic diffusion theory of random matrices and discuss its potential applications in vibroacoustic analysis. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

5 Research objectives Aim: Introduce methods of free probability theory in the context of stochastic diffusion theory of random matrices and discuss its potential applications in vibroacoustic analysis. Notice: M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

6 Research objectives Aim: Introduce methods of free probability theory in the context of stochastic diffusion theory of random matrices and discuss its potential applications in vibroacoustic analysis. Notice: the formalism is applied to additive matrix diffusion ie. a matrix analogue of a Brownian motion. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

7 Research objectives Aim: Introduce methods of free probability theory in the context of stochastic diffusion theory of random matrices and discuss its potential applications in vibroacoustic analysis. Notice: the formalism is applied to additive matrix diffusion ie. a matrix analogue of a Brownian motion. heavily relies on Random Matrix Theory. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

8 Research objectives Aim: Introduce methods of free probability theory in the context of stochastic diffusion theory of random matrices and discuss its potential applications in vibroacoustic analysis. Notice: the formalism is applied to additive matrix diffusion ie. a matrix analogue of a Brownian motion. heavily relies on Random Matrix Theory. the formalism is originally applied to large matrices of size N N with N but works well in N > 10 M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

9 Dynamic systems analysis and challenges Major challenges in the analysis of the dynamic response of structural and dynamic systems at higher frequencies M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

10 Dynamic systems analysis and challenges Major challenges in the analysis of the dynamic response of structural and dynamic systems at higher frequencies computational efficiency (short wavelengths that appear at high frequencies, how can the response of the structure be computed without using an excessive number of degrees of freedom? M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

11 Dynamic systems analysis and challenges Major challenges in the analysis of the dynamic response of structural and dynamic systems at higher frequencies computational efficiency (short wavelengths that appear at high frequencies, how can the response of the structure be computed without using an excessive number of degrees of freedom? robustness: Given that the local response at higher frequencies is very sensitive to wave scattering caused by spatial variations in geometry, material properties, and boundary conditions, how can the uncertainty of the response, due to the very many uncertain model parameters that affect it, be quantified? M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

12 Dynamic systems analysis and challenges Major challenges in the analysis of the dynamic response of structural and dynamic systems at higher frequencies computational efficiency (short wavelengths that appear at high frequencies, how can the response of the structure be computed without using an excessive number of degrees of freedom? robustness: Given that the local response at higher frequencies is very sensitive to wave scattering caused by spatial variations in geometry, material properties, and boundary conditions, how can the uncertainty of the response, due to the very many uncertain model parameters that affect it, be quantified? evolution of an acousting system in time? M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

13 Standard approaches pros and cons M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

14 Standard approaches pros and cons Deterministic element-based methods such as the finite element method (FEM) and the boundary element method (BEM) in their standard form have a high computation cost even for a single deterministic model. Randomization increases computation cost even more. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

15 Standard approaches pros and cons Deterministic element-based methods such as the finite element method (FEM) and the boundary element method (BEM) in their standard form have a high computation cost even for a single deterministic model. Randomization increases computation cost even more. Statistical energy analysis- The system is decomposed into subsystems homogenous and weakly coupled.the response for a subsystem is a single random variable, its total vibrational energy, and the interaction between subsystems is described by means of a power balance. Computationally very efficient (the response of each subsystem is global and due to a nonparametric model of uncertainty, which implies that the detailed statistics of the underlying physical parameters are not needed for computing the energy statistics. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

16 SEA limitations M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

17 SEA limitations only the marginal probability distribution of the response can be predicted with reasonable accuracy, and only when all random subsystems have sufficiently high modal overlap and when they are loaded in a specific way. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

18 SEA limitations only the marginal probability distribution of the response can be predicted with reasonable accuracy, and only when all random subsystems have sufficiently high modal overlap and when they are loaded in a specific way. The probability distribution of the response is needed for computing, e.g., the probability that a certain response level is exceeded, or for computing confidence intervals on system response quantities. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

19 SEA limitations only the marginal probability distribution of the response can be predicted with reasonable accuracy, and only when all random subsystems have sufficiently high modal overlap and when they are loaded in a specific way. The probability distribution of the response is needed for computing, e.g., the probability that a certain response level is exceeded, or for computing confidence intervals on system response quantities. fundamental assumption that all subsystems are weakly coupled to each other and failure of SEA has been observed for systems that are strongly coupled according to some measure M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

20 What is a Random Matrix and why do we care? M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

21 What is a Random Matrix and why do we care? The matrix is everywhere" M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

22 What is a Random Matrix and why do we care? The matrix is everywhere" A random matrix is a matrix whose elements are random variables populated from certain distribution. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

23 What is a Random Matrix and why do we care? The matrix is everywhere" A random matrix is a matrix whose elements are random variables populated from certain distribution. Take N N matrix (N possibly large) and populate its elements from eg. Gaussian distribution M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

24 What is a Random Matrix and why do we care? The matrix is everywhere" A random matrix is a matrix whose elements are random variables populated from certain distribution. Take N N matrix (N possibly large) and populate its elements from eg. Gaussian distribution look at eigenvalues of such a matrix M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

25 What is a Random Matrix and why do we care? The matrix is everywhere" A random matrix is a matrix whose elements are random variables populated from certain distribution. Take N N matrix (N possibly large) and populate its elements from eg. Gaussian distribution look at eigenvalues of such a matrix What happens with eigenvalues if N goes larger? How does it scale with N? M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

26 What is a Random Matrix and why do we care? The matrix is everywhere" A random matrix is a matrix whose elements are random variables populated from certain distribution. Take N N matrix (N possibly large) and populate its elements from eg. Gaussian distribution look at eigenvalues of such a matrix What happens with eigenvalues if N goes larger? How does it scale with N? How spectrum of eigenvalues evolve in time? M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

27 Connection with dynamical systems M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

28 Connection with dynamical systems Historical example: Sinai billiard is a chaotic dynamical system. Even a tiny small exterior force (like the gravitational force of a person watching the ball) influences the trajectory essentially after a few impacts. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

29 Connection with dynamical systems Historical example: Sinai billiard is a chaotic dynamical system. Even a tiny small exterior force (like the gravitational force of a person watching the ball) influences the trajectory essentially after a few impacts. the description of the system is nontrivial M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

30 Connection with dynamical systems Historical example: Sinai billiard is a chaotic dynamical system. Even a tiny small exterior force (like the gravitational force of a person watching the ball) influences the trajectory essentially after a few impacts. the description of the system is nontrivial Assume no initial knowledge of the system. Then instead of looking at the levels/excitations give the statistical description of data and look at the distribution of resonances (Wigner, 1951) M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

31 Connection with dynamical systems Historical example: Sinai billiard is a chaotic dynamical system. Even a tiny small exterior force (like the gravitational force of a person watching the ball) influences the trajectory essentially after a few impacts. the description of the system is nontrivial Assume no initial knowledge of the system. Then instead of looking at the levels/excitations give the statistical description of data and look at the distribution of resonances (Wigner, 1951) Probabilistic approach (due to Wigner): unknown random source is sending the streams of signal with certain probabilities. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

32 RMT based approach M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

33 RMT based approach Also employs a nonparametric model of uncertainty for spatial variations in geometry, material properties and boundary conditions. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

34 RMT based approach Also employs a nonparametric model of uncertainty for spatial variations in geometry, material properties and boundary conditions. the response of a nonparametric random subsystem is characterized by its full displacement field and not total energy. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

35 RMT based approach Also employs a nonparametric model of uncertainty for spatial variations in geometry, material properties and boundary conditions. the response of a nonparametric random subsystem is characterized by its full displacement field and not total energy. it is possible to couple a nonparametric random subsystem to any other type of subsystem and no additional assumptions are needed for computing the probability distribution of the response. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

36 RMT based approach Also employs a nonparametric model of uncertainty for spatial variations in geometry, material properties and boundary conditions. the response of a nonparametric random subsystem is characterized by its full displacement field and not total energy. it is possible to couple a nonparametric random subsystem to any other type of subsystem and no additional assumptions are needed for computing the probability distribution of the response. use of the fact that, when a homogenous elastodynamic system is highly sensitive to wave scattering caused by spatial variations in geometry, material properties, and boundary conditions, the statistics of its undamped eigenvalues and mode shapes saturate to universal distributions. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

37 RMT based approach Also employs a nonparametric model of uncertainty for spatial variations in geometry, material properties and boundary conditions. the response of a nonparametric random subsystem is characterized by its full displacement field and not total energy. it is possible to couple a nonparametric random subsystem to any other type of subsystem and no additional assumptions are needed for computing the probability distribution of the response. use of the fact that, when a homogenous elastodynamic system is highly sensitive to wave scattering caused by spatial variations in geometry, material properties, and boundary conditions, the statistics of its undamped eigenvalues and mode shapes saturate to universal distributions. the statistics of the local spacings between the eigenvalues saturate to those of the Gaussian orthogonal ensemble (GOE) matrix from random matrix theory. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

38 Concept of freeness RMT can be viewed as a generalization of the classical probability calculus, where the concept of probability density distribution for a one-dimensional random variable is generalized onto an averaged spectral distribution of the ensemble of large, non-commuting random matrices. Such a structure exhibits several phenomena known in classical probability theory, including central limit theorems Actually, the celebratedwigner semicircle lawfor the eigenvalue distribution of large matrices with a finite second spectral moment is nothing but the matrix-analogue for similar limiting distribution in the classical probability theory, i.e. the Gaussian distribution. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

39 Free probability theory and freeness M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

40 Free probability theory and freeness The original idea of Free Probability Theory originated in the context of non commuting random variables ( situation, when the order of operands changes the results of calculation). M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

41 Free probability theory and freeness The original idea of Free Probability Theory originated in the context of non commuting random variables ( situation, when the order of operands changes the results of calculation). In the matrix case when calculating covariance between entries of two random matrices one uses unital algebra A (complex von Neumann algebra C of continuous linear operators on also complex Hilbert space) equipped with a noncommutative expectation (linear functional): such that φ : A C φ(1) = 1. Two matrices are then freely independent if and only if the expectation of the product a 1..., a n is zero. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

42 Free probability theory and freeness The original idea of Free Probability Theory originated in the context of non commuting random variables ( situation, when the order of operands changes the results of calculation). In the matrix case when calculating covariance between entries of two random matrices one uses unital algebra A (complex von Neumann algebra C of continuous linear operators on also complex Hilbert space) equipped with a noncommutative expectation (linear functional): such that φ : A C φ(1) = 1. Two matrices are then freely independent if and only if the expectation of the product a 1..., a n is zero. Mixed moments are combinations of products of individual moments and do not factorize into separate moments. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

43 Free Probability calculus in a nutshell Classical Probability independence x - random variable, p(x) pdf characteristic function logarithm of characteristic function f.generating moments addition law multiplication law Central Limit Theorem Gaussian FRV freeness H - random matrix, P(H) spectral density ϱ(λ)dλ Green s function (Stieltjes transform) 1 G(z) = N Tr 1 z 1 H = 1 n=0 1 z n+1 N TrHn R transform 1 N TrHn = dλϱ(λ)λ n R 1+2 (z) = R 1 (z) + R 2 (z) whereg [ R(z) + 1 ] z = z S 1 2 (z) = S 1 (z) S 2 (z) Free Central Limit Theorem Semicircle M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

44 Diffusion in the matrix case Remind the diffusion process, where stochastic variable y undergoes an evolution dy = µdt + σdx t where dx t is the Wiener process obeying dx t = 0, dx 2 t = dt and µ represents the drift. After averaging over independent identical distributions (iid) of the Gaussian variables, we recover the well-known solution for the probability density fulfilling the heat equation. We would like to parallel the above construction, in the case when the independent increments in the diffusion process are replaced by free, large and non-commuting random matrices. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

45 Construction of diffusion process M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

46 Construction of diffusion process Lets start with division of time interval [0, t] into K infinitesimal steps and X i belong to independent free GOEs. Natural, finite time-interval definition of the Gaussian increment is Y i = t/kx i. Each matrix measure fulfils Y = 0, Y 2 dt where dt = t/k, behaves as an analogue of the Wiener measure. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

47 Construction of diffusion process Lets start with division of time interval [0, t] into K infinitesimal steps and X i belong to independent free GOEs. Natural, finite time-interval definition of the Gaussian increment is Y i = t/kx i. Each matrix measure fulfils Y = 0, Y 2 dt where dt = t/k, behaves as an analogue of the Wiener measure. We can now study the problem, how some initial spectral distribution (e.g. Wigner semicircle) behaves under a series of identical, free increments of Gaussian origin. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

48 Construction of diffusion process Lets start with division of time interval [0, t] into K infinitesimal steps and X i belong to independent free GOEs. Natural, finite time-interval definition of the Gaussian increment is Y i = t/kx i. Each matrix measure fulfils Y = 0, Y 2 dt where dt = t/k, behaves as an analogue of the Wiener measure. We can now study the problem, how some initial spectral distribution (e.g. Wigner semicircle) behaves under a series of identical, free increments of Gaussian origin. note that the Green s function of the matrix ensemble t/kx is related to the R-transform R(z) = (t/k)z, it corresponds to the rescaling of the variance. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

49 Additive diffusion ctd. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

50 Additive diffusion ctd. The algebraic Brownian walk is then realized trivially as a sum of subsequent R transformations R(z) = lim R K (z) = lim R i = K t K K K z = tz and the corresponding Green s function G(z) = 1 2τ (z z 2 4t) fulfils the complex Burgers equation t G(z, t) + G(z, t) z G(z, t) = 0 suplemented by the initial condition G(z, t = 0) = 1 2 (z z 2 4) K M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

51 Additive diffusion ctd. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

52 Additive diffusion ctd. By taking the imaginary part, we get the evolving-in-time semi-circle law. The edges of the spectrum diffuse with time, i.e. λ 2 t, represents diffusion in the space of the eigenvalues of the ensemble. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

53 Additive diffusion ctd. By taking the imaginary part, we get the evolving-in-time semi-circle law. The edges of the spectrum diffuse with time, i.e. λ 2 t, represents diffusion in the space of the eigenvalues of the ensemble. Hence the complex Burgers equation is a FRV analogue of the heat equation, corresponding to the standard convolution of one-dimensional Gaussian variables. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

54 References: Reynders, E., Legault, J., Langley, R. S. (2014). An efficient probabilistic approach to vibro-acoustic analysis based on the Gaussian orthogonal ensemble. The Journal of the Acoustical Society of America, 136(1), Biane, P., Speicher, R. (1998). Stochastic calculus with respect to free Brownian motion and analysis on Wigner space. Probability theory and related fields, 112(3), Gudowska-Nowak, E., Janik, R. A., Jurkiewicz, J., Nowak, M. A. (2005). On diffusion of large matrices. New Journal of Physics, 7(1), 54. M. Snarska (UEK) Matrix Diffusion Grodek, February / 18

55 Thank you! M. Snarska (UEK) Matrix Diffusion Grodek, February / 18