Numerical approximation of stochastic evolution equations: Convergence in scale of Hilbert spaces
The paper is devoted to the numerical approximation of a very general stochastic nonlinear evolution equation in a separable Hilbert space . Examples of such equations which fall into our framework include the GOY and sabra shell models and a nonlinear heat equation. The space-time numerical scheme is defined in terms of a Galerkin approximation in space and by a semi implicit finite difference approximation in time (Rothe approximation). We prove the convergence in probability of our scheme. This is shown by means of an estimate of the error on a localized set of arbitrary large probability. Let us mention that our error estimate is shown to hold in a more regular space with and that the speed of convergence depends on this parameter . Also, an explicit rate is given as a consequence.
Throughout this paper we fix a complete filtered probability space with the filtration satisfying the usual conditions. Throughout this paper we fix a separable Hilbert space equipped with a scalar product with the associated norm . We fix also another separable Hilbert space .
In this paper, we analyze numerical approximations of the following stochastic evolution equation
where is a self-adjoint positive operators on , , and are nonlinear maps satisfying several technical assumptions to be specified later and a -valued Wiener process.
The abstract equation can describe several problems from different fields including mathematical finance, electromagnetism, and fluid dynamic. Stochastic models are well fitted to describe small fluctuations or perturbations which arises in nature. The stochastic Navier–Stokes equations is used to describe fully turbulent fluids via the celebrated K41 theory [?], see also [?] and reference therein. The reader interested to other models involving the introduction of a stochastic term are invited to consult [?].
Due to the nonlinearity , it is very difficult to derive an exact solution (or closed-form solution) stochastic evolution equations such as , a numerical resolution is then inevitable. The field of numerical analysis for SPDEs or stochastic evolution equations is still young and the numerical approximation of these equations is not trivial when the nonlinear term is not Lipschitz which is the case for stochastic Navier–Stokes equations type. Recent works involving a bilinear term have been done including [?]. In [?] a weak martingale to the incompressible Navier-Stokes equations with Gaussian multiplicative noise is constructed from a convergent finite element based on space-time discretization, and the authors of [?] proved the convergence rates of an explicit and an implicit numerical schemes by means of a Gronwall argument. The main issue when the term is not Lipschitz lies on its interplay with the stochastic forcing, which prevents a Gronwall argument in the context of expectations. This issue is for example solved in [?] by the introduction of a weight, which when carefully chosen will contribute to remove unwanted terms and allow to use Gronwall lemma. In [?], the authors use different approach by computing the error estimates on a sample subset with large probability. In particular, the set is carefully chosen so that the random variables are bounded as long as the events are taken in , and . Then, standard machinery using the Gronwall lemma are performed.
In this paper, we discretize using a coupled Galerkin method and (semi-)implicit Euler scheme and show convergence with rates. Regardless of the approach, this work is similar to [?] but in a smaller space, i.e. we use a Gronwall argument and utilize Hölder continuity in time of in some Hilbert space to arrive at convergence in probability in with rate where is arbitrary. In contrast to the nonlinear term of Navier–Stokes equations with periodic boundary condition treated in [?], our nonlinear term is allowed to non-bilinear and it does not satisfy the property which plays a crucial role in the analysis in [?]. Examples of semilinear equations which fall into our framework include the GOY and sabra shell models. These models are created in view of simulating turbulence, but it seems that our work is the first one rigorously addressing their numerical analysis. Our result also confirm that, in term of numerical analysis, the shell models behave far better than the Navier-Stokes. Another example such as a nonlinear heat equation are described in Section 5. We should also note that we also give a new and simple proof of the existence of solutions to stochastic shell models driven by Gaussian multiplicative noise.
This paper is organized as follows: In Section 2, we introduce the necessary notations and the standing assumptions that will be used in the present work. In Section 3, we present our numerical scheme and also discuss the stability and existence of solution at each time step. The convergence of the proposed method in the mean-square sense is presented in Section 4. In Section 5 we present the stochastic shell models for turbulence and stochastic nonlinear heat equation as a motivating examples.
2Notations, assumptions, preliminary results and the main theorem
In this section we introduce the necessary notations and the standing assumptions that will be used in the present work. We will also introduce our numerical scheme and state our main result.
2.1Assumptions and notations
Throughout this paper we fix a separable Hilbert space with norm and a fixed orthonormal basis . We assume that we are given a linear operator which is a self-adjoint and positive operator such that the fixed basis satisfies
for an increasing sequence of positive numbers with . as . It is clear that is the infinitesimal generator of an analytic semigroup , on . For any the domain of denoted by is a separable Hilbert space when equipped with the scalar product
The norm associated to this scalar product will be denoted by , . In what follows we set .
Next, we consider a nonlinear map satisfying the following set of assumptions, where hereafter denotes the dual of the Banach space .
There exists a constant such that for any and satisfying , we have
If , then we assume that there exists a constant such that holds with replaced by .
In addition to the above, we also assume that for any there exists a constant such that
There exists a positive number such that for any
Without of loss of generality we will assume that for the remaining part of the paper.
We assume that for any we have
Note that Assumptions ? and ? imply
There exists a constant such that for any numbers and satisfying , we have
If , then we assume that there exists a real number such that holds with replaced by .
Now, let be a complete filtered probability space where the filtration satisfies the usual condition. Let be a sequence of mutually independent and identically distributed standard Brownian motions on . Let be separable Hilbert space and be the space of all trace class operators on . Recall that if is a symmetric , positive operator and is an orthonormal basis of consisting of eigenvectors of , then the series
where are the eigenvalues of , converges in and it defines an -valued Wiener process with covariance operator . Furthermore, for any positive integer there exists a constant such that
for any with . Before proceeding further we recall few facts about stochastic integral. Let be a separable Hilbert space, be the space of all bounded linear -valued operators defined on , be the space of all equivalence classes of -progressively measurable processes satisfying
If is a symmetric, positive and trace class operator then and for any we have , where (with ) is the Hilbert space of all operators satisfying
Furthermore, from the theory of stochastic integration on infinite dimensional Hilbert space, for any the process defined by
is a -valued martingale. Moreover, we have the following Itô isometry
and the Burkholder-Davis-Gundy inequality
Now, we impose the following set of conditions on the nonlinear term and the Wiener process .
Let be a separable Hilbert space. We assume that the driving noise is a -valued Wiener process with a positive and symmetric covariance operator .
We assume that for the nonlinear function maps to and that there exists a constant such that for any , we have
In this subsection we recall and derive some preliminary results that we will be using in the remaining part of the paper. To this end, we first define the notion of solution of .
Next, we recall the following result.
Let us first prove the existence of a local mild solution. For this purpose, we study the properties if in order to apply a contraction principle as in [?]. Let be the mapping defined by for any . Let . Using Assumptions ? with , , we derive that
for any . Since, by [?], coincide with the complex interpolation , we infer from the interpolation inequality [?] and that
for any . Now, we denote by the Banach space endowed with the norm
We recall the following classical result (see [?])
Thus, thanks to , and Assumption ? we can apply [?] to infer the existence of a unique local mild solution with lifespan of (we refer to [?] for the definition of local solution). Let be an increasing sequence of stopping times converging almost surely to the lifespan . Using the equivalence lemma in [?] we can easily prove that the local mild solution is also a local weak solution satisfying with replaced by , . Now, we can prove by arguing as in [?] or [?] that the local solution satisfies uniformly w.r.t. . With this observation along with an argument similar to [?] we conclude that admits a global solution satisfying and almost-surely. The proof of is very similar to that of , hence below we will only show .
To prove it suffices to treat the case , in particular . To start with we will formally apply the Itô formula to . The following calculations can be rigorously justified by using the Galerkin truncation.
Applying the Itô formula to the functional
which along with the inequality , where the norm is understood as the norm of a bilinear map, implies
Since the embedding is continuous for any , we can use Assumptions ? and the Cauchy inequality to infer that
for some . By complex interpolation inequality in [?] we have
Plugging the latter inequality into , using the assumption on we obtain
Taking the supremum over , then raising both sides of the resulting inequality to the power , taking the mathematical expectation, using the Burkholder-Davis-Gundy inequality yield
Here we have used the fact that for any integer and we can find a constant such that
for a sequence of non-negative numbers .
Invoking , , the assumption on we derive that and the Gronwall lemma we have the following chain of inequalities
Now, we apply Gronwall’s inequality for the function to infer that
from which along with the former estimate we readily complete the proof of proposition.
An -adapted process -a.s. is called a mild solution to if for every and -a.s,
We will use this remark later on to prove a very important lemma for our analysis, see Lemma ?.
2.3The numerical scheme and the main result
Let be a positive integer, the linear space spanned by , and the orthogonal projection of to the finite dimensional subspace . The projection of by is denoted by
for and The Galerkin projection of the SPDEs reads
Due to the assumption ?- ? and ?, we can use Proposition ? to prove that has global weak solution.
To derive an approximation of the exact solution of we construct an approximation of the Galerkin solution. To this end, let be a positive integer and an equidistant grid of mesh-size covering . Now, for any we look for a sequence of -adapted random variables , such that for any
where , , is an independently and identically distributed random variables. We will justify in the following proposition that for a given the numerical scheme admits at least one solution , and that is stable in and .
The detailed proofs of the existence, measurability and the estimates and will be given in Section 3. Thanks to the assumption ?, the proof of the inequalities - is very similar to the proof of [?], so we omit it.
Now, we proceed to the statement of the main result of this paper.
The proof of this theorem will be given in Section 4.
To shorten notation let us set and
for any positive numbers and . Let be as in the statement of Theorem ?. Owing to , , and the Chebychev-Markov, we can find a constant such that
Letting , then and finally in the last line we easily conclude the proof of the corollary.
To close this section let us make some few remarks. Instead of the scheme we could also use a fully-implicit scheme. More precisely, for any we look for a -measurable random variable such that for any
where , . We have the following theorem
The arguments for the proof of this theorem are very similar to those of the proofs of Proposition ?, Theorem ? and Corollary ?, thus we omit them.
3Existence and stability analysis of the scheme: Proof of Proposition
In this section we will show that for any the numerical scheme admits at least one solution . We will also show that is stable in provided that . The precise statement of these facts was already done in Proposition ?, thus we proceed directly to their proofs.
As we mentioned in subSection 2.3 we will only prove the existence, measurability and the estimates and . The proof of the inequalities - will be omitted because it is very similar to the proof of [?] (see also [?]).
Proof of the existence. We first establish that for any there exists satisfying the numerical scheme . To this end, let us fix and for a given consider the map defined by
for any . Note that since the map is well-defined. From assumptions ? and ? and the linearity of it is clear that for given the map is continuous. Furthermore, using Hölder’s inequality the fact that , and assumptions ? and ? we derive that
Since and, by Assumption ?, , the constant is positive and whenever . Thus, we have for any where is an arbitrary constant. Since is given, we can conclude from the above observations and Brouwer fixed point theorem (see, for instance, [?] ) that there exists at least one satisfying
In a similar way, assuming that we infer that there exists at least one such that
Therefore, we have just prove by induction that given and a -valued Wiener process , for each there exists a sequence satisfying the algorithm .
Proof of the measurability. In order to prove the -measurability of it is sufficient to show that for each one can find a Borel measurable map such that . In fact, if such claim is true then by exploiting the -measurability of one can argue by induction and show that if is -measurable then , hence , is -measurable. Thus, it remains to prove the existence of . For this purpose we will closely follow [?]. Let be the set of subsets of and consider a multivalued map such that for each , denotes the set of solution of . From the existence result abovewe deduce that maps to nonempty closed subsets of . Furthermore, since we are in finite dimensional space , we can prove, by using the assumptions ? and ? and the sequential characetrization of the closed graph theorem, that the graph of is closed. From these last two facts and [?] we can find a univocal map such that and is measurable when and are equipped with their respective Borel -algebra. This completes the proof of the measurability of the solutions of . Proof of -. Thanks to the assumption ?, the proof of the inequalities - is very similar to the proof of [?], so we omit it and we directly proceed to the proof of the estimates and .
Proof of . Taking in , using the Cauchy-Schwarz inequality and the identity
Using Assumption ?, the complex interpolation inequality in [?] and the Young inequality
Using the continuous embedding we obtain
which implie sthat
Since is a constant adapted and hence progressively measurable, it is not difficult to prove that
Using and with and respectively, we easily prove that there exists a constant , depending only on , and the norm of ), such that