数值方法：求解数学物理逆问题

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"本书《Numerical Methods for Solving Inverse Problems of Mathematical Physics》主要探讨了数学物理方程的反问题的主要类别以及它们的数值求解方法，旨在为应用数学、计算数学和数学建模领域的研究生和专家提供指导。" 在数学物理中，反问题是研究的一种核心领域，它涉及从观测数据推断出模型参数或初始条件的问题。这些问题往往与实际应用紧密相关，如地球物理勘探、医学成像和图像处理等。本书属于"Inverse and Ill-Posed Problems Series"，表明其关注的是那些可能没有唯一解或者稳定性差的问题，即不适定问题。书中涵盖的关键点包括： 1. 反问题：这涉及到如何从实验数据中恢复物理过程的特性，如分布、形状或速度等。 2. 数值方法：书中介绍了各种数值技术，用于处理数学物理方程的反问题。这些方法对于解决实际问题至关重要，因为解析解通常是不可行的。 3. 边界值问题：这是数学物理方程中的常见类型，通常需要找到满足特定边界条件的解。 4. 常微分方程：这些方程在描述连续系统的动态行为时发挥着关键作用。 5. 抛物型和椭圆型方程：这两种类型的偏微分方程广泛应用于热传导、流体动力学和电磁学等领域。 6. 右端项识别：在反问题中，确定模型的输入或源项是常见的任务。 7. 进化型反问题：这些问题涉及到随时间变化的系统，如扩散过程或振动分析。 8. 不适定问题和正则化方法：由于反问题的不稳定性，需要采用正则化技术，如Tikhonov正则化，来稳定求解过程。 9. 对偶梯度法：这是一种优化方法，常用于求解反问题，以最小化误差函数。 10. 差分原理：用于判断数值解的精度和调整算法的准则。 11. 线性代数方法：如有限差分法和有限元法，这些都是数值求解偏微分方程的常用工具。数学主题分类2000年代码显示，这本书涉及的领域包括： - 65-02：数值分析的一般著作 - 65F22：线性方程组的迭代方法 - 65J20：偏微分方程的数值解 - 65L09：离散时间的数值方法 - 65M32：偏微分方程的数值反问题 - 65N21：椭圆和抛物型方程的有限元素方法该书还遵循了酸性免费纸张的标准，以确保长期保存和耐用性，并且在德意志国家图书馆有详细记录，方便进一步的学术检索和研究。《Numerical Methods for Solving Inverse Problems of Mathematical Physics》是一本深入探讨数学物理反问题数值求解策略的专著，对研究者和专业人士来说是一份宝贵的资源。通过学习和应用书中的理论和方法，读者能够更有效地处理实际世界中的复杂问题。

1 Inverse mathematical physics problems

We associate direct mathematical physics problems with classical boundary value

problems often encountered in mathematical physics. In a direct problem, it is required

to ﬁnd a solution that satisﬁes some given partial differential equation and some initial

and boundary conditions. In inverse problems, the master equation and/or initial con-

ditions and/or boundary conditions are not fully speciﬁed but, instead, some additional

information is available. So separating out inverse mathematical physics problems,

we can speak of coefﬁcient inverse problems (in which the equation is not speciﬁed

completely as some equation coefﬁcients are unknown), boundary inverse problems

(in which boundary conditions are unknown), and evolutionary inverse problems (in

which initial conditions are unknown). Very often, inverse problems are problems

ill-posed in the classical sense. A typical feature here is the violated requirement of

solution continuity on input data. An ill-posed problem can be moved up into the class

of well-posed problems by narrowing the class of admissible solutions.

1.1 Boundary value problems

The core of applied mathematical models is made up by partial differential equations.

Here, the solution can be found from mathematical physics equations and some ad-

ditional relations. Such additional relations are, ﬁrst of all, boundary and initial con-

ditions. In courses on mathematical physics equations, equations most important for

applications are second-order equations. Among such equations are elliptic, parabolic,

and hyperbolic equations.

1.1.1 Stationary mathematical physics problems

As an example, consider several two-dimensional boundary value problems. The so-

lution u(x), x = (x

, x

) is to be found in some bounded domain  with a sufﬁciently

smooth boundary ∂. This solution is deﬁned by the second-order elliptic equation

−



α=1

∂

∂x



k(x)

∂u

∂x



+ q(x)u = f (x), x ∈ . (1.1)

Normally, the constraints imposed on the equation coefﬁcients look as

k(x) ≥ κ>0, q(x) ≥ 0, x ∈ .

A typical example of an elliptic equation (1.1) is given by the Poisson equation

−u ≡−



α=1

∂

∂x

= f (x), x ∈ , (1.2)

4 Chapter 1 Inverse mathematical physics problems

1.2 Well-posed problems for partial differential equations

Here, we introduce the notion of well-posed boundary value problem, related with

the existence of a unique solution that continuously depends on input data. Results

on stability of classical boundary value problems for partial differential equations are

presented.

1.2.1 The notion of well-posedness

Boundary and initial conditions are formulated to identify, among the whole set of

possible solutions of a partial differential equation, a desired solution. These additional

conditions must be not too numerous (solutions must exist), nor they must be few

in number (solutions must be not numerous). With this circumstance, the notion of

well-posed statement of a problem is related. Let us dwell ﬁrst on the notion of well-

posedness of a problem according to J. Hadamard (well-posedness in the classical

sense).

A problem is well-posed if:

1) the solution of the problem does exist;

2) the solution is unique;

3) the solution continuously depends on input data.

It is the third condition for well-posedness that is of primary signiﬁcance here. This

condition provides for smallness of the solution changes resulting from small input-

data variation. The input data are the equation coefﬁcients, the right-hand side and

the boundary and initial conditions, taken from an experiment and always known to

some limited accuracy. In fact, solution stability with respect to small perturbations of

initial and boundary conditions, coefﬁcients, and right-hand side justiﬁes the problem

statement itself, as well as its cognitive essence, and makes the whole study valuable.

In consideration of boundary value problems for mathematical physics equations,

the existence, uniqueness and stability theorems taken as a whole provide a complete

study of well-posedness of a posed problem. Of course, conditions for well-posedness

must be rendered concrete considering each particular problem. The latter is related

with the fact that the solution of the problem and the input data are considered as

elements in a certain fully deﬁned functional space. That is why a given problem can

be ill-posed with one choice of spaces and well-posed with another choice of spaces.

Hence, a statement that this or that problem is a well- (ill-)posed one is never global:

such statements must be supplemented with necessary amendments.

1.2.2 Boundary value problem for the parabolic equation

Some fundamental points in a consideration of well- or ill-posedness of a boundary

value mathematical physics problem can be illustrated with the example of a sim-

plest boundary value problem for the one-dimensional parabolic equation (1.9)–(1.11).

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