{'fraction_non_alphanumeric': 0.05730791937252256, 'fraction_numerical': 0.016614658007927805, 'mean_word_length': 3.98005039899202, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 9, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Speed of convergence of Chernoff approximations for two model examples: heat equation and transport equation

Paul Chernoff in 1968 proposed his approach to approximations of one-parameter operator semigroups while trying to give a rigorous mathematical meaning to Feynman's path integral formulation of quantum mechanics. In early 2000's Oleg Smolyanov noticed that Chernoff's theorem may be used to obtain approximations to solutions of initial-value problems for linear partial differential equations (LPDEs) of evolution type with variable coefficients, including parabolic equations, Schr\"odinger equation, and some other. Chernoff expressions are explicit formulas containing variable coefficients of LPDE and the initial condition, hence they can be used as a numerical method for solving LPDEs. However, the speed of convergence of such approximations at the present time is understudied which makes it risky to employ this class of numerical methods. In the present paper we take two equations with known solutions (heat equation and transport equation) and study both analytically and numerically the speed of decay of the norm of the difference between Chernoff approximations and exact solutions. We also provide graphical illustrations of convergence and its rate. These model examples, being relatively simple, allow to demonstrate general properties of Chernoff approximations. The observations obtained build a base for the future employment of the approach based on Chernoff's theorem to the problem of construction of new numerical methods for solving initial-value problem for parabolic LPDEs with variable coefficients.


Introduction
The Chernoff theorem [3] is an effective tool of functional analysis which allows to construct approximations to C 0 -semigroups that, in turn, provide solutions to some evolution equations (e.g., heat equation, parabolic equation with variable coefficients, Schrödinger equation, etc.); standard textbooks on the topic are [5,12,8]. If one finds so-called operator-valued Chernoff function then Chernoff's theorem generates a sequence of functions, also referred to as Chernoff approximations, that converges to the solution of a Cauchy problem for the evolution equation. In many cases constructing Chernoff function is the only method that allows to express the solution in terms of both variable coefficients of the equation and the chosen initial condition. At the time the subject area has a decent amount of data on different methods to construct Chernoff functions. For instance, the members of Smolyanov's group implemented the use of integral operators to find the solutions to parabolic equations in a plethora of cases during the last 15 years (refer to pioneering papers [18,17], overview [19,2] and some other results [16,4,21]). The solutions obtained in the above mentioned studies were represented in the form of Feynman formula, i.e., as a limit of multiple integral when multiplicity goes to infinity. In turn, Schrödinger equation also belongs to the class of evolution equations, and the use of the same technique allows to represent the solution in the form of Feynman and quasi-Feynman integral formulas as well, see [15,13]. When dealing with expressions provided by the Chernoff theorem, one might be interested in obtaining the estimate for the rate of decay of approximation error as n tends to infinity. Some results regarding this topic were already proposed for the case of Schrödinger equation in [11]. Similarly, the question of convergence speed arises for the Trotter product formula e A+B = lim n→∞ (e A/n e B/n ) n which we do not exploit in current paper but which is a particular exemplar of Chernoff's theorem for G(t) = e tA e tB (see recent papers [9,10,22]). In a less specific setting, other than within the framework of Chernoff's theorem, one may refer to the results presented in [6,1,7]. Nevertheless, this area is not well studied at the present time in general. Starting from [20] members of I.D. Remizov's group examine the problem systematically. So, this paper is dedicated to the analysis of the convergence speed for some set of initial conditions and Chernoff functions. We suggest few analytic and computational methods in order to obtain the information on the order of approximation subspaces to which the examined model examples belong to as well as demonstrate the dependence of the order of the approximation on the choice of the initial condition. It is also shown on the model example that if the initial condition does not belong to the domain of the generator of the corresponding C 0 -semigroup then the convergence speed can be slower than the one attained on the functions from the domain. Conducting this study implies the use of basic methods of infinite-dimensional functional analysis and operator semigroups, as well as the use of Matlab software under student license in order to perform numerical experiments and visualise the results.

Preliminaries
2.1 C 0 -semigroups and Chernoff 's theorem This section is solely dedicated to the introduction of basic concepts that will be used throughout the rest of the paper. During the paper we will work with the system of equations given by Definition 1. Let X be an infinite set, and F be a Banach space of all number-valued functions on X. Moreover, assume having dom(L) ⊂ F and a closed linear operator L : dom(L) → F. Then the system of equations where x ∈ X, u 0 ∈ F, and u(t, ·) ∈ F for all t ≥ 0, is called a Cauchy problem for the evolution equation.
Remark. In case of existence of the C 0 -semigroup (G(t)) t≥0 with the generator (L, dom(L)) the solution of (1) exists and is given by the equality u(t, x) = (e tL u 0 )(x) for all t ≥ 0 and x ∈ X.
Definition 2. If u 0 ∈ dom(L) then u(t, ·) ∈ dom(L) for all t ≥ 0 and u is called a classical solution.
Remark. For an arbitrary u 0 ∈ F the solution of (1) exists only as a solution of the corresponding integral equation u(t, ·) = L t 0 u(s, ·)ds + u 0 . After defining a Cauchy problem, it appears necessary to provide a common definition of a C 0semigroup [5].
Definition 3. Let F be a Banach space, and L (F) be a space of all linear bounded operators in F. Let a mapping V : [0, +∞) → L (F) also be given, i.e., for a fixed t ≥ 0 we have V (t) being a linear bounded operator. Then V is called a C 0 -semigroup, or the same as strongly continuous one-parametric semigroup of linear bounded operators, if the following conditions hold: 3. V is continuous in L (F) endowed with the strong operator topology, i.e., for all ϕ ∈ F a mapping t → V (t)ϕ is continuous on [0, +∞).
The concept of a C 0 -semigroup is closely related to the notion of its generator [5]. So, we propose the following Definition 4. Let (V (t)) t≥0 be a C 0 -semigroup on a Banach space F. Then the linear operator L defined as on a linear subspace is called an infinitesimal generator of a C 0 -semigroup (V (t)) t≥0 . We also say that operator L : dom(L) → F generates a C 0 -semigroup and denote V (t) = e tL .
Definition 5. Let dom(L) ⊂ F, and L : dom(L) → F be a linear operator. Then L is called • closable (or admitting closure) if the closure of its graph Γ L in space F × F with the norm · F ×F is a graph of some operator (L, dom(L)), meaning Γ L = Γ L .
Remark. If operator L exists then it is a linear closed operator extending operator L, i.e., dom(L) ⊂ dom(L), and L| dom(L) = L| dom(L) .
The next set of definitions is provided in the wording of I.D. Remizov [14]. Remark. Conditions (CT3) and (CT4) together mean that there exists a core D for L.
Classical wording of the next theorem can be found in [3,5]. Based on the notion of Chernoff tangency we provide Theorem 1 (The Chernoff theorem). Let F be a Banach space, and L (F) be a space of all linear bounded operators in F. Let a mapping G : [0, +∞) → L (F), or the same as a family of linear bounded operators (G(t)) t≥0 in F, also be given. Moreover, assume having a closed linear operator L : dom(L) ⊂ F → F. Suppose that the following conditions are met: (E). there exists a C 0 -semigroup (e tL ) t≥0 with generator (L, dom(L)); (CT). function G is Chernoff tangent to operator L; (N). there exists number ω ∈ R such that G(t) ≤ e ωt for all t ≥ 0.
Then for each f ∈ F we have (G(t/n)) n f

Conjectures on convergence speed for Chernoff approximations
During the analysis of the rate of convergence both for the transport and heat equations, our aim is to check the validity of the following conjectures for several model examples of initial conditions. Conjecture 1 (I. Remizov, 2018). Let (e tL ) t≥0 be a C 0 -semigroup in a Banach space F with generator (L, dom(L)), and G be a Chernoff function for operator L. Moreover, assume t 0 ≥ 0 and f ∈ F are given, and suppose that for all t ∈ [0, t 0 ] we have f ∈ dom(G (t)) ∩ dom(G (t)) and functions t → G (t)f and t → G (t)f are continuous. Then there exists a number C ≥ 0 such that for all t ∈ [0, t 0 ) and all n ∈ N the estimate G t n n f − e tL f ≤ C n holds. Conjecture 2 (I. Remizov, 2018). Let (e tL ) t≥0 be a C 0 -semigroup in a Banach space F with generator (L, dom(L)), and G be a Chernoff function for operator L. Moreover, assume number t 0 ≥ 0 and f ∈ F are given, and suppose that for all t ∈ [0, t 0 ] we have function f being from the intersection of the domains of G (t), 2 , having that each of these operators is continuous for all t ∈ [0, t 0 ]. Then there exists a number C ≥ 0 such that for all t ∈ [0, t 0 ) and all n ∈ N the following inequality is valid: For the further convenience of the interpretation of numerical studies we provide the definition of an approximation subspace proposed in [20].
Remark. The approximation subspace A τ ψ is indeed a linear subspace of F. Hence, we now state the previous Conjecture 2 in the new wording in Conjecture 3. Let (e tL ) t≥0 be a C 0 -semigroup in a Banach space F with generator (L, dom(L)), and G be a Chernoff function for operator L. More than that, assume number t 0 ≥ 0 is given, and denote the intersection of the domains of G (t), . Assume that D(t 0 ) is dense in F and each of these operators are continuous in t on all vectors from D(t 0 ) for each t ∈ [0, t 0 ]. Then function G has a dense approximation subspace A [0,t 0 ) ζ of order ζ(n) = 1 n 2 , and this subspace is a subset of D(t 0 ). Despite the fact that only fast converging approximations have a practical value for the research as a whole, it is worth mentioning that Chernoff functions can be likewise constructed in a way to provide an arbitrary slow convergence rate. This result is presented in the next

Exact formulas and estimates for convergence speed
During this section we will propose the results on the approximation speed for some model initial conditions and both one-dimensional transport and heat equations.

Transport equation
As a preliminary we will commence with the notion of the translation semigroup that gives the solution for one-dimensional transport equation. The examined system of equations is stated in Definition 9. The system of equations defined as where x ∈ R, u 0 ∈ U C b (R) for all t ≥ 0, is called a Cauchy problem for the first order partial differential equation on the real line, also referred to as the transport equation.
Remark. The solution of the system (5) and, hence, the corresponding C 0 -semigroup is given by the formula u(t, x) = (e tL u 0 )(x) = u 0 (x + t). This C 0 -semigroup is called the translation semigroup.
First, let us verify that the generator of the translation semigroup is indeed the differentiation operator.
Proof. Using Definition 4 of the generator of a C 0 -semigroup we get where · denotes the standard supremum norm in U C b (R). Applying the definition of the C 0 -semigroup, we obtain Now, let us show that f is uniformly continuous. Take a sequence (t n ) n∈N such that lim n→∞ t n = 0 and t n ≥ 0 for all n ∈ N. Then it follows that Since functions given by the formula f (x+tn)−f (x) tn are uniformly continuous for all n ∈ N, by Weierstrass theorem f is also uniformly continuous. Next, we will show that f is bounded. By the definition of uniform continuity we have that For small |t| the former term is small due to (7), and the latter one due to f being uniformly continuous. So, we get that Hence, if the limit given by (6) exists, meaning f ∈ dom(L), i.e., f ∈ dom(L). So, we proved that L is generator of this semigroup and dom(L) = U C 1 b (R). Prior to the numerical studies we will find the composition degree of some Chernoff functions G for an arbitrary initial condition u 0 ∈ U C b (R).
for some fixed a, k > 0. Then its n-th composition degree is given by the formula Proof. By the definition of Chernoff function we get for all x ∈ R and t ≥ 0.

Now, using Taylor expansion sin
which proves that u 0 = [x → sin(x)] belongs to an approximation subspace of order 1 n k .

Analysis of slow converging approximations for
At last, we will generalize Theorem 5 for an arbitrary function w : R + → R + .
Theorem 6. Let (e tL ) t≥0 be the translation semigroup on the real line, and G be the Chernoff function such that for all f ∈ U C b (R) we have where w : R + → R + such that lim n→∞ w(n) = 0. Then function u 0 = [x → sin(x)] belongs to the approximation subspace of order w, and the error is given by the formula G t n n u 0 − e tL u 0 = 2 |sin(t/2 · w(n/t))| .
Proof. Similarly to Proposition 4 we first obtain n-th composition degree for Chernoff function G. We get It now suffices to set τ = t · w(n/t) and use the same technique as of Theorem 5. Hence, since w(n) n→∞ − −− → 0, we have ξ(n) = 2 |sin(t/2 · w(n/t))| = t · w(n/t) + o w 2 (n) as n → ∞, which proves that u 0 = [x → sin(x)] belongs to an approximation subspace of order w.
Later in Section 4 we will check the convergence speed attained for the slow converging Chernoff function G given by (10) with w(t) = 1 t γ for different values of 0 < γ < 1 applied to approximate the solution for u 0 = [x → sin(x)].

Heat equation
The next differential equation we are going to analyse is the heat equation. Now, we will state the corresponding Cauchy problem and give an explicit formula for the solution in a form of a C 0 -semigroup. Definition 10. The system of equations defined the following way where a > 0, x ∈ R, and u 0 ∈ U C b (R) for all t ≥ 0, is called a Cauchy problem for the heat equation on the real line with constant coefficient of thermal conductivity.
Remark. The solution of the system (12), and, hence, the corresponding C 0 -semigroup, is given by the Poisson integral where We will analyse the performance of approximations for the following three Chernoff functions: proposed by I. Remizov in [14], proposed by A. Vedenin in [20], and (17) also proposed by A. Vedenin in oral communication to the author of the present paper in 2020.
To simplify the complexity of computational algorithms one might need a general formula for the n-th composition degree for the above Chernoff functions. So we present the following Proposition 7. Let G be the Chernoff function given by (15) for all f ∈ U C b (R) and fixed a > 0. Then its n-th composition degree is given by the formula where for each n ∈ N and p ∈ {−n, . . . , n} coefficients α p,n are defined as ) for all t > 0, where I is an identity operator and S τ is a shift operator given by (S τ f )(x) = f (x + τ ) for some τ ∈ R. It then yields that Similarly, one may derive the next Proposition 8. Let G be the Chernoff function given by (16) for all f ∈ U C b (R) and fixed a > 0. Then its n-th composition degree is given by the formula where for each n ∈ N and p ∈ {−n, . . . , n} coefficients β p,n are defined as Analysis of convergence rate for the initial condition Let us now find the solution of the above Cauchy problem (12) for . Then the solution of the system (12) with initial condition u 0 is given by the formula u(t, x) = (e tL u 0 )(x) = e −a 2 t sin(x).
Proof. Using (13) and substituting u 0 = [x → sin(x)], we get Now, let us apply the following identity sin(x) = e ix −e −ix

2i
. So, we obtain The former integral now takes the form Similarly, the latter integral is equal to exp (−ix − a 2 t), i.e., the solution can now be expressed as So, we found the solution of the Cauchy problem given by (12) for Prior to analysing the convergence rate for Chernoff function (15) and u 0 = [x → sin(x)] let us compute its n-th composition degree.
Proof. Substituting the initial condition into (15), we get So, for all t ≥ 0 operator G(t) is a function multiplication operator, meaning its n-th composition degree has the following form Finally, let us find the convergence speed for u 0 = [x → sin(x)] and the above derived Chernoff function (21). The main result of this subsection is presented in the next Theorem 11. Let (e tL ) t≥0 be the heat C 0 -semigroup, and G be the Chernoff function given by (15) for all f ∈ U C b (R) and fixed a > 0. Then function u 0 = [x → sin(x)] belongs to the approximation subspace of order 1 n . Proof. Applying the definition of a standard norm in U C b (R), we get This proves that u 0 = [x → sin(x)] belongs to the approximation subspace of order 1 n . Similarly, applying the same technique as of Theorem 11 one may propose the next series of results containing the estimates of the convergence rate for Chernoff functions (16) and (17), starting with Proposition 12. Let G be the Chernoff function given by (16) (respectively (17)) for all f ∈ U C b (R) and fixed a > 0. Then its n-th composition degree for u 0 = [x → sin(x)] is given by Theorem 13. Let (e tL ) t≥0 be the heat C 0 -semigroup, and G be the Chernoff function given by (16) (respectively (17)) for all f ∈ U C b (R) and fixed a > 0. Then function u 0 = [x → sin(x)] belongs to the approximation subspace of order 1 n 2 and 1 n 3 respectively.
Finding the exact solution for the initial condition u 0 = [x → exp(−|x|)] Here we will find the solution of the Cauchy problem given by (12) for u 0 = [x → e −|x| ].
. Then the solution of the system (12) with initial condition u 0 is given by the formula where Proof. First, let us plug the formula giving the initial condition into the solution defined in (13) and (14). We have Now, change the variables within the integral by denoting x − y = 2z √ t. It then yields that y = x − 2z √ t and dy = 2 √ tdz. So, Now, set z − t = w and z + t = w for the former respectively latter integral. Hence, we obtain Besides, as erfc(x) x→+∞ − −−− → 0, then for each t > 0 one has u(t, x) ∼ e t−x as x → +∞ and, moreover, u(t, x) ∼ e t−|x| as x → ∞.
As analysing the convergence speed for the obtained solution is technically a sophisticated task, we will present only the numerical estimates of the norm decay later on. Even though it is not a rigorous proof, it gives hope that one can obtain a formal reasoning.

Numerical experiments on model examples
This chapter is devoted to the results of numerical experiments performed for the previously analysed initial conditions u 0 = [x → sin(x)] and u 0 = [x → exp(−|x|), and different Cauchy problems (5) and (10).

Transport equation
First, we will analyse the convergence speed for u 0 = [x → sin(x)]. As the initial condition as well as the Chernoff approximations have period equal to 2π, the uniform norm in U C b (R) is reached on the interval of this period. Note that the domain of the graphs presented below varies significantly and is chosen in favor of visual clarity. So, Figure 1 containes the graphs of the initial condition and the corresponding solution for t = 2. Whereas, Figures 2 and 3 represent some examples of Chernoff approximations with respect to the initial condition u 0 = [x → sin(x)] and t = 2 with composition degree n = 1 and n = 5 respectively.  Figure 4 represents the graphs of convergence rate, i.e., the decay in the norm of the difference of the Chernoff approximations and the corresponding solution depending on the growth of composition degree.  As the initial condition and the Chernoff approximations are decaying outside of some segment, we assume that the value of the uniform norm of the difference is reached on the interval of correctly selected length (e.g., [−5, 5]). So, Figure 9 provides a graph of convergence rate, i.e., the decay of the norm of the difference depending on the growth of composition degree. Figure 9: Graphs of estimates of the convergence speed (t = 1) Figure 10: Graphs of estimates of the convergence speed in log-log scale (t = 1) More than that, Figures 9 and 10, using both standard and log-log scale, illustrate the results of a non-linear regression applied to analyse the approximation subspaces to which the examined initial condition belongs to for Chernoff functions from Theorem 5.
Numerical demonstration of slow convergence for u 0 = [x → sin(x)] Third, we will analyse the convergence rate for u 0 = [x → sin(x)] and Chernoff function G (10) given in Theorem 6 with w(t) = 1 t γ for γ = 1 2 , γ = 1 3 , and γ = 1 6 . Figure 11: Graphs of approximations to the solution of the transport equation (t = 2, n = 1) Here, we will again exploit the periodic nature of the solution and the Chernoff approximations and suggest that the uniform norm is reached on the properly selected interval. So, Figure 13 provides a graph of convergence rate when t = 2. Apart from that, the below series of figures also illustrates the order of approximation subspaces to which the examined initial condition belongs to. Thus, Figures 13 and 14, using both standard and log-log scale, represent the convergence rate and the corresponding obtained regression results for Chernoff functions from Theorem 6 with γ = 1 2 , γ = 1 3 , and γ = 1 6 . Figure 13: Graphs of estimates of the convergence speed (t = 1) Figure 14: Graphs of estimates of the convergence speed with log-log scale (t = 1) One can clearly notice that, independent of the considered initial value of the equation, all of the approximations performed as if the examined model function belonged to the domain of generator, i.e., even u 0 = [x → exp(−|x|)] / ∈ U C 1 b (R) belongs to the higher order approximation subspaces. We will examine later in this chapter that for the heat equation this is not the case already.

Heat equation
Convergence speed for u 0 = [x → sin(x)] Here, we will analyse the convergence speed of Chernoff approximations defined by (15), (16), and (17) for u 0 = [x → sin(x)] to the exact solution given by (20). The graphs of the initial condition and investigated solution for t = 1 are shown in Figure 15.  Here, we will analyse the convergence rate for the initial condition u 0 = [x → e −|x| ]. The graphs of the initial condition and solution given by the C 0 -semigroup for t = 1 are shown in Figure 20.  Let us once again refer to the decaying property of the initial condition and the Chernoff approximations. We propose that the standard norm in U C b (R) is reached on the appropriately defined interval (e.g., [−5, 5]). So, Figure 23 contains two graphs of convergence rate for t = 1. Figure 24: Graphs of estimates of the convergence speed with log-log scale (t = 1) The above series of figures illustrates the results of a non-linear regression applied to analyse the approximation subspaces to which the examined initial condition belongs to. So, Figures 23 and 24, using both standard and logarithmic scale, represent the convergence rate and the corresponding obtained regression results for the heat equation. One can easily derive now that the behaviour of the Chernoff approximations depends on the choice of initial condition, e.g., for u 0 = [x → exp(−|x|)] we have it belonging to the approximation subspaces of the same order 1 n when applying expressions via different Chernoff functions (15), (16), and (17).