The Hypotheses on Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity and Their Partial Proof

In this review article we collected more than ten theorems on expansions of iterated Ito and Stratonovich stochastic integrals, which have been formulated and proved by the author. These theorems open a new direction for study of iterated Ito and Stratonovich stochastic integrals. The expansions based on multiple and iterated Fourier-Legendre series as well as on multiple and iterated trigonomectic Fourier series converging in the mean and pointwise are presented in the article. Some of these theorems are connected with the iterated stochastic integrals of multiplicities 1 to 5. Also we consider two theorems on expansions of iterated Ito stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized multiple Fourier series converging in the sense of norm in Hilbert space $L_2([t, T]^k)$ as well as two theorems on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized iterated Fourier series converging pointwise. On the base of the presented theorems we formulate 3 hypotheses on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized multiple Fourier series converging in the sense of norm in Hilbert space $L_2([t, T]^k).$ The mentioned iterated Stratonovich stochastic integrals are part of the Taylor-Stratonovich expansion. Moreover, the considered expansions from these 3 hypotheses contain only one operation of the limit transition and substantially simpler than their analogues for iterated Ito stochastic integrals. Therefore, the results of the article can be useful for the numerical integration of Ito stochastic differential equations. Also, the results of the article were reformulated in the form of theorems of the Wong-Zakai type for iterated Stratonovich stochastic integrals.Comment: 35 pages. Section 12 was added. arXiv admin note: text overlap with arXiv:1712.09516, arXiv:1712.08991, arXiv:1802.04844, arXiv:1801.00231, arXiv:1712.09746, arXiv:1801.0078

APPROXIMATION PROPERTIES OF SOME DISCRETE FOURIER SUMS FOR PIECEWISE SMOOTH DISCONTINUOUS FUNCTIONS

Denote by Ln, N (f, x) a trigonometric polynomial of order at most n possessing the least quadratic deviation from f with respect to the system tk = u + 2πk N N−1 k=0 , where u ∈ R and n 6 N/2. Let D1 be the space of 2π-periodic piecewise continuously differentiable functions f with a finite number of jump discontinuity points −π = ξ1 < . . . < ξm = π and with absolutely continuous derivatives on each interval (ξi , ξi+1). In the present article, we consider the problem of approximation of functions f ∈ D1 by the trigonometric polynomials Ln, N (f, x). We have found the exact order estimate |f(x) − Ln, N (f, x)| 6 c(f, ε)/n, |x − ξi | > ε. The proofs of these estimations are based on comparing of approximating properties of discrete and continuous finite Fourier series

Energy-based comparison between the Fourier--Galerkin method and the finite element method

The Fourier-Galerkin method (in short FFTH) has gained popularity in numerical homogenisation because it can treat problems with a huge number of degrees of freedom. Because the method incorporates the fast Fourier transform (FFT) in the linear solver, it is believed to provide an improvement in computational and memory requirements compared to the conventional finite element method (FEM). Here, we systematically compare these two methods using the energetic norm of local fields, which has the clear physical interpretation as being the error in the homogenised properties. This enables the comparison of memory and computational requirements at the same level of approximation accuracy. We show that the methods' effectiveness relies on the smoothness (regularity) of the solution and thus on the material coefficients. Thanks to its approximation properties, FEM outperforms FFTH for problems with jumps in material coefficients, while ambivalent results are observed for the case that the material coefficients vary continuously in space. FFTH profits from a good conditioning of the linear system, independent of the number of degrees of freedom, but generally needs more degrees of freedom to reach the same approximation accuracy. More studies are needed for other FFT-based schemes, non-linear problems, and dual problems (which require special treatment in FEM but not in FFTH).Comment: 24 pages, 10 figures, 2 table